Source: High school physics (Chinese)
Problem Sets:
Problem
A block of mass $m$ starts from rest at point A, the top of a quarter-circular track of radius $R$. It slides down to point B at the bottom of the track, reaching a final speed $v$.
We use the work-energy theorem, which states that the net work done on an object equals its change in kinetic energy, $W_{net} = \Delta K$. The forces doing work are gravity and friction. The normal force does no work as it is always perpendicular to the displacement.
The work done by gravity as the block moves through a vertical displacement of $R$ is:
$$W_g = mgR$$The change in the block's kinetic energy is:
$$\Delta K = K_f - K_i = \frac{1}{2}mv^2 - 0 = \frac{1}{2}mv^2$$The net work is the sum of the work done by gravity and friction:
$$W_{net} = W_g + W_f$$Equating net work to the change in kinetic energy:
$$mgR + W_f = \frac{1}{2}mv^2$$Solving for the work done by friction, $W_f$:
$$W_f = \frac{1}{2}mv^2 - mgR$$