Animated Solution for Physics - Work, Energy, and Power: A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle. The motion of the particle takes place in a plane, it follows that

The problem presents a classic scenario in physics: a particle moving under the influence of a force that is always perpendicular to its velocity. This simple condition unlocks a wealth of physical insights, particularly regarding work and energy.

Imagine a particle zipping through space. At any given moment, it has a velocity vector $\vec{v}$. Now, a force $\vec{F}$ is applied to it. The crucial constraint here is that $\vec{F}$ is always perpendicular to $\vec{v}$.

What does this mean physically? A force perpendicular to velocity cannot speed up or slow down the particle; it can only change its direction. This is the hallmark of centripetal force, which causes uniform circular motion.

To understand the energy dynamics, we look at the power delivered by the force. Power $P$ is the rate at which work is done, defined mathematically as the dot product of force and velocity:

Because the force is always perpendicular to the velocity, the angle $\theta$ between them is exactly $90^{\circ }$. The dot product expands to:

Since $\cos (90^{\circ })=0$, the power delivered is zero. Consequently, the total work done by this force over any time interval is also zero.

Now, we bring in the heavy hitter: the Work-Energy Theorem. This theorem states that the net work done on an object equals its change in kinetic energy ($\Delta K$).

Since we've established that $W_{\text{net}}=0$, it immediately follows that:

This means the kinetic energy $K$ of the particle does not change. It remains perfectly constant throughout the motion.

While the kinetic energy (and thus the speed) is constant, the velocity is not, because its direction is continuously changing. Similarly, the acceleration is not constant because the force vector is continuously rotating to stay perpendicular to the velocity. The path traced by the particle is a circle, not a straight line. Therefore, the only correct conclusion is that the kinetic energy is constant.