A body of mass slides down a plane inclined at an angle . The coefficient of friction is . Find rate at which kinetic plus gravitational potential energy is dissipated at any time .
Motion exists along the surface of the plane only (call this direction horizontal). Analyse the weight force in two components: the vertical and horizontal forces, and respectively. Along the horizontal direction, two forces are applied: and the friction . Call the velocity at time . By Newton's law, , and solve to get (assume the body was resting at ).
If is the height and the length traveled at time , then a simple argument shows . Now, the total energy is . Differentiate to get ...
(Sorry for being so analytic. I was so bored today )