A body of mass $m$ slides down a plane inclined at an angle $\alpha$. The coefficient of friction is $\mu$. Find the rate at which kinetic plus gravitational potential energy is dissipated at any time $t$.

Kinetic plus potential energy is the total mechanical energy.

$\therefore$Rate of dissipation of (kinetic energy + potential energy)

$=-$[rate of work done by friction]

$=-\left[-f·v\right]=fv$

Here $f=$frictional force $=\mu mg \cos \alpha$
and $v=$ velocity of body at time $t=at$
where $a=g \sin \alpha -\mu g \cos \alpha$

$\therefore v=\left(g \sin \alpha -\mu g \cos \alpha \right)t$

$\therefore$ The desired rate

$$\begin{gathered} =\mu mg \cos \alpha (g \sin \alpha -\mu g \cos \alpha ) t \\ =\mu m{g}^2 \cos \alpha (\sin \alpha -\mu \cos \alpha ) t \end{gathered}$$