The work done by a variable force of the form F = kx is given by which expression?

The work done by a variable force such as F = kx can be determined using the concept of integration. In this case, the force varies linearly with displacement, where k is a constant and x represents the displacement from a reference point.

To find the work done, we need to integrate the force over the displacement from an initial position (0) to a final position (x). The work done, W, can be calculated using the formula:

[ W = \int_{0}^{x} F , dx = \int_{0}^{x} kx , dx. ]

Carrying out the integration gives:

[ W = k \int_{0}^{x} x , dx = k \left[\frac{x^2}{2}\right]_{0}^{x} = k \left(\frac{x^2}{2} - 0\right) = \frac{1}{2} k x^2. ]

This derived expression, (\frac{1}{2} k x^2), represents the total work done by the variable force as it moves an object from the starting position to a specific position x.

This result reflects the energy imparted to the system by the