# Torque, Angular, Momentum and Equilibrium

## Torque or Moment of Force

Definition

If a body is capable of rotating about an axis, then force applied properly on this body will rotate it about the axis (axis of rotating). This turning effect of the force about the axis of rotation is called torque.

Torque is the physical quantity which produces angular acceleration in the body.

Explanation

Consider a body which can rotate about O (axis of rotation). A force F acts on point P whose position vector w.r.t O is r.

Diagram Coming Soon

F is resolved into F1 and F2. θ is the angle between F and extended line of r.

The component of F which produces rotation in the body is F1.

The magnitude of torgue (π) is the product of the magnitudes of r and F1.

Equation (1) shows that torque is the cross-product of displacement r and force F.

Torque → positive if directed outward from paper

Torque → negative if directed inward from paper

The direction of torque can be found by using Right Hand Rule and is always perpendicular to the plane containing r & F.

Thus

Clockwise torque → negative

Counter-Clockwise torque → positive

Alternate Definition of Torque

π = r x F

|π| = r F sin θ

|π| = F x r sin θ

But r sin θ = L (momentum arm) (from figure)

Therefore,

|π| = F L

Magnitude of Torque = Magnitude of force x Moment Arm

Note

If line of action of force passes through the axis of rotation then this force cannot produce torque.

The unit of torque is N.m.

Couple

Two forces are said to constitute a couple if they have

1. Same magnitudes

2. Opposite directions

3. Different lines of action

These forces cannot produces transiatory motion, but produce rotatory motion.

Moment (Torque) of a Couple

Consider a couple composed of two forces F and -F acting at points A and B (on a body) respectively, having position vectors r1 & r2.

If π1 is the torque due to force F, then

π1 = r1 x F

Similarly if π2 is the torque due to force - F, then

π2 = r2 x (-F)

The total torque due to the two forces is

π = π1 + π2

π = r1 x F + r2 x (-F)

π = r1 x F - r2 x (-F)

π = (r1 - r2) x F

π = r x F

where r is the displacement vector from B to A.

The magnitude of torque is

π = r F sin (180 - θ)

π = r F sin θ .................... {since sin (180 - θ) = sin θ}

Where θ is the angle between r and -F.

π = F (r sin θ)

But r sin θ is the perpendicular distance between the lines of action of forces F and -F is called moment arm of the couple denoted by d.

π = Fd

Thus

[Mag. of the moment of a couple] = [Mag. of any of the forces forming the couple] x [Moment arm of the couple]

Moment (torque) of a given couple is independent of the location of origin.

## Centre of Mass

Definition

The centre of mass of a body, or a system of particles, is a point on the body that moves in the same way that a single particle would move under the influence of the same external forces. The whole mass of the body is supposed to be concentrated at this point.

Explanation

During translational motion each point of a body moves in the same manner i.e., different particles of the body do not change their position w.r.t each other. Each point on the body undergoes the same displacement as any other point as time goes on. So the motion of one particle represents the motion of the whole body. But in rotating or vibrating bodies different particles move in different manners except one point called centre of mass. The centre of mass of a body or a system of particle is a point which represents the movement of the entire system. It moves in the same way that a single particle would move under the influence of same external forces.

Centre of Mass and Centre of Gravity

In a completely uniform gravitational field, the centre of mass and centre of gravity of an extended body coincides. But if gravitational field is not uniform, these points are different.

Determination of the Centre of Mass

Consider a system of particles having masses m1, m2, m3, ................. mn. Suppose x1, and z1, z2, z3 are their distances on z-axis, all measured from origin.

## Equilibrium

A body is said to be in equilibrium if it is

1. At rest, or

2. Moving with uniform velocity

A body in equilibrium possess no acceleration.

Static Equilibrium

The equilibrium of bodies at rest is called static equilibrium. For example,

1. A book lying on a table

2. A block hung from a string

Dynamic Equilibrium

The equilibrium of bodies moving with uniform velocity is called dynamic equilibrium. For example,

1. The jumping of a paratrooper by a parachute is an example of uniform motion. In this case, weight is balanced by the reaction of the air on the parachute acts in the vertically upward direction.

2. The motion of a small steel ball through a viscous liquid. Initially the ball has acceleration but after covering a certain distance, its velocity becomes uniform because weight of the ball is balanced by upward thrust and viscous force of the liquid. Therefore, ball is in dynamic equilibrium.

Angular Momentum

Definition

The quantity of rotational motion in a body is called its angular momentum. Thus angular momentum plays same role in rotational motion as played by linear momentum in translational motion.

Mathematically, angular momentum is the cross-product of position vector and the linear momentum, both measured in an inertial frame of reference.

ρ = r x P

Explanation

Consider a mass 'm' rotating anti-clockwise in an inertial frame of reference. At any point, let P be the linear momentum and r be the position vector.

ρ = r x P

ρ = r P sinθ ........... (magnitude)

ρ = r m V sinθ .......... {since P = m v)

where,

V is linear speed

θ is the angle between r and P

θ = 90º in circular motion (special case)

The direction of the angular momentum can be determined by the Righ-Hand Rule.

Also

ρ = r m (r ω) sin θ

ρ = m r2 ω sin θ

Units of Angular Momentum

The units of angular momentum in S.I system are kgm2/s or Js.

1. ρ = r m V sin θ

= m x kg x m/s

= kg.m2/s

2. ρ = r P sin θ

= m x Ns

= (Nm) x s

= J.s

Dimensions of Angular Momentum

[ρ] = [r] [P]

= [r] [m] [V]

= L . M . L/T

= L2 M T-1

Relation Between Torque and Angular Momentum

OR

Prove that the rate of change of angular momentum is equal to the external torque acting on the body.

Proof

We know that rate of change of linear momentum is equal to the applied force.

F = dP / dt

Taking cross product with r on both sides, we get

R x F = r x dP / dt

τ = r x dP / dt ............................. {since r x P = τ}

Now, according to the definition of angular momentum

ρ = r x P

Taking derivative w.r.t time, we get

dρ / dt = d / dt (r x P)

=> dρ / dt = r x dP / dt + dr / dt x P

=> dρ / dt = τ + V x P .................. {since dr / dt = V}

=> dρ / dt = τ + V x mV

=> dρ / dt = τ + m (V x V)

=> dρ / dt = τ + 0 ................. {since V x V = 0}

=> dρ / dt = τ

Or, Rate of change of Angular Momentum = External Torque ................. (Proved)

# Motion in two Dimension

## Projectile Motion

A body moving horizontally as well as vertically under the action of gravity simultaneously is called a projectile. The motion of projectile is called projectile motion. The path followed by a projectile is called its trajectory.

Examples of projectile motion are

1. Kicked or thrown balls

2. Jumping animals

3. A bomb released from a bomber plane

4. A shell of a gun.

Analysis of Projectile Motion

Let us consider a body of mass m, projected an angle θ with the horizontal with a velocity V0. We made the following three assumptions.

1. The value of g remains constant throughout the motion.

2. The effect of air resistance is negligible.

3. The rotation of earth does not affect the motion.

Horizontal Motion

Acceleration : ax = 0

Velocity : Vx = Vox

Displacement : X = Vox t

Vertical Motion

Acceleration : ay = - g

Velocity : Vy = Voy - gt

Displacement : Y = Voy t - 1/2 gt2

Initial Horizontal Velocity

Vox = Vo cos θ ...................... (1)

Initial Vertical Velocity

Voy = Vo sin θ ...................... (2)

Net force W is acting on the body in downward vertical direction, therefore, vertical velocity continuously changes due to the acceleration g produced by the weight W.

There is no net force acting on the projectile in horizontal direction, therefore, its horizontal velocity remains constant throughout the motion.

X - Component of Velocity at Time t (Vx)

Vx = Vox = Vo cos θ .................... (3)

Y - Component of Velocity at Time t (Vy)

Data for vertical motion

Vi = Voy = Vo sin θ

a = ay = - g

t = t

Vf = Vy = ?

Using Vf = Vi + at

Vy = Vo sin θ - gt .................... (4)

Range of the Projectile (R)

The total distance covered by the projectile in horizontal direction (X-axis) is called is range

Let T be the time of flight of the projectile.

Therefore,

R = Vox x T .............. {since S = Vt}

T = 2 (time taken by the projectile to reach the highest point)

T = 2 Vo sin θ / g

Vox = Vo cos θ

Therefore,

R = Vo cos θ x 2 Vo sin θ / g

R = Vo2 (2 sin θ cos θ) / g

R = Vo2 sin 2 θ / g .................. { since 2 sin θ cos θ = sin2 θ}

Thus the range of the projectile depends on

(a) The square of the initial velocity

(b) Sine of twice the projection angle θ.

The Maximum Range

For a given value of Vo, range will be maximum when sin2 θ in R = Vo2 sin2 θ / g has maximum value. Since

0 ≤ sin2 θ ≤ 1

Hence maximum value of sin2 θ is 1.

Sin2 θ = 1

2θ = sin(-1) (1)

2θ = 90º

θ = 45º

Therefore,

R(max) = Vo2 / g ; at θ = 45º

Hence the projectile must be launched at an angle of 45º with the horizontal to attain maximum range.

Projectile Trajectory

The path followed by a projectile is referred as its trajectory.

We known that

S = Vit + 1/2 at2

For vertical motion

S = Y

a = - g

Vi = Voy = Vo sin θ

Therefore,

Y = Vo sinθ t - 1/2 g t2 ....................... (1)

Also

X = Vox t

X = Vo cosθ t ............ { since Vox = Vo cosθ}

t = X / Vo cos θ

(1) => Y = Vo sinθ (X / Vo cos θ) - 1/2 g (X / Vo cos θ)2

Y = X tan θ - gX2 / 2Vo2 cos2 θ

For a given value of Vo and θ, the quantities tanθ, cosθ, and g are constant, therefore, put

a = tan θ

b = g / Vo2 cos2θ

Therefore

Y = a X - 1/2 b X2

Which shows that trajectory is parabola.

## Uniform Circular Motion

If an object moves along a circular path with uniform speed then its motion is said to be uniform circular motion.

Recitilinear Motion

Displacement → R

Velocity → V

Acceleration → a

Circular Motion

Angular Displacement → θ

Angular Velocity → ω

Angular Acceleration → α

Angular Displacement

The angle through which a body moves, while moving along a circular path is called its angular displacement.

The angular displacement is measured in degrees, revolutions and most commonly in radian.

Diagram Coming Soon

s = arc length

r = radius of the circular path

θ = amgular displacement

It is obvious,

s ∞ θ

s = r θ

θ = s / r = arc length / radius

It is the angle subtended at the centre of a circle by an arc equal in length to its radius.

Therefore,

When s = r

θ = 1 radian = 57.3º

Angular Velocity

When a body is moving along a circular path, then the angle traversed by it in a unit time is called its angular velocity.

Diagram Coming Soon

Suppose a particle P is moving anticlockwise in a circle of radius r, then its angular displacement at P(t1) is θ1 at time t1 and at P(t2) is θ2 at time t2.

Average angular velocity = change in angular displacement / time interval

Change in angular displacement = θ2 - θ1 = Δθ

Time interval = t2 - t1 = Δt

Therefore,

ω = Δθ / Δt

Angular velocity is usually measured in rad/sec.

Angular velocity is a vector quantity. Its direction can be determined by using right hand rule according to which if the axis of rotation is grasped in right hand with fingers curled in the direction of rotation then the thumb indicates the direction of angular velocity.

Angular Acceleration

It is defined as the rate of change of angular velocity with respect to time.

Thus, if ω1 and ω2 be the initial and final angular velocity of a rotating body, then average angular acceleration "αav" is defined as

αav = (ω2 - ω1) / (t2 - t1) = Δω / Δt

The units of angular acceleration are degrees/sec2, and radian/sec2.

Instantaneous angular acceleration at any instant for a rotating body is given by

Angular acceleration is a vector quantity. When ω is increasing, α has same direction as ω. When ω is decreasing, α has direction opposite to ω.

Relation Between Linear Velocity And Angular Velocity

Consider a particle P in an object in X-Y plane rotating along a circular path of radius r about an axis through O, perpendicular to the plane of figure as shown here (z-axis).

If the particle P rotates through an angle Δθ in time Δt,

Then according to the definition of angular displacement.

Δθ = Δs / r

Dividing both sides by Δt,

Δθ / Δt = (Δs / Δt) (1/r)

=> Δs / Δt = r Δθ / Δt

For a very small interval of time

Δt → 0

Alternate Method

We know that for linear motion

S = v t .............. (1)

And for angular motion

S = r θ ................. (2)

Comparing (1) & (2), we get

V t = r θ

v = r θ/t

V = r ω ........................... {since θ/t = ω}

Relation Between Linear Acceleration And Angular Acceleration

Suppose an object rotating about a fixed axis, changes its angular velocity by Δω in Δt. Then the change in tangential velocity, ΔVt, at the end of this interval is

ΔVt = r Δω

Dividing both sides by Δt, we get

ΔVt / Δt = r Δω / Δt

If the time interval is very small i.e., Δt → 0 then

Alternate Method

Linear acceleration of a body is given by

a = (Vr - Vi) / t

But Vr = r ω r and Vi = r ω i

Therefore,

a = (r ω r - r ω i) / t

=> a = r (ωr- ωi) / t

a = r α .................................... {since (ωr = ωi) / t = ω}

Time Period

When an object is rotating in a circular path, the time taken by it to complete one revolution or cycle is called its time period, (T).

We know that

ω = Δθ / Δt OR Δt = Δθ / ω

For one complete rotation

Δθ = 2 π

Δt = T

Therefore,

T = 2 π / ω

If ω = 2πf ........................ {since f = frequency of revolution}

Therefore,

T = 2π / 2πf

=> T = 1 / f

Tangential Velocity

When a body is moving along a circle or circular path, the velocity of the body along the tangent of the circle is called its tangential velocity.

Vt = r ω

Tangential velocity is not same for every point on the circular path.

Centripetal Acceleration

A body moving along a circular path changes its direction at every instant. Due to this change, the velocity of the body 'V' is changing at every instant. Thus body has an acceleration which is called its centripetal acceleration. It is denoted by a(c) or a1 and always directed towards the centre of the circle. The magnitude of the centripetal acceleration a(c) is given as follows

a(c) = V2 / r, ........................... r = radius of the circular path

Prove That a(c) = V2 / r

Proof

Consider a body moving along a circular path of radius of r with a constant speed V. Suppose the body moves from a point P to a point Q in a small time Δt. Let the velocity of the body at P is V1 and at Q is V2. Let the angular displacement made in this time be ΔO .

Since V1 and V2 are perpendicular to the radial lines at P and Q, therefore, the angle between V1 and V2 is also Δ0, Triangles OPQ and ABC are similar.

Therefore,

|ΔV| / |V1| = Δs / r

Since the body is moving with constant speed

Therefore,

|V1| = |V2| = V

Therefore,

ΔV / V = Vs / r

ΔV = (V / r) Δs

Dividing both sides by Δt

Therefore,

ΔV / Δt = (V/r) (V/r) (Δs / Δt)

taking limit Δt → 0.

Proof That a(c) = 4π2r / T2

Proof

We know that

a(c) = V2 / r

But V = r ω

Therefore,

a(c) = r2 ω2 / r

a(c) = r ω2 ...................... (1)

But ω = Δθ / Δt

For one complete rotation Δθ = 2π, Δt = T (Time Period)

Therefore,

ω = 2π / T

(1) => a(c) = r (2π / T)2

a(c) = 4 π2 r / T2 .................. Proved

Tangential Acceleration

The acceleration possessed by a body moving along a circular path due to its changing speed during its motion is called tangential acceleration. Its direction is along the tangent of the circular path. It is denoted by a(t). If the speed is uniform (unchanging) the body do not passes tangential acceleration.

Total Or Resultant Acceleration

The resultant of centripetal acceleration a(c) and tangential acceleration a(t) is called total or resultant acceleration denoted by a.

Centripetal Force

If a body is moving along a circular path with a constant speed, a force must be acting upon it. Direction of the force is along the radius towards the centre. This force is called the centripetal force by F(c).

F(c) = m a(c)

F(c) = m v(2) / r ..................... {since a(c) = v2 / r}

F(c) = mr2 ω2 r ....................... {since v = r ω}

F(c) = mrω2

# Motion

Definition

If an object continuously changes its position with respect to its surrounding, then it is said to be in state of motion.

Rectilinear Motion

The motion along a straight line is called rectilinear motion.

## Velocity

Velocity may be defined as the change of displacement of a body with respect time.

Velocity = change of displacement / time

Velocity is a vector quantity and its unit in S.I system is meter per second (m/sec).

Average Velocity

Average velocity of a body is defined as the ratio of the displacement in a certain direction to the time taken for this displacement.

Suppose a body is moving along the path AC as shown in figure. At time t1, suppose the body is at P and its position w.r.t origin O is given by vector r2.

Diagram Coming Soon

Thus, displacement of the body = r2 - r1 = Δr

Time taken for this displacement - t2 - t1 = Δt

Therefore, average velocity of the body is given by

Vav = Δr / Δt

Instantaneous Velocity

It is defined as the velocity of a body at a certain instant.

V(ins) = 1im Δr / Δt

Where Δt → 0 is read as "Δt tends to zero", which means that the time is very small.

Velocity From Distance - Time Graph

We can determine the velocity of a body by distance - time graph such that the time is taken on x-axis and distance on y-axis.

## Acceleration

Acceleration of a body may be defined as the time rate of change velocity. If the velocity of a body is changing then it is said to posses acceleration.

Acceleration = change of velocity / time

If the velocity of a body is increasing, then its acceleration will be positive and if the velocity of a body is decreasing, then its acceleration will be negative. Negative acceleration is also called retardation.

Acceleration is a vector quantity and its unit in S.I system is meter per second per second. (m/sec2 OR m.sec-2)

Average Acceleration

Average acceleration is defined as the ratio of the change in velocity of a body and the time interval during which the velocity has changed.

Suppose that at any time t1 a body is at A having velocity V1. At a later time t2, it is at point B having velocity V2. Thus,

Change in Velocity = V2 - V1 = Δ V

Time during which velocity has changed = t2 - t1 = Δ t

Instantaneous Acceleration

It is defined as the acceleration of a body at a certain instant

a(ins) = lim Δ V / Δ t

where Δt → 0 is read as "Δt tends to zero", which means that the time is very small.

Acceleration from Velocity - Time Graph

We can determine the acceleration of a body by velocity - time graph such that the time is taken on x-axis and velocity on y-axis.

Equations of Uniformly Accelerated Rectilinear Motion

There are three basic equations of motion. The equations give relations between

Vi = the initial velocity of the body moving along a straight line.

Vf = the final velocity of the body after a certain time.

t = the time taken for the change of velocity

a = uniform acceleration in the direction of initial velocity.

S = distance covered by the body.

Equations are

1. Vf = Vi + a t

2. S = V i t + 1/2 a t2

3. 2 a S = V f2 - V i 2

Motion Under Gravity

The force of attraction exerted by the earth on a body is called gravity or pull of earth. The acceleration due to gravity is produced in a freely falling body by the force of gravity. Equations for motion under gravity are

1. Vf = Vi + g t

2. S = V i t + 1/2 g t2

3. 2 g S = Vr2 - Vi2

where g = 9.8 m / s2 in S.I system and is called acceleration due to gravity.

## Law of Motion

Isaac Newton studied motion of bodies and formulated three famous laws of motion in his famous book "Mathematical Principles of Natural Philosophy" in 1687. These laws are called Newton's Laws of Motion.

## Newton's First Law of Motion

Statement

A body in state of rest will remain at rest and a body in state of motion continues to move with uniform velocity unless acted upon by an unbalanced force.

Explanation

This law consists of two parts. According to first part a body at rest will remain at rest will remain at rest unless some external unbalanced force acts on it. It is obvious from our daily life experience. We observe that a book lying on a table will remain there unless somebody moves it by applying certain force. According to the second part of this law a body in state of uniform motion continuous to do so unless it is acted upon by some unbalanced force.

This part of the law seems to be false from our daily life experience. We observe that when a ball is rolled in a floor, after covering certain distance, it stops. Newton gave reason for this stoppage that force of gravity friction of the floor and air resistance are responsible of this stoppage which are, of course, external forces. If these forces are not present, the bodies, one set into motion, will continue to move for ever.

Qualitative Definition of Net Force

The first law of motion gives the qualitative definition of the net force. (Force is an agent which changes or tends to change the state of rest or of uniform motion of a body).

First Law as Law of Inertia

Newton's first law of motion is also called the Law of inertia. Inertia is the property of matter by virtue of which is preserves its state of rest or of uniform motion. Inertia of a body directly related to its mass.

## Newton's Second Law of Motion

Statement

If a certain unbalanced force acts upon a body, it produces acceleration in its own direction. The magnitude of acceleration is directly proportional to the magnitude of the force and inversely proportional to the mass of the body.

Mathematical Form

According to this law

f ∞ a

F = m a → Equation of second law

Where 'F' is the unbalanced force acting on the body of mass 'm' and produces an acceleration 'a' in it.

From equation

1 N = 1 kg x 1 m/sec2

Hence one newton is that unbalanced force which produces an acceleration of 1 m/sec2 in a body of mass 1 kg.

Vector Form

Equation of Newton's second law can be written in vector form as

F = m a

Where F is the vector sum of all the forces acting on the body.

## Newton's Third Law of Motion

Statement

To every action there is always an equal and opposite reaction.

Explanation

For example, if a body A exerts force on body B (F(A) on B) in the opposite direction. This force is called reaction. Then according to third law of motion.

Examples

1. When a gun is fired, the bullet flies out in forward direction. As a reaction of this action, the gun reacts in backward direction.

2. A boatman, when he wants to put his boat in water pushes the bank with his oar, The reaction of the bank pushes the boat in forward direction.

3. While walking on the ground, as an action, we push the ground in the backward direction. As a reaction ground pushes us in the forward direction.

4. In flying a kite, the string is given a downward jerk and is then released. Thereupon the reaction of the air pushes the kite upward and makes it rise higher.

Tension in a String

Consider a body of weight W supported by a person with the help of a string. A force is experienced by the hand as well as by the body. This force is known as Tension. At B the hand experiences a downward force. So the direction of force at point B is downward. But at point A direction of the force is upward.

These forces at point A and B are tensions. Its magnitude in both cases is same but the direction is opposite. At point A,

Tension = T = W = mg

## Momentum of a Body

The momentum of a body is the quantity of motion in it. It depends on two things

1. The mass of the object moving (m),

2. The velocity with which it is moving (V).

Momentum is the product of mass and velocity. It is denoted by P.

P = m V

Momentum is a vector quantity an its direction is the same as that of the velocity.

Unit of Momentum

Momentum = mass x velocity

= kg x m/s

= kg x m/s x s/s

= kg x m/s2 x s

since kg. m/s2 is newton (N)

momentum = N-s

Hence the S.I unit of momentum is N-s.

Unbalanced or Net Force is equal to the Rate of Change of Momentum

i.e., F = (mVf = mVi) / t

Proof

Consider a body of mass 'm' moving with a velocity Vl. A net force F acts on it for a time 't'. Its velocity then becomes Vf.

Therefore

Initial momentum of the body = m Vi

Final momentum of the body = m Vf

Time interval = t

Unbalanced force = F

Therefore

Rate of change of momentum = (m Vf - m Vi) / t ....................... (1)

But

(Vf - Vi) / t = a

Therefore,

Rate of change of momentum = m a = F ..................... (2)

Substituting the value of rate of change of momentum from equation (2) in equation (1), we get

F = (m Vf - m Vi) / t ............................. Proved

## Law of Conservation of Momentum

Isolated System

When a number of bodies are such that they exert force upon one another and no external agency exerts a force on them, then they are said to constitute and isolated system.

Statement of the Law

The total momentum of an isolated system of bodies remains constant.

OR

If there is no external force applied to a system, then the total momentum of that system remains constant.

Elastic Collision

An elastic collision is that in which the momentum of the system as well as the kinetic energy of the system before and after collision, remains constant. Thus for an elastic collision.

If P momentum and K.E is kinetic energy.

P(before collision) = P(after collision)

K.E(before collision) = K.E(after collision)

Inelastic Collision

An inelastic collision is that in which the momentum of the system before and after the collision remains constant but the kinetic energy before and after the collision changes.

Thus for an inelastic collision

P(before collision) = P(after collision)

Elastic Collision in one Dimension

Consider two smooth non rotating spheres moving along the line joining their centres with velocities U1 and U2. U1 is greater than U2, therefore the spheres of mass m1 makes elastic collision with the sphere of mass m2. After collision, suppose their velocities become V1 and V2 but their direction of motion is along same line as before.

## Friction

When two bodies are in contact, one upon the other and a force is applied to the upper body to make it move over the surface of the lower body, an opposing force is set up in the plane of the contract which resists the motion. This force is the force of friction or simply friction.

The force of friction always acts parallel to the surface of contact and opposite to the direction of motion.

Definition

When one body is at rest in contact with another, the friction is called Static Friction.

When one body is just on the point of sliding over the other, the friction is called Limiting Friction.

When one body is actually sliding over the other, the friction is called Dynamic Friction.

Coefficient of Friction (μ)

The ratio of limiting friction 'F' to the normal reaction 'R' acting between two surfaces in contact is called the coefficient of friction (μ).

μ = F / R

Or

F = μ R

Fluid Friction

Stoke found that bodies moving through fluids (liquids and gases) experiences a retarding force fluid friction or viscous drag. If the moving bodies are spheres then fluid friction F is given by

F = 6 π η r v

Where η is the coefficient of viscosity,

Where r is the radius of the sphere,

Where v is velocity pf the sphere.

Terminal Velocity

When the fluid friction is equal to the downward force acting on the sphere, the sphere attains a uniform velocity. This velocity is called Terminal velocity.

## The Inclined Plane

A plane which makes certain angle θ with the horizontal is called an inclined plane.

Diagram Coming Soon

Consider a block of mass 'm' placed on an inclined plane making certain angle θ with the horizontal. The forces acting on the block are

1. W, weight of the block acting vertically downward.

2. R, reaction of the plane acting perpendicular to the plane

3. f, force of friction which opposes the motion of the block which is moving downward.

Diagram Coming Soon

Now we take x-axis along the plane and y-axis perpendicular to the plane. We resolve W into its rectangular components.

Therefore,

Component of W along x-axis = W sin θ

And

Component of W along y-axis = W cos θ

1. If the Block is at Rest

According to the first condition of equilibrium

Σ Fx = 0

Therefore,

f - W sin θ = 0

Or

f = W sin θ

Also,

Σ Fy = 0

Therefore,

R - W cos θ = 0

Or

R = W cos θ

2. If the Block Slides Down the Inclined Plane with an Acceleration

Therefore,

W sin θ > f

Net force = F = W sin θ - f

Since F = m a and W = m g

Therefore,

m a = m g sin θ - f

3. When force of Friction is Negligible

Then f ≈ 0

Therefore,

equation (3) => m a = m g sin ≈ - 0

=> m a = m g sin ≈

or a = g sin ≈ ............. (4)

Particular Cases

Case A : If the Smooth Plane is Horizontal Then 0 = 0º

Therefore,

Equation (4) => a = g sin 0º

=> a = g x 0

=> a = 0

Case B : If the Smooth Plane is Vertical Then θ = 90º

Therefore,

Equation (4) => a = g sin 90º

=> a = g x 1

=> a = g

This is the case of a freely falling body.

## Scalars

Physical quantities which can be completely specified by

1. A number which represents the magnitude of the quantity.

2. An appropriate unit

are called Scalars.

Scalars quantities can be added, subtracted multiplied and divided by usual algebraic laws.

Examples

Mass, distance, volume, density, time, speed, temperature, energy, work, potential, entropy, charge etc.

## Vectors

Physical quantities which can be completely specified by

1. A number which represents the magnitude of the quantity.

2. An specific direction

are called Vectors.

Special laws are employed for their mutual operation.

Examples

Displacement, force, velocity, acceleration, momentum.

Representation of a Vector

A straight line parallel to the direction of the given vector used to represent it. Length of the line on a certain scale specifies the magnitude of the vector. An arrow head is put at one end of the line to indicate the direction of the given vector.

The tail end O is regarded as initial point of vector R and the head P is regarded as the terminal point of the vector R.

Diagram Coming Soon

Unit Vector

A vector whose magnitude is unity (1) and directed along the direction of a given vector, is called the unit vector of the given vector.

A unit vector is usually denoted by a letter with a cap over it. For example if r is the given vector, then r will be the unit vector in the direction of r such that

r = r .r

Or

r = r / r

unit vector = vector / magnitude of the vector

Equal Vectors

Two vectors having same directions, magnitude and unit are called equal vectors.

Zero or Null Vector

A vector having zero magnitude and whose initial and terminal points are same is called a null vector. It is usually denoted by O. The difference of two equal vectors (same vector) is represented by a null vector.

R - R - O

Free Vector

A vector which can be displaced parallel to itself and applied at any point, is known as free vector. It can be specified by giving its magnitude and any two of the angles between the vector and the coordinate axes. In 3-D, it is determined by its three projections on x, y, z-axes.

Position Vector

A vector drawn from the origin to a distinct point in space is called position vector, since it determines the position of a point P relative to a fixed point O (origin). It is usually denoted by r. If xi, yi, zk be the x, y, z components of the position vector r, then

r = xi + yj + zk

Diagram Coming Soon

Negative of a Vector

The vector A. is called the negative of the vector A, if it has same magnitude but opposite direction as that of A. The angle between a vector and its negative vector is always of 180º.

Multiplication of a Vector by a Number

When a vector is multiplied by a positive number the magnitude of the vector is multiplied by that number. However, direction of the vector remain same. When a vector is multiplied by a negative number, the magnitude of the vector is multiplied by that number. However, direction of a vector becomes opposite. If a vector is multiplied by zero, the result will be a null vector.

The multiplication of a vector A by two number (m, n) is governed by the following rules.

1. m A = A m

2. m (n A) = (mn) A

3. (m + n) A = mA + nA

4. m(A + B) = mA + mB

Division of a Vector by a Number (Non-Zero)

If a vector A is divided by a number n, then it means it is multiplied by the reciprocal of that number i.e. 1/n. The new vector which is obtained by this division has a magnitude 1/n times of A. The direction will be same if n is positive and the direction will be opposite if n is negative.

## Resolution of a Vector Into Rectangular Components

Definition

Splitting up a single vector into its rectangular components is called the Resolution of a vector.

Rectangular Components

Components of a vector making an angle of 90º with each other are called rectangular components.

Procedure

Let us consider a vector F represented by OA, making an angle O with the horizontal direction.

Draw perpendicular AB and AC from point on X and Y axes respectively. Vectors OB and OC represented by Fx and Fy are known as the rectangular components of F. From head to tail rule of vector addition.

OA = OB + BA

F = Fx + Fy

Diagram Coming Soon

To find the magnitude of Fx and Fy, consider the right angled triangle OBA.

Fx / F = Cos θ => Fx = F cos θ

Fy / F = sin θ => Fy = F sin θ

Addition of Vectors by Rectangular Components

Consider two vectors A1 and A2 making angles θ1 and θ2 with x-axis respectively as shown in figure. A1 and A2 are added by using head to tail rule to give the resultant vector A.

Diagram Coming Soon

The addition of two vectors A1 and A2 mentioned in the above figure, consists of following four steps.

Step 1

For the x-components of A, we add the x-components of A1 and A2 which are A1x and A2x. If the x-components of A is denoted by Ax then

Ax = A1x + A2x

Taking magnitudes only

Ax = A1x + A2x

Or

Ax = A1 cos θ1 + A2 cos θ2 ................. (1)

Step 2

For the y-components of A, we add the y-components of A1 and A2 which are A1y and A2y. If the y-components of A is denoted by Ay then

Ay = A1y + A2y

Taking magnitudes only

Ay = A1y + A2y

Or

Ay = A1 sin θ1 + A2 sin θ2 ................. (2)

Step 3

Substituting the value of Ax and Ay from equations (1) and (2) respectively in equation (3) below, we get the magnitude of the resultant A

A = |A| = √ (Ax)2 + (Ay)2 .................. (3)

Step 4

By applying the trigonometric ratio of tangent θ on triangle OAB, we can find the direction of the resultant vector A i.e. angle θ which A makes with the positive x-axis.

tan θ = Ay / Ax

θ = tan-1 [Ay / Ax]

Here four cases arise

(a) If Ax and Ay are both positive, then

θ = tan-1 |Ay / Ax|

(b) If Ax is negative and Ay is positive, then

θ = 180º - tan-1 |Ay / Ax|

(c) If Ax is positive and Ay is negative, then

θ = 360º - tan-1 |Ay / Ax|

(d) If Ax and Ay are both negative, then

θ = 180º + tan-1 |Ay / Ax|

## Addition of Vectors by Law of Parallelogram

According to the law of parallelogram of addition of vectors, if we are given two vectors. A1 and A2 starting at a common point O, represented by OA and OB respectively in figure, then their resultant is represented by OC, where OC is the diagonal of the parallelogram having OA and OB as its adjacent sides.

Diagram Coming Soon

If R is the resultant of A1 and A2, then

R = A1 + A2

Or

OC = OA + OB

But OB = AC

Therefore,

OC = OA + AC

β is the angle opposites to the resultant.

Magnitude of the resultant can be determined by using the law of cosines.

R = |R| = √A1(2) + A2(2) - 2 A1 A2 cos β

Direction of R can be determined by using the Law of sines.

A1 / sin γ = A2 / sin α = R / sin β

This completely determines the resultant vector R.

1. Commutative Law of Vector Addition (A+B = B+A)

Consider two vectors A and B as shown in figure. From figure

OA + AC = OC

Or

A + B = R .................... (1)

And

OB + BC = OC

Or

B + A = R ..................... (2)

Since A + B and B + A, both equal to R, therefore

A + B = B + A

Diagram Coming Soon

2. Associative Law of Vector Addition (A + B) + C = A + (B + C)

Consider three vectors A, B and C as shown in figure. From figure using head - to - tail rule.

OQ + QS = OS

Or

(A + B) + C = R

And

OP + PS = OS

Or

A + (B + C) = R

Hence

(A + B) + C = A + (B + C)

Diagram Coming Soon

Product of Two Vectors

1. Scalar Product (Dot Product)

2. Vector Product (Cross Product)

1. Scalar Product OR Dot Product

If the product of two vectors is a scalar quantity, then the product itself is known as Scalar Product or Dot Product.

The dot product of two vectors A and B having angle θ between them may be defined as the product of magnitudes of A and B and the cosine of the angle θ.

A . B = |A| |B| cos θ

A . B = A B cos θ

Diagram Coming Soon

Because a dot (.) is used between the vectors to write their scalar product, therefore, it is also called dot product.

The scalar product of vector A and vector B is equal to the magnitude, A, of vector A times the projection of vector B onto the direction of A.

If B(A) is the projection of vector B onto the direction of A, then according to the definition of dot product.

Diagram Coming Soon

A . B = A B(A)

A . B = A B cos θ {since B(A) = B cos θ}

Examples of dot product are

W = F . d

P = F . V

Commutative Law for Dot Product (A.B = B.A)

If the order of two vectors are changed then it will not affect the dot product. This law is known as commutative law for dot product.

A . B = B . A

if A and B are two vectors having an angle θ between then, then their dot product A.B is the product of magnitude of A, A, and the projection of vector B onto the direction of vector i.e., B(A).

And B.A is the product of magnitude of B, B, and the projection of vector A onto the direction vector B i.e. A(B).

Diagram Coming Soon

To obtain the projection of a vector on the other, a perpendicular is dropped from the first vector on the second such that a right angled triangle is obtained

In Δ PQR,

cos θ = A(B) / A => A(B) = A cos θ

In Δ ABC,

cos θ = B(A) / B => B(A) = B cos θ

Therefore,

A . B = A B(A) = A B cos θ

B . A = B A (B) = B A cos θ

A B cos θ = B A cos θ

A . B = B . A

Thus scalar product is commutative.

Distributive Law for Dot Product

A . (B + C) = A . B + A . C

Consider three vectors A, B and C.

B(A) = Projection of B on A

C(A) = Projection of C on A

(B + C)A = Projection of (B + C) on A

Therefore

A . (B + C) = A [(B + C}A] {since A . B = A B(A)}

= A [B(A) + C(A)] {since (B + C)A = B(A) + C(A)}

= A B(A) + A C(A)

= A . B + A . C

Therefore,

B(A) = B cos θ => A B(A) = A B cos θ1 = A . B

And C(A) = C cos θ => A C(A) = A C cos θ2 = A . C

Thus dot product obeys distributive law.

Diagram Coming Soon

2. Vector Product OR Cross Product

When the product of two vectors is another vector perpendicular to the plane formed by the multiplying vectors, the product is then called vector or cross product.

The cross product of two vector A and B having angle θ between them may be defined as "the product of magnitude of A and B and the sine of the angle θ, such that the product vector has a direction perpendicular to the plane containing A and B and points in the direction in which right handed screw advances when it is rotated from A to B through smaller angle between the positive direction of A and B".

A x B = |A| |B| sin θ u

Where u is the unit vector perpendicular to the plane containing A and B and points in the direction in which right handed screw advances when it is rotated from A to B through smaller angle between the positive direction of A and B.

Examples of vector products are

(a) The moment M of a force about a point O is defined as

M = R x F

Where R is a vector joining the point O to the initial point of F.

(b) Force experienced F by an electric charge q which is moving with velocity V in a magnetic field B

F = q (V x B)

Physical Interpretation of Vector OR Cross Product

Area of Parallelogram = |A x B|

Area of Triangle = 1/2 |A x B|

### Class XI, ENGLISH, "Punctuation"

1. Androcles who had no arms of any kind now gave himself up for lost what shall I do said he I have no spear or sword no not so much of a stick to defend myself with.

A. Androcles, who had no arms of any kind, now gave himself up for lost. “What shall I do?” aid he, “I have no spear or sword. No, not so much of a stick to defend myself with?”

2. May I ask the name of this building said the president kindly it is called the department of justice.

A. “May I ask the name of this building,” said the president kindly. “It is called the Department of Justice.”

3. Think a hundred times before you take a decision Jinnah told the league at lahore but once the decision is taken stand by it as one man.

A. “Think a hundred times before you take a decision,” Jinnah told the League at Lahore, but once the decision is taken, stand by it as one man.”

4. What a lazy boy you are said the teacher to saleem you never pay attention to your studies I am too weak to pull on with the class replied saleem.

A. “What a lazy boy you are!” said the teacher to Saleem. “You never pay attention to your studies.” I am too weak to pull on with the class replied Saleem.”

5. You say said the judge that the bag you lost contained one hundred and ten pounds yes your honour said the miser then as this bag contains one hundred pounds only it cannot be yours said the judge.

A. “You say,” said the judge, “that the bag you lost contained one hundred and ten pounds.” “Yes, your Honour,” said the miser. “Then, as this bag contains one hundred pounds only, it cannot be yours,” said the judge.

6. Why have you come late he asked don’t you know that the school begins at eight I am very sorry sir replied the body my bus broke down on the way I had to walk more than a mile to reach here.

A. “Why have you come late?” he asked. “Don’t you know that the school beings at eight?” “I am very sorry, sir,” replied the boy. “My bus broke down on the way. I had to walk more than a mole to reach here.”

7. At the end of the play she turned to me and said quite naturally oh did you get my letter didn’t you I said well I got an envelope this morning with a ticket in it.

A. At the end of the play, she turned to me and said quite naturally, “Oh! Did you get my letter, didn’t you?” I said, “Well I got an envelope this morning, with a ticket in it.”

8. Are you aware my dear husband that there is an old arab custom never to break bread in the house of an enemy.

A. “Are you aware, my dear husband, that there is an old Arab custom never to break bread in the house of an enemy?”

9. I shall give her notice as soon as the new year festivities are over said the baroness till then I shall be too busy to manage without her.

A. “I shall give her notice as soon as the New Year festivities are over,” said the Baroness, “till then, I shall be too busy to manage without her.”

10. You are very young my son and games are more suited to your years than battles but you are as strong as an elephant.

A. “You are very young, my son and games are more suited to your years than battles. But you are as strong as an elephant.”

### Class XI, ENGLISH, "Prepositions"

1. I come to the university ______ bus.(By)

2. He is afraid ______ snakes.(of)

3. The author beings ______ asking four questions.(By)

4. All science beings ______ the knowledge of our ignorance.(With)

5. I complimented him ______ his success.(on)

6. Send this letter ______ this address.(To)

7. He managed to get a seat ______ the bus.(In)

8. He does not devote much time ______ his studies.(To)

9. He is an authority ______ Malaysian history.(on)

10. She had a good reason ______ being angry.(For)

11. I have very little faith ______ his judgement.(In)

12. Do not laugh ______ the old man.(At)

13. He comes ______ a noble family.(of)

14. He did not agree ______ me.(With)

15. I was angry ______ his behaviour.(At)

16. His views are not clear ______ me.(To)

17. She is kind ______ me.(To)

18. My brother is angry ______ me.(With)

19. She felt great joy ______ her success.(In)

20. She is good ______ English.(In)

21. Put ______ the lights.(out)

23. The man is blind ______ one eye.(With)

24. My sister stays ______ home.(At)

25. Take care ______ your health.(Of)

26. Write ______ the margin of your book.(In)

27. His office is adjacent ______ the mosque.(To)

28. She is very proficient ______ English.(In)

29. I am not ashamed ______ anything I have done.(Of)

30. She has always felt inferior ______ her sister.(To)

31. He is frequently absent ______ school.(From)

32. He is ______ the height of his career.(At)

33. The poor beggar was crushed ______ a car.(By)

34. What the chances ______ your success in the examination?(Of)

35. I got eighty marks ______ of hundred.(Out)

36. What is the table made ______?(Of)

37. They are laughing ______ him.(At)

38. We went ______ the seaside by car.(To)

39. I bought this car ______ Rs. 20/-.(For)

40. He was accused ______ stealing the money.(Of)

41. He stood ______ me in crisis.(By)

42. Do not blame me ______ this.(For)

43. He has promised to look ______ the matter.(Into)

44. He has acceded ______ my request.(To)

45. He told me that he was familiar ______ that subject.(With)

46. Do not cry ______spilt mill.(At)

47. The children have been playing ______ 5O’clock.(Since)

48. Charity begins ______ home.(At)

49. The revolt was put ______ at the right time.(Out)

50. He is a man ______ iron will.(Of)

51. A man ______ need is a friend indeed.(In)

52. Will you take care ______ this packet and keep it ______ you.(Of,with)

53. Is he an authority ______ this subject.(On

54. I prefer foot ball ______ to baseball.(To)

55. The committee was opposed ______ the proposal.(With)

56. He is very much interested ______ world affairs.(In)

57. She is not afraid ______ death.(Of)

58. Wait ______ me ______ the station.(For,at)

59. They went home ______ foot.(On

60. The house, he lives ______ is very old.(In)

61. Smoking is injurious ______ health.(For)

62. He will tell you ______ his result.About)

63. He is trying to change this wood ______ coal.(Into)

65. Isn’t the road ______ construction.(For)

66. I arrived ______ the station ______ the train had left.(At,after)

67. The shop was ______ fire and the people were crying ______ help.(On,for)

68. What a pity! We don’t adhere ______ our own principles.(To)

69. He stood ______ me.(By)

70. He ran ______ me.(After)

71. She does her work ______ night.(At)

72. There were hills all ______ the way.(Along)

73. I have some pain ______ my chin.(Under)

74. The end is ______ hand.(At)

75. I am tired ______ doing nothing.(Of)

76. He succeeded ______ secuing some votes.(In)

77. That portrait is very much true ______ life.(To)

78. I differ ______ you entirely.(With)

79. I was thinking ______ you.(of)

80. It is of no use ______ me.(To)

81. She was standing ______ the two houses.(Between)

82. A girl ______ blue eyes has just gone ______ the door.(With,off)

83. He complained ______ pain in her back.(of)

84. She is glad ______ my success.(At)

85. Always be good ______ others.(To)

86. They walked ______ the railway line.(Along)

87. Turn left ______ the next intersection.(At)

88. He was wearing a fine shirt ______ his coat.(Underneath)

89. There is a stream ______ the bridge.(Under)

90. He spoke ______ me in English.(To)

91. I have applied ______ the post.(For)

92. He is interested ______ buying my car.(In)

93. He has not replied ______ my letter.(To)

94. He is angry ______ me.(With)

95. Are you sure ______ the facts.(Of)

96. I have great regard ______ my father.(For)

97. The hunt ______ knowledge is a long-life task.(For)

98. He was shot dead ______ his enemy.(By)

99. The house is ______ fire.(On)

### Class XI, ENGLISH, "Articles"

1. ______ apple ______ day keeps ______ doctor away. (An, a, the)

2. He has gone to ______ hospital for ______ operation. (The, an)

3. He is ______ honourable man. (An)

4. ______ bird can fly very high in ______ sky.(a, the)

5. Space travel has now become ______ reality.(a)

6. John is ______ only student who didn’t pass ______ test.(The,the)

7. I plan to buy ______ expensive camera.(An)

8. He will leave after ______ day or two.(A)

9. ______ old gardener is watering ______ plants.(An,the)

10. He is always helpful to ______ people.(The)

11. I want ______ glass of milk.(A)

12. Ahmed is ______ tallest boy of the class.(The)

13. There is ______ fly in ______ ointment.(A,the)

14. She is ______ honourable woman.(An)

15. For ______ first time on a trip, we had ______ difference of opinion as to how ______ best, this should be done for ______ better result.(The,the,the)

16. At ______ little distance from ______ college, there is ______ old building in which ______ honest and hardworking man lives on ______ top floor.(A,the,an,an,the)

17. He is ______ very brilliant boy.(A)

18. They are ______ best out of the whole lot.(The)

19. There is ______ shop round the corner.(A)

20. I must have ______ extra key for the front door.(An)

21. He broke ______ leg in the skiing accident. It’s still in the plaster.(A)

22. Quaid-e-Azam was ______ honourable man.(An)

23. ______ independence of United States of America dates from ______ 4th of July, 1776.(The,the)

24. ___the___ sun went down below the horizon.(The)

25. ______ stitch in time saves time.(A)

26. ______man is mortal.(The)

27. Honesty is ______ best policy.(The)

28. I saw Ahmed in ______ hotel. I saw him ______ hour ago.(A,an)

29. The more you work, ______ better will your result.(The

30. They received ______ telegram in the after noon.(A)

31. He worked hard, as he had ______ object to work for.(An)

32. You should eat ______ apple ______ day.(An,a)

33. My hen laid ______ egg Yesterday.(An)

34. ______ few of then manged to touch ______ shore.(A,the)

35. ______ indus has flooded ______ village.(The,a)

36. ______ apple has ______ sweet taste.(An,a)

37. the knife is made of ______ metal.(A)

38. He is ______ M.A in English.(An)

39. Iron is ______ metal.(A)

40. A red and ______ white cow are grazing in the field.(A)

41. ______ sky was dark and it seemed that ______ storm was coming.(The,a)

42. I need ______ umbrella.(An)

43. ______ Ravi is not ______ longest river in Pakistan.(The,the)

44. ______ dust man comes only a week.(The)

45. Karachi is ______ biggest city in Pakistan.(The)

46. ______ Quran is a holy book.(The)

47. ______ Jhelum is a big river.(The)

48. Man is ______ mortal being.(A)

50. The lion is ______ noble beast.(A)

51. We started late in ______ afternoon.(The)

52. ______ Indus is a great river.(The)

53. Quran is sacred book of ______ Muslims.(The)

54. ______ horse is a useful animal.(The)

55. ______ luggage is on ______ platform.(The,the)

56. He showed ______ courage worthy of ______ old man.(A,an)

57. I cannot forget ______ kindness with which he treated me.(The)

58. I went to ______ hospital to see my uncle.(The)

59. Copper is ______ useful metal.(A)

60. I have ______ black and ______ white dog.(A,a)

61. ______ more I think about the idea ______ less like it.(The,the)

62. I should like ______ house in ______ country.(A,the)

63. ______ rich are happy.(The)

64. Which is ______ longest river in Pakistan?(The)

65. I hae been ill for ______ last two days.(The)

66. ______ honest man is ______ noblest work of God.(An,the)

### Class XI, ENGLISH, "Idiomatic Structures"

1. AT SIXES AND SEVENS: Home ruler, who were all at sixes and sevens among themselves agreed only upon the one thing and that was the freedom of India.

2. ALL IN ALL: The Head clerk is all in all in this office.

3. ALL THE SAME: It is all the same to me whether the pull over is home-made or bazaar-made.

4. AT LARGE: The culprits are still at large.

5. BY FITS AND STARTS: He works by fits and starts and does not apply him steadily.

6. BLACK SHEEP: We should be aware of the black sheep in our society.

7. A BONE OF CONTENTION: This property is a bone of contention between the two brothers.

8. TO BREAK THE ICE: We all wanted to talk on this subject by no one willing to break the ice.

9. A BURNING QUESTION: Kashmir is a burning question of the day.

10. TO BACK OUT: He promised to help me but backed out at the eleventh hour.

11. TO BEAT ABOUT THE BUSH: Stop beating about the bush; say exactly what you mean.

12. BED OF ROSES: A military life is not bed of roses.

13. IN COLD BLOOD: He murdered the merchant in cold blood.

14. TO FALL TO THE GROUND: The theory has fallen to the ground.

15. GO HAND IN HAND: Diligence and prosperity go hand in hand.

16. LEAVE NO STONE UNTURNED: Shah Faisal left no stone unturned to bring about unity in the Islamic world.

17. LIVE FROM HAND TO MOUTH: Our middle class people live generally from hand to mouth.

18. LOOK DOWN UPON: He is so proud of his promotion that he looks down upon all his former friends.

19. AT A LOSS: He is never at a loss for an appropriate word.

20. TO PAY BACK IN THE SAME COIN: If a person rude towards you, it does not mean that you should pay him in the same coin.

21. TO KEEP PACE WITH: Agriculture in the states has kept pace with manufacture, but it has far out stepped commerce. 22. RED TAPE: Florence Nightingale was a sworn enemy of red tape.

23. TO SPEAK VOLUMES: The murders spoke volumes about political conditions before Indian elections.

24. UP TO THE MARK: You don’t look quite up to the mark today.

25. TO GET INTO HOT WATER: Do not quarrel with your officers or you will soon get into hot water.

26. TIME AND AGAIN: Time and again proverbs come to be true.

27. CUT OFF: The supplies were cut off from the soldier due to snow fall.

28. RUN AGAINST: Zuhair Akram Nadeem was running against Dr. Farooq Sattar in the elections 89.

29. TO TURN OVER A NEW LEAF: The teacher pardoned the boy on the condition that he promised to turn over a new leaf in future.

30. TO NIP IN THE BUD: The plot to overthrow the Government was detected and nipped in the bud.

31. TO FEEL LIKE A FISH OUT OF WATER: Being the only educated person in that village, I felt like a fish out of water.

32. TO SHED CROCODILE TEARS: Don’t be deceived by the beggar’s crying. They are only crocodile’s tears.

33. LION SHARE: The stronger person generally gets the lions share of the property.

34. TO CRY OVER SPILT MILK: The damage has been done but instead of crying over spilt milk do something to repair it.

35. IT IS HIGH TIME: The exams begin next month so it is high time to study seriously.

36. TO SAVE SOMETHING FOR THE RAINY DAY: He wasted his savings and has kept nothing for the rainy day.

37. WITH A HIGH HAND: He is the most unpopular because he decides matters with a high hand.

38. DAY IN AND DAY OUT: I have been warning you day in and day out.

39. TO MAKE THE MOST OF: He let me use his bicycle for a week and I am going to make the most of it.

40. TO MAKE THE FUN OF: We should not make fun of handicaps.

41. TO MAKE ROOM FOR: They made room for more guests as all seats were full.

42. TO GO THROUGH: He went through the whole book within a week.

43. IN ALL: He got 782 marks in all.

44. ALL ALONE: Yesterday night she was all alone in her house.

45. TO PUT INTO PRACTICE: The Holy Prophet (P.B.U.H) put into practice what he preaches.

46. A WILD GOOSE CHASE: The robbers fled away and the police gave them a wild goose chase.

47. TO END IN SMOKE: All his efforts ended in smoke because they were not made sincerely.

48. WITH FLYING COLOURS: If you work hard you will pass your examination with flying colours.

49. ODDS AND ENDS: The shopkeeper does not sell any particular article, but deals in odds and ends.

50. UNDER ONE’S NOSE: The police were on the look out for the culprit who was hiding under their nose.

51. TO POKE ONE’S NOSE INTO: One should not poke one’s nose into others affairs.

52. TO KICK UP A ROW: It is useless kicking up a row when the matters can be decided peacefully.

53. TO WIND UP: He is winding up his business in the city, as he going abroad.

54. IN BLACK AND WHITE: I want your statement in black and white.

55. A RED LETTER DAY: 14th August is a red letter day in the history of Pakistan.

56. TO RUN INTO: Last night my friend ran into a cheat who deprived him of his brief case by changing it with an empty one.

57. TO BRING TO LIGHT: A number of facts were brought to light by the Prime Minister in the recent Press Conference.

58. AT THE ELEVENTH HOUR: The president postponed his meeting with the journalists due to visit of the French delegation at the eleventh hour.

59. TO COME ACROSS: In the wedding party, she come across he two very close friends of the University life.

60. TO GIVE UP: The doctor has strictly advised him to give up drinking and smoking for the sake of his life.

62. TO LOOK AFTER: All the parents have to look after their children during the early period of the school life.

63. TO BREAK UP: The two partners have decided to break up the partnership and divide the assets equally.

64. TO GET RID OF: Pakistan must get rid of that type of foreign aid, which puts on her, undue political pressure.

65. AT A STRETCH: Saeed Anwer played an aggressive inning and continued to score runs at a stretch.

66. TO GIVE IN: Imran Khan and Miandad were real fighters and they would never give in till the last ball.

67. TO LET DOWN: The rich feel proud of their wealth and usually let down the poor.

68. ONCE IN A BLUE MOON: I am not so fond of movies and watch some fine art movie once in a blue moon.

69. TO FALL OUT: A short tempered football player fell out with his opponents and got wounded.

70. TO CALL ON: The winners of 1994 World Cup called on the President, with their captain.

71. TO CALL OFF: The University students finally decided to call off the strike as their demands were accepted.

72. TO BRING HOME TO: Rizwan brought home to her all the important aspects of the matter.

73. TO GET OVER: The Indian Government made all possible efforts to get over the epidemic of plague.

74. TO GET ACROSS: The news of Mr. Eddhi’s self-exile got across the country within no time.

75. TO MAKE UP FOR: The Government and people of Iraq are working day and night to make up the loss caused by the Gulf war.

76. TO MAKE OFF: The robbers made off through the back door just as the security guard started firing into air.

77. TO BRING OUT: The telephone Corporation has brought a decent Directory in three volumes.

78. TO BRING UP: Abraham Lincoln was brought up by his parents in a state of very limited financial resources.

79. TO TAKE OFF: The Hajj flight will take off every morning during the next couple of weeks.

80. TO TAKE PLACE: The wedding of my cousin will take place in the first week of November, next.

81. TO KEEP UP: Our cricket team must go through an extensive training and practice session to keep up their position in the next world cup.

82. TO STIR UP: The statement given by Mr. Abdul Sattar Eddhi caused great stir up in the political circles.

83. TO GO OFF: While the police man was cleaning his rifle, it suddenly went off because it was loaded.

84. TO LET OFF: Finally, the defaulter was let off by the civil authorities in view of his undertaking to abide by the rules in future.

85. TO BEG FOR: The Quaid-e-Azam begged for peace and friendship with his former enemies, the Congress leaders.

86. TO FURNISH WITH: The chief justice was furnished with all the documentary proofs against the accused.

87. TO LOOK FOR: After the panic had subsided, people started looking for their misplaced baggage.

88. TO RUN AFTER: According to Einstein, ordinary people run after ordinary objects such as property and luxury.

89. TO TURN DOWN: The secretary was taking down the main points to prepare a summary of the Seminar on pollution.

90. TO WATCH OVER: Sensible parents make it a point to watch over the outdoor activities of their growing up children.

91. TO BANK ON: Never bank on a fair weather friend because he will certainly cheat you.

92. TO BLOW HOT AND COLD: It is part of his nature to blow hot and cold as he favours this political party today the other party tomorrow.

93. TO BREAK THE NEWS: It was really very hard to break the shocking news of her husband’s accidental death to her.

94. TO CALL NAMES: He is such loose tempered man that he often begins to call names to his neighbours.

95. TO TURN THE TABLES: The pace attack by Wasim Akram and Waqar turned the tables against India and our cricket team got victory.

96. TO HOLD WATER: The judge will give a favourable verdict only when you lawyer’s arguments hold water.

97. TO FACE THE MUSIC: Those who are responsible for terrorism in the city must face the music and be dealt with.

98. TO BE UNDER THE CLOUD: These days, the opposition leaders are under a cloud and being tortured by the Government.

99. BY HOOK OR BY CROOK: The corrupt politicians try to win in every general election by hook or by crook.

100. TO RUN SHORT OF: These days most areas in Karachi are running short of water supply.

101. TO KEEP AN EYE ON: Wise and responsible parents always keep and eye on the outdoor activities of their children.

102. TO BUILD CASTLES IN THE AIR: It is a favourite hobby of day dreamers and idealists to build castles in the air.

103. TO TAKE TO HEELS: Just as the mobile of Rangers approached, the robbers jumped over the gate and took to their heels.

104. BY LEAPS AND BOUNDS: In the 21st century, Pakistan is expected to make progress by leaps and bound.

105. TO TURN DEAF EAR TO: He turned a deaf ear to his father’s advice and as a result, fell into trouble.

106. AT THE NICK OF TIME: Medical aid was provided to the injured passengers at the nick of time and it proved effective.

107. TO BELL THE CAT: All the office workers are annoyed with the attitude of the M.D but no body dares to bell the cat.

108. TO HAVE AN AXE TO GRIND: He certainly had an axe to grind behind his sympathetic attitude.

109. TO BURRY THE HATCHET: At last the two combatant groups agreed to bury the hatchet and restore peace.

110. TO BEAR WITH: During our lifetime we have to bear with many sorrows and sufferings.

111. TO BEAR OUT: As a witness, he bore out in the court that the man was innocent.

112. TO BEAR IN MIND: Always bear in the advice of your elders.

113. TO BREAK INTO: The Dakotas broke into the bank and took away a large sum of money.

114. TO BREAK OFF: Pakistan has broken off with Israel since the last two decades.

115. TO BREAK DOWN: If my car had not broken down on the way, I would have reached in time.

116. TO BREAK THE HEART: Don’t break the heart by rejecting the offer.

117. TO BLOW OUT: On the occasion of his birthday, he blow out the candle on cake.

118. TO BLOW UP: Four bombs blew up at different places simultaneously.

119. TO BLOW ONE’S OWN TRUMPET: I always try to avoid such people who keep on blowing their own trumpet.

120. TO BRING IN: Imran Khan has brought in a large amount for setting up the cancer hospital.

121. TO BRING ABOUT: The fight between the two political parties can bring about another martial law.

122. TO BRING ROUND: By presenting a very logical argument, he was able to bring round all the members of committee.

123. TO BRING TO BOOK: All those who kidnap people for ransom money should be brought to book.

124. TO CARRY ON: Let me carry on my work without any disturbance.

125. TO CARRY THROUGH: If we work altogether like a lean, we can easily carry through our mission with any difficulty.

126. TO CALL ON: I shall call on your brother next week.

127. TO CALL AT: I shall call at your office tomorrow.

128. TO CALL FOR: You careless and rude behavior call for an explanation.

129. TO CALL IT A DAY: As we are tired after a hard day but let it call it a day.

130. TO CALL TO MIND: I can call to mind when I saw you last.

131. TO COME OF: Although she comes of a rich family, she is not proud of her wealth.

132. TO COME OFF: The annual meeting of the Board of Directors will come off next month.

133. TO COME BY: It is difficult to understand how did he come by all that money.

134. TO COME ROUND: He comes round after I had presented my views in a logical way.

135. TO COME TO LIGHT: Once the facts come to light, we will know who is responsible for creating such a situation.

136. TO COME TO BLOWS: Very often, student belonging to different groups come to blows on silly matters.

137. TO COME OVER: With faith in God and confidence in your self you can come over all you problems.

138. TO DO WITHOUT: No living creature can do without air.

139. TO DO AWAY WITH: It is the duty of the young people to do away with all the evil customs and traditions of the society.

140 TO DIE OFF: In the under developed countries, a large number of people die off.

141. TO DIE IN HARNESS: Once he had lost all his money at stakes he died in harness.

142. TO DEAL WITH: He has the knack of dealing with all kinds of people and situation.

143. TO DEAL IN: As he deals in auto-parts, he has a good knowledge of different kind of car.

144. TO DEAL OUT: He dealt out the card after shuffling the cards.

145. TO FALL SHORT OF: The performance of Indian Cricket team fell short of the expectations of the spectators.

146. TO FALL A PREY TO: The poor and the deprived always fall a prey to cruelty and injustice.

147. TO GIVE AWAY: At the end of the function, the prizes were given way by the chief guest.

148. TO KEEP IN THE DARK: The patient was kept in the dark about the nature of his illness.

149. TO KEEP BODY AND SOUL TOGETHER: With the price spiral, it is becoming difficult for the common man to keep body and soul together.

150. TO LOOK FORWARD TO: We are looking forward to this visit next month.

151. TO MAKE OFF WITH: The robber make off with a large amount from the super market.

152. TO MAKE FOR: The Birkenhead met with a disaster when it was making for South Africa.

153. TO MAKE BOTH ENDS MEET: With his limited income, it is really very difficult to make both ends meet.

154. TO MAKE UP THE MIND: Once you make your mind then stick to your decision.

155. TO PUT IN A NUT SHELL: At the end of his lecture, he put all his arguments in a nut shell.

156. TO PUT DOWN: The revolt against the king was put down by the royal forces.

157. TO PUT OFF: The debate, which was put off last week, is scheduled for tomorrow.

158. TO STAND BY: I shall stand by you whenever you are in trouble.

159. TO TAKE AFTER: Children very often take after their parents

160. TO TAKE UP: He has decided to take up the profession of teaching.

## Introduction

This is an interesting story about supernatural forces and strange happenings written by Saki H.H. Munro. The author is well known for his tales of mystery and magical powers. The story is about an old castle and its owners, the Cernogratz family who had to sell off their family castle, when their fortune turned against them. Thus, they abondoned their ancestral family castle. As time went by, the castle was purchased by Gruebel family.

## Summary

One of the last days of December, the Baroness, the new owner of the Cernogratz Castle, was engaged in a friendly conversation with her guests. She was telling them about a strange legend attached to the castle. She said that whenever someone died in the castle, all the wild beasts and wolves would appear from nowhere and start howling all night long. But she quickly brushed this legend aside by saying that, it is only a gimmick to enhance the value of castle. She also said that she did not believe in the legend as she had proof that nothing of the sort ever happened. When her old mother-in-law died in the castle, nothing of the sort happened and no wolves appeared. According to her it was utter rubbish and that there was no truth in the legend. That the people had merely invented a story so as to give cheap publicity to the place. On hearing this account, Amalie von Cernogratz, an old governess employed in the house, remarked:

“THE STORY IS NOT AS YOU HAVE TOLD IT. IT IS NOT WHEN ANY ONE DIES IN THE CASTLE, THAT THE HOWLING IS HEARD. IT IS ONLY HEARD, WHEN A MEMBER OF THE CERNOGRATZ FAMILY, DIES IN HIS FAMILY CASTLE THAT THE WOLVES APPEAR IN THEIR THOUSANDS AND START HOWLING.”

The Governess strongly protested and insisted that the legend was quite true. She knew the family legend very well, as she was the last of the great Cernogratz family. The old Governess repeated the actual legend in a note of defiance, almost in contempt. She made it quite clear that no howling was heard if a stranger died in the castle. But if a cernogratz died in his family castle, not only did the wolves would howl in chorus, but also a large tree would crash in the park as the soul of the dying one left its body. Naturally the company showed its disbelief. They thought that the old lady is pretending to be an important person. She knows that she will be soon past work and she wants to appeal to our sympathies. That the old Amalie is making a false claim to be a Cernogratz. When the old Governess left the room, the Baroness and her guests were convinces that, she is an ordinary woman and had some how learnt the Cernogratz legend from the peasants living in the vicinity. Later on, some mysterious happenings proved every word of the old governess. It so happened that the old governess fell ill and confined herself to her small, cheerless room. Just then the company heard the howling of wolves. Moved by some impulse, the Baroness went to the Governess’s room. To her horror, she found, all the windows open, despite the biting cold, while the old lady was lying on the couch terribly sick. The Baroness rushed forward to shut the windows. But the Governess forbade her in a very stern manner. She asked the Baroness to leave and let the windows be open, as she wanted to hear the “Death Music” of her family. The Baroness announced to her guests that the old governess was dying. While the guests were talking, they heard a loud noise of a tree splitting and then crashing down, with a loud thud. At that moment, the governess breathed her last. The news of Amalie-von-Cernogratz’s death and her affiliation to the Cernogratz family was confirmed in the newspaper, the following day. Amalie, the valued fried of the Baron and Baroness Gruebel had passed away in her old family castle.

## Introduction

My Bank Account is written by Stephen Leacock. He is one of the most popular mockers and article writer. His witty articles are the best example of sardonic Mockery. The most of his stories like Brown Eyes, Freedom Cost, Behind the Table are written in first person style.

“CONFIDENCE WITH FEAR LEAD TO MOCKERY.” ___________________________________ Stephen

## Summary

Our author had a particular kind of fear about banks. Every time he entered a bank, to do business, he felt awfully frightened. The author was afraid of banks but he had to go to a bank, as he received a raise in his salary. As he entered the bank his fear overlook him. In this panic, the author made number of stupid mistakes. He entered the bank and demanded to see the manager alone. The author was taken to a private room. The manager was convinced that the author was millionaire who wished to deposit millions of dollars. But the manager was very upset, when he learn that the author wanted to deposit, fifty six dollars and that he would deposit fifty dollars every month. Quite obviously the manager was irritated and directed him to the accountant, so that his account could be opened, which was duly done. After having deposited fifty-six dollars. The author wanted to withdraw six dollars for his current use. In his nervousness, the author made a blunder. Instead of writing a cheque for six dollars he had written fifty-six dollars. The accountant looked at him in astonishment and asked if he wanted to withdraw all his money. The author realized his mistake, but wanted to cover up his stupidness. So he replied he wanted to draw all his money. Author was feeling miserable and he want to rush out of the bank sooner than he received his money, he rushed out of the bank as he did so, a loud burst of laughter went up, to the roof of the bank.

## Introduction

The Hostile Witness is a detective story by D.Y. Morgan. In this story he depicts the character of a person who was not satisfied with the performance of the state Police, but he changed his views when witnesses the performance of the police in arresting a dangerous murderer at large.

## Summary

Norman Charlton is a robber and a murderer. He lives as a servant in White Hart Hotel. This hotel is in Kirby, a small Yorkshire town about 32 kilometers south of Darlington. He robbed the weekly wages of two people at New Castle, who were going to their factory carrying thousands of pounds. While committing the robbery, he faces protest from the two persons. In reply he shoots them. One of them, named John Edward Robson, gets killed while the other becomes seriously wounded. Charlton escapes in a stolen sports car from the place of robbery. After reaching a safe place, he leaves the car and asks for a lift at about a hundred yards from that point. Fortunately, he gets a lift from Mr. Earnest King who was on his way to White Hart Hotel. When they reach the hotel, Charlton transfers the stolen bag from Mr. King’s car into the hotel and places it behind the reception desk. He also puts King’s luggage in a room and gets dressed in his white jacket. He leaves the weapon of murder under the mattress. Mr. King sits in the lounge and orders for cold drink. While enjoying cold drink, Mr. Abott and two other guests, Mr. Cartor and Johnson enter the room. They exchange greetings and then Mr. King gets informed about the robbery and the murder. Mr. Cartor also tells about the checking of his car by the police. While they were talking about the robbery, three persons, Detective-Sergeant Manning and Police Constable Stevens and Edwards, enter the hotel. They told the people about the security and said that their duty was to check all the hotels in the North of England. Detective-Sergeant Manning asks for the keys of the rooms so that he could check the luggage of the guests. Everyone gives them the keys except Mr. King, who thinks that this action was unrespectable. But after some discussion he also gives the keys. the detective then go up to check the rooms. When they return, one of them holds a pistol in his hand, which had been fired recently. He tells that this pistol was laid under the mattress in Mr. King’s room. He suspects Mr. King for the murder. Charlton takes advantages of this situation and gives the description of the murderer – medium height and build, fresh complexion, dark hair, moustache, horn-rimmed glasses, Grey suit and Grey hat. This description exactly suited the dressing of Mr. King, which made him more mistrustful and every body stares at him. Charlton also gives the bag of money having massive locks from behind the reception desk and says that this bag was carried by Mr. King. Suddenly, Manning turns at Charlton and orders the constable to arrest him. He suspected him of the spirit gum and fake moustaches that were left on his upper lip. The lift which he had taken from Mr. King made him more suspicious. He also asks him of horn-rimmed glasses and receives the answer in positive. Manning then arrests him and tells the people how Charlton had committed the crime. He hid the pistol under the mattress in Mr. King’s room while he as keeping the luggage and threw away his fake moustache soon after killing the person. He had then taken lift from Mr. King at a bus stop near Great North Road and gotten away safely from the police. This entire work carried out by the police really impressed Mr. King and the killer was caught.

## Introduction

Birkenhead Drill is a story of extravagant deed of heroism and shvarism shown by the soldiers of British 963Army. The Birkenhead Drill was a troop ship, which come across with sudden accident which has brought a dramatic change in the life of soldiers.

Birkenhead Drill means Women and Children First is the order followed on all ships that are in danger. Birkenhead Drill means today to stand be still facing certain death so that the weaker ones may have a chance of life.

“Some people born great

Some people have greatness

Some people achieve greatness

We should make our lifes sublime”

## Summary

Birkenhead Drill was a military ship. In February 1851, it was going to South Africa with troops and their families. There were a total of six hundred and thirty people on board out of which one hundred and seventy were women and children. The rest were inexperienced military men and officers. At 2 a.m. on 25th February, when this ship was near Cape Town, it strokes a massive rock that was not shown on any of the maps. This sudden jerk broke the ship into two pieces. The front half soon sank but the hind half stayed afloat. Most of the people manage to reach the rear part. There were only 3 lifeboats left undamaged with a capacity of 60 persons per boat. Only 180 people could be saved in this way. The others would draw because the damaged military ship could not float much longer. This situation should have caused a panic on the ship. Unlike other happenings, there was no panic or confusion in the boat. The trained soldiers managed complete discipline and were successful in transferring some women and children into the lifeboats. The captain and soldiers stood line by line as if they were on their daily drill. Every one was loyal to his duty and himself. A commander set an example by giving his life for two young soldiers. When the commander was hanging on to some piece of wood, he saw two young soldiers struggling in the water. He allowed them to hold on wreckage. But the commander realized that the wreckage could not bear the weight of the three persons so he let go his hold and went into the water for ever. Due to complete management of discipline, 194 people were saved but 436 men drowned forever in the deep waters. Moral The undiminished order of the soldiers presented in the article reflects the quality of sacrifice and devotion in mankind. We should stand till facing certain fate so that the weak ones may have a chance to live. This has been known as Birkenhead Drill since then.

“DUTY, THAT WHICH STERNLY IMPELS US IN THE DIRECTION OF PROFIT, ALONG THE LINE OF DESIRE.”

## Introduction

The poem The Abbot of Canterbury, included in our book, is a ballad. Ballads have crude language because fine writing would not be suitable for the telling of this straightforward and amusing folk story. King John ruled England from 1199 to 1219, was a very unpleasant man and a thoroughly bad kind.

## Structure of Poem

The Abbot of Canterbury is a ballad of unknown poet telling an ancient story consisting of 100 lines of regular rhythm.

## Summary

There was a king of England whose name was King John. He ruled England very cruelly and he always did the wrong thing. Once he came to know that there was an Abbot of Canterbury who was leading life better than the king. He had one hundred servants and each one there wore fifty gold chains and velvet coat. They where always ready to serve the Abbot. For very minor things they used to go to the city of London.

This way of leading life angered the king and his sent for the Abbot to explain his position before the king. The Abbot said to him that he was spending the money so luxuriously because he had inherited a lot of money from his fore fathers. The king disbelieved him and charged him to be a traitor.

The king said to Abbot that his life and properties would be confiscated by the government if he could not answer his three questions. There was given three weeks time to answer those questions.

• The first question was what his worth and value was when he was having such a precious crown on his head.
• The second question was how soon he could make the journey of the whole world.
• The third question was that he was thinking at that time when he was talking to the Abbot of Canterbury.

After listening three questions the Abbot was very much confuse. He confessed that he had no mind to answer those difficult questions. He went to many universities but no one was able to answer him.

Then the shepherd of the Abbot offered him that he would imposter the Abbot prodded he was given the relevant dress of the Abbot.

In reply to the first question he said that his value was twenty-nine pence. In response to the second question he said if he rose with the sun and rode with the same he would complete the journey of the world in 24 hours. In response to the third question he said that he thinking that he was talking to the Abbot by he was not the Abbot but the shepherd. The king was very happy, he rewarded him and pardoned the Abbot.

## Introduction of the Poet

The poem The Character of a Happy Life, has been written by Sir Henry Wotton. He was born in Kent and the son of country gentleman. In this poem Wotton has described the characteristics of a person who can truly be called a happy man.

## Introduction of the Poem

We find Wotton’s poem is a sharp sense of contrast between the uneasy life of the ambitious man and the contented life of man satisfied to live an obscure life of peaceful virtue. Sir Henry Wotton wrote from experience; for he was a distinguished servant of the crown who had seen for himself the rise and fall of ambitious men. The poem consisting of six stanzas of regular couplet rhymed.

“Nature is the peace not the Land.”

__________________________________ John Keats

## Summary

A person who has freedom of will and thought leads a happy life. He does not act according to other people’s wishes. His only weapon is his simplicity and truth.

Such an upright man is not a slave of his desires. He is always prepared for death. This man is not concerned about being famous or in what people say about him publicly or privately.

This upright man is not jealous of chance or foul play. He knows that flattery gives the deepest wounds. He does not abide by the rules of the society which compel a person to do unwanted deeds. He follows the rules of goodness which will lead him to the right path.

A happy man’s life is free from numerous when he is sad he retires in his comfort of clear conscience. He hasn’t got a high position. Therefore people don’t flatter him or nick him at his down fall. A happy man prays to God regularly. In this prayer he does not ask for rocks but he asks God to be gracious and merciful on him. His favourite positive is good book or friend. A man who is truly happy is free from slavery of his desires. He isn’t ambitious. He does not expect too much therefore his hopes aren’t shattered. He doesn’t fear a downfall. This man hasn’t got lands or wealth and yet he has everything. He has got the greatest wealth of contentment and happiness.

## Conclusion

The poet described the characteristics of a noble and happy man, who is honest, simple, not slave of his desires. He follows the rules of goodness, which will lead him to the right path.

### Class XI, ENGLISH, Summary, "The Toys"

Introduction of the Poet

The Toys in one of the beautiful poems composed by Conventry Patmore. Conventry Patmore was born in London in 1823. He had a religious bend of mind deeply associated with the everyday happenings of life. His publications are The Angel in the House and The Unknown Eros and Other Odes. Patmore died in 1896.

## Introduction of the Poem

The poem conveys the idea of mercy of God through an incident in the Poet’s house. The verses of this poem are different from other poems. The lines are uneven and are rhymed in an irregular manner.

## Summary

One night, the poet scolded his son for disobeying orders and talking in loud voice. The child went quietly to his bedroom with a gloomy heart. After some time, the poet realized that he had made a mistake, as his son was lonely and his mother had died. No one was there to cheer him up.

The poet, went into his son’s bedroom thinking that his child must be weeping and trying to sleep. But when he entered the room, he saw the boy calmly sleeping in bed. The poet noticed marks of tears on his cheeks. He also saw some toys neatly arranged on a table besides his son, which were kept to comfort the sad heart. The sight of the room was very uncomfortable for the poet. He felt sorry for his attitude and learning the importance of toys for the child.

The poet was inspired by the whole incident. He kissed his son and cleared the tears in his eyes. He understood that God loves his fellowmen more than a father loves his son. Then why should not God forgive the people who commit mistakes. He also realized that as the toys were of no importance to him, this world has no worth before God. We only console our hearts by the beauty of this world. This thought gave the poet a New Hope. He prayed all night with the feeling that God is merciful and would forgive him.

## Moral

We should forgive the mistakes of people and live with a friendly atmosphere. God is merciful on us and he forgives those people who forgive the mistakes of human beings.

“Let me be a little kindness, let me be a little blinder to the faults of those around me.”