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ü        We all know that it is hard work lifting a heavy box from one platform to another in a railway station. Similarly we know that children need lots of energy as they grow up.

ü        We feel tired if we run a hundred meters in twenty seconds, whereas we could walk that distance easily in a couple of minutes.

ü        These are some common sense notions of work and energy which can however be precisely defined and measured in physics.

ü        These definitions are measurements can be used consistently to describe and predict the behaviour of bodies and thus can form very powerful tool for analysis of physical systems.

OBJECTIVE

ü        In the previous chapter we studied the Newton’s laws of motion to understand how objects move under the influence of force acting on them or the relation between the force and the acceleration produced by this force.

ü        In this chapter however, we will study the situation where one is not interested in such an exhaustive study of object’s motion rather desires to relate the final velocity of an object to the forces acting on it without going into the deeper details how the object acquired that velocity. The work-energy theorem solves this purpose.

PRE-REQUISITE

Velocity

ü        Rate of change of position of an object is known as its velocity.

V = dx /dt

Acceleration

ü        Rate of change of velocity of an object is known as its acceleration.

a= dv / dt = d²x / dt²

Force

ü        Force is a push or pull which tends to change the position of the object on which it is applied.

Newton’s first law of motion

ü        It states that every object persists in its natural state of motion i.e. continues to be at rest or moves in a straight line with uniform (constant) velocity. (This is what is meant by natural state of motion); In the absence of a net external force acting (impressed) on it.

Newton’s Second law of motion

Newton’s Third Law of Motion

ü        When ever a body exerts a force on a second body, the second body always exerts a force of equal magnitude but opposite in direction on the first one.

Conservation of Energy

ü        Whenever one form of energy is transformed into other forms, the total amount of energy before transformation is always equal to the amount of energy after transformation, i.e., the total amount of energy remains conserved.

Centre of mass of system of the discrete particles

The centre of mass of an object is a point that represents the entire body and move in the same way as a point mass having mass equal to that of the object, when subjected to the same external forces that act on the object. That is, if the resultant force acting on an object (or system of objects) of mass m is F, the acceleration of the centre of mass of the object (or system) is given by

Where the sums extend over all masses composing the object. In a uniform gravitational field, the centre of mass and the centre of gravity coincide.

Centre of mass of continuous distribution of particles

Centre of mass of a body having continuous distribution of particles (mass) is given by

Velocity & Acceleration of centre of mass

Position vector of the centre of mass of a system of particle is given by

Conservation of Linear Momentum

System of particles

Conservation of motion of centre of mass

In absence of a net external resultant force along a certain direction on a particulars system will retain its state of motion along that direction. If it had been at rest, it will remain at rest. If it had been moving with uniform velocity then it will continue moving with uniform velocity

Characteristic of linear momentum

If depends on the frame of reference, e.g., the linear momentum of a body at rest in a moving train, is zero relative to a person sitting in the train while it is not zero for a person standing on the ground.

Two bodies of same mass and moving with same speed will have different momenta unless their directions of motion are same.

Relation between kinetic energy and momentum

Each object around us has certain shape and size. When we study motion of such objects, we will have to study motion of all the different particles of the object. Practically speaking, this is not possible and we will have to consider alternative ways to study motion of the object. Scientists have defined a concept called `Centre of Mass’ to take care of this situation. To consider motion of an object having finite size, they consider motion of the centre of mass of the object. The basic objective of this chapter is to understand this concept.

OBJECTIVE

We will attempt to define the concept of a point where the mass of the whole body can be assumed to be concentrated. This helps us in analysis of problems that involve application of forces that result in subsequent motion. We will study systems which have a non uniform distribution of mass. And finally we will understand the motion of centre of mass as the body moves.

PRE-REQUISITE

The following pre-requisites apply

Þ         Vector Representation.

Þ         Integration in one variable.

Þ         Force & Acceleration.

CORE CONCEPTS

Classically each and every object has some finite size, but we have assumed them as particles i.e. having mass but no size so far.

In translatory motion each point on an object undergoes the same displacement as any other point as time goes on, so that the motion of one particle represents the motion of whole object. Thus, the translatory motion of an object of finite size can be studied by analysing the motion of any constituent particle.

But, in cases where the motion is not translatory, rather than considering any point on the object, a point known as the centre of mass is defined, and the motion of this point is studied. The motion of the centre of mass leads to the analysis of the object as a whole

Conceptually, the point where the whole mass of body or system can be assumed to be concentrated for simplified study of its motion is called the centre of mass.

For a discrete system of particles the positions (see figure given below) of the centre of mass is

Now we shall discuss another example of two-dimensional motion that is motion of a particle on a circular path. This type of motion is called circular motion.

Consider a particle P is moving on circle of radius r on X – Y plane with origin O as centre.

The position of the particle at a given instant may be described by angle q, called angular position of the particle, measured in radian. As the particle moves on the path, its angular position (q) changes. The rate of change of angular position is called angular velocity (w) measured in radian per second.

The rate of change of angular velocity is called angular acceleration, measured in rad/s². Thus, the angular acceleration is

RELATION BETWEEN DIFFERENT PARAMETERS OF CIRCULAR MOTION

ü        It is easy to derive the equations of rotational kinematics for the case of constant angular acceleration with fixed axis of rotation. These equations are of the same form as those for one-dimensional translational motion.

w = w₀ +αt                                                             …(i)

Φ = Φ₀ + w₀t + αt²/2                                 …(ii)

w² = w₀² + 2α (Φ- Φ₀)                             …(iii)

Φ= Φ₀ + (w₀ +w)/(2t)                               …(iv)

Here,Φ is the initial angle and w₀ is the initial angular speed.

Illustrations

1.     (a) What is the angular velocity of the minute and hour hands of a clock?

(b) Suppose the clock starts malfunctioning at 7 AM which decelerates the minute hand at the rate of 4p radians/day. How much time would the clock loose by 7 AM next day?

Sol. Angular speed of

minute hand : wmh = 2Π rad/hr

= 48Π rad/day = (Π/1800) rad/sec

hour hand : whh = (Π/6) rad/hr

= 4Π rad/day = (Π/21600) rad/sec

(b)     Assume at t = 0, Φ₀ = 0, when the clock begins to malfunction. Use equation (ii) to get the angle covered by the minute hand in one day.

Hence the minute hand complete 23 revolutions, so the clock losses 1 hour.