Work, Energy and Power

• Work done by a constant force and a variable force

• Kinetic energy

• Work-energy theorem

• Power

• Notion of potential energy

• Potential energy of a spring

• Conservative forces

• Conservation of mechanical energy (kinetic and potential energies)

• Non-conservative forces

• Motion in a vertical circle

• Elastic and inelastic collisions in one and two dimensions

SUMMARY

1. The work-energy theorem states that the change in kinetic energy of a body is the work done by the net force on the body.

Kf - Ki = Wnet

2. A force is conservative if (i) work done by it on an object is path independent and depends only on the end points {xi , xj }, or (ii) the work done by the force is zero for an arbitrary closed path taken by the object such that it returns to its initial position. 3. For a conservative force in one dimension, we may define a potential energy function V(x).

4. The principle of conservation of mechanical energy states that the total mechanical energy of a body remains constant if the only forces that act on the body are conservative.

5. The gravitational potential energy of a particle of mass m at a height x about the earth’s surface is V(x) = m g x where the variation of g with height is ignored.

6. The scalar or dot product of two vectors A and B is written as A.B and is a scalar quantity given by : A.B = AB cos θ, where θ is the angle between A and B. It can be positive, negative or zero depending upon the value of θ. The scalar product of two vectors can be interpreted as the product of magnitude of one vector and component of the other vector along the first vector.

POINTS TO PONDER

1. The phrase ‘calculate the work done’ is incomplete. We should refer (or imply clearly by context) to the work done by a specific force or a group of forces on a given body over a certain displacement.

2. Work done is a scalar quantity. It can be positive or negative unlike mass and kinetic energy which are positive scalar quantities. The work done by the friction or viscous force on a moving body is negative.

3. The work done by a force can be calculated sometimes even if the exact nature of the force is not known. This is clear from Example 6.2 where the WE theorem is used in such a situation.

4. The WE theorem is not independent of Newton’s Second Law. The WE theorem may be viewed as a scalar form of the Second Law. The principle of conservation of mechanical energy may be viewed as a consequence of the WE theorem for conservative forces.

5. The WE theorem holds in all inertial frames. It can also be extended to noninertial frames provided we include the pseudoforces in the calculation of the net force acting on the body under consideration.

6. The potential energy of a body subjected to a conservative force is always undetermined upto a constant. For example, the point where the potential energy is zero is a matter of choice. For the gravitational potential energy mgh, the zero of the potential energy is chosen to be the ground.

7. Every force encountered in mechanics does not have an associated potential energy. For example, work done by friction over a closed path is not zero and no potential energy can be associated with friction.

8. During a collision : (a) the total linear momentum is conserved at each instant of the collision ; (b) the kinetic energy conservation (even if the collision is elastic) applies after the collision is over and does not hold at every instant of the collision. In fact the two colliding objects are deformed and may be momentarily at rest with respect to each other.