7.10:
Work-Energy Theorem for Motion Along a Curve
The work done to displace an object along a curved path equals the change in the kinetic energy of the object. This is the work-energy theorem.
Consider a ball attached to a string of negligible mass, initially at rest. As the ball is dropped, it swings, tracing a circular path.
A free-body diagram can be drawn to understand the forces acting on the ball at any arbitrary point on its path.
The work done can be found by integrating the product of the force component along the displacement and the displacement.
Putting the force and the displacement as a function of angular displacement and integrating the expression from the initial to the final position, the expression of work done is obtained.
The work equals the change in kinetic energy of the ball.
As the ball is initially at rest, its initial kinetic energy is zero. So, the final kinetic energy at the lowest point is the same as the potential energy of the ball at its initial position.
7.10:
Work-Energy Theorem for Motion Along a Curve
The work-energy theorem can be generalized to the motion of a particle along any curved path. The simple argument here is that the curved path can be considered a sum of many infinitesimal paths, each of which is a straight path. The force on the particle can be considered constant along any such infinitesimal path so that the work-energy theorem can be applied along it. So, it is also valid for the sum of these paths. The net work done is the integral of the work done along the infinitesimal paths, and the total change in kinetic energy is the sum of all the changes.
It should be noted that for motion along a curved path, the force may not necessarily be along the curve. That is, the force may have two components: one along the tangent of the curve and one perpendicular to the tangent. The force that does work on the particle and, as a result, changes its kinetic energy, is the tangential force. The line integral of the dot-product of the force and the infinitesimal line element gives the work.
Suggested Reading
- Young, H.D and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson: section 6.3; pages 191–192.