Introduction
In physics, the concept of 'work' has a specific mathematical definition. Often, our intuition about work (effort) contradicts the scientific definition. A classic example is a body moving in a circular path at a constant speed.
The Physics of Work
Work done ($W$) is defined as the dot product of the force vector ($\vec{F}$) and the displacement vector ($\vec{d}$):
$$W = \vec{F} \cdot \vec{d} = F d \cos(\theta)$$
Where:
- $F$ is the magnitude of the force.
- $d$ is the magnitude of the displacement.
- $\theta$ is the angle between the force and the displacement vectors.
Analyzing Circular Motion
When a body moves along a circular path of radius $r$ at a constant velocity $v$, it is subject to a centripetal force.
- Direction of Centripetal Force: By definition, the centripetal force is always directed toward the center of the circular path.
- Direction of Displacement: The displacement of the object at any given instant is tangent to the circular path.
- The Angle: Because the radius of a circle is always perpendicular to the tangent at any point, the angle $\theta$ between the centripetal force (radial) and the displacement (tangential) is exactly $90^\circ$.
Calculation
Since $\theta = 90^\circ$:
$$W = F d \cos(90^\circ)$$
Knowing that $\cos(90^\circ) = 0$, we get:
$$W = F d (0) = 0$$
Conclusion
Therefore, the work done by the centripetal force on a body revolving in a circular path is zero. This is because the centripetal force does not change the speed of the object, only its direction. Since there is no change in kinetic energy ($KE = \frac{1}{2}mv^2$), according to the Work-Energy Theorem, the net work done must be zero.