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…
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.
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Q1: How does the work-energy theorem apply to motion along a curved path?
The work-energy theorem states that work done on an object equals its change in kinetic energy, even along curved paths. A curved path can be treated as many infinitesimal straight segments where the force remains approximately constant. Applying the theorem to each segment and summing them gives the total work and total kinetic energy change for the entire curved motion.
Q2: What role does tangential force play in doing work on a moving particle?
Only the tangential force component does work on a particle moving along a curved path. While the total force may have both tangential and perpendicular components, only the tangential component changes the particle's kinetic energy. The perpendicular component changes the direction of motion but does no work.
Q3: How is work calculated when force and displacement vary along a curved path?
Work is calculated using the line integral of the dot product of force and infinitesimal displacement elements along the path. By integrating the force component along the displacement direction from initial to final position, you obtain the total work done. This mathematical approach handles variable forces and curved paths systematically.
Q4: Why is the initial kinetic energy zero for a ball starting from rest?
A ball at rest has zero velocity, and kinetic energy depends on velocity. Since kinetic energy equals one-half mass times velocity squared, an object with zero velocity has zero kinetic energy. This initial condition is essential for applying the work-energy theorem to find final kinetic energy.
Q5: How does potential energy relate to kinetic energy in a swinging ball?
When a ball swings downward along a curved path, its gravitational potential energy converts to kinetic energy. At the lowest point, the final kinetic energy equals the initial potential energy at the starting position. This energy transformation demonstrates conservation principles underlying the work-energy theorem.
Q6: Can the work-energy theorem be applied to forces that are not along the direction of motion?
Yes, but only the force component along the direction of motion does work. When force has components perpendicular to the path, only the tangential component contributes to the dot product in the line integral. The perpendicular components change direction but do not alter kinetic energy.
Q7: How does integrating force over angular displacement help find work done?
For circular or curved motion, expressing force and displacement as functions of angular displacement allows integration from initial to final angular position. This mathematical technique converts the work calculation into a manageable integral form, yielding the total work and confirming it equals the change in kinetic energy.