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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

It states that rate of change of momentum of a body is equal to the force applied on it, in terms of the magnitude as well as in the sense of direction. Here the momentum is defined as the product of mass and velocity i.e. .

Therefore we can write mathematically.

F = d(mv)/ dt, if ‘m’ remains constant then

Thus acceleration is rate of change of velocity.

Since direction of a is same as F, we can write

F = ma , which is mathematically Newton’s second law of motion.

Here, if F = 0 then we find a = 0. This reminds us of first law of motion. That is, if net external force  is absent, then there will be no change in state of motion, that means its acceleration is zero.

Further we can extend second law of motion, (in fact its decomposition) to three mutually perpendicular directions as per our coordinate system.

If components in x,y, and z directions are Fx, Fy, & Fz respectively, the three accelerations produced when Fx, Fy, & Fz act simultaneously) in the body are ,  Now,

If we add three forces then resultant is called net external force.

Similarly

is called net acceleration produced in the body.

Illustrations

1.             In Figure 3.1. Let us have M = 10 kg and a new net external force in the direction as shown in figure 3.2 is 150 N. Find its acceleration.

Sol.

• The position of object can change on a straight line (like on x-axis with respect to origin) or on a plane with respect to some fixed point or frame. So we can define motion as follows:
• An object or a body is said to be in motion if its position continuously changes with time with reference to a fixed point (or fixed frame of reference)
• When the position of object changes on a straight line i.e. motion of object along straight line is called motion in one dimension.
• To understand the essential concepts of one dimensional motion we have to go through some basic definitions.
• One can see the platform from a running train, and it seems that all the objects placed on platform are continuously changing their position. But one, who is on platform, concludes that the objects on the platform are at rest. It means if we will take the train as reference frame the objects are not stationary and taking reference frame as platform the objects are stationary. So the study of motion is a combined property of the object under study and the observer. Hence there is a need to define a frame of reference under which we have to study the motion of an object. We can define the frame of reference as follows:
• A frame of reference is a set of coordinate axes which is fixed with respect to a space point (a body or an object can also be treated as a point mass and therefore it can become a site for fixing a reference frame), which we have arbitrarily chosen as per our observer’s requirement. The essential requirement for a frame of reference is that, it should be rigid.
• The position of an object is defined with respect to some frame of reference. As a convention, we define position of a point (essentially we treat body as a point mass) with the help of three co-ordinates X, Y and Z. Hence (X, Y, Z) is a set of coordinate axes representing a 3-dimensional space and each point in this space can be uniquely defined with the help of a set of X, Y and Z coordinate, all three axes being mutually perpendicular to each other. The line drawn from origin to the point represents the position vector of that point.

In the figure 2.1, the position of a point P is specified and

is called the position vector.

• Consider a case in which the position of an object changes with time. Suppose at certain instant ‘t’ the position of an object is x₁ along the x axis and some other instant ‘T’ the position is x ₂ then the displacement Δx is defined as

Δx = x₂ – x₁

• It can be seen in the figure 2.2 where x₁ and x₂ are instantaneous position of the object at time t and T respectively.
• Now consider the motion of a point A with respect to a reference point O. The motion of point A makes its radius vector vary in the general case both in magnitude and in direction as shown in figure 2.3. Suppose the point A travels from point 1 to point 2 in the time interval Δt.

Distance and Displacement

• To understand the difference between distance and displacement, we study the motion of vertical throw of a ball with respect to point O to height h.
• After some time it will come again to the same point O. The displacement of ball is zero but there is some distance traversed by the ball. It’s because distance is a scalar quantity but displacement is a vector quantity.

Speed and Velocity

• Speed is the rate of change of distance without regard to directions. Velocity is the rate at which the position vector of a particle changes with time. Velocity is a vector quantity whereas speed is scalar quantity but both are measured in the same unit m/sec.