Work-Energy Theorem for Motion Along a Curve

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.

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.

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.

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.

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.

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.

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.