Question

A uniform rod of mass $M$ and length $L$ has been placed on a horizontal surface.

Friction is present only in the region from $x=0$ to $x=L$ and the coefficient of friction varies according to relation $\mu=kx$, where $k$ is a positive constant. A constant force $F$ is applied to pull the rod. Initial acceleration of the rod is nearly zero. Find the total heat generated in the time where the rod crosses the rough region.

Answer 1

$\frac{kMgL^2}{3}$

Answer 2

The heat generated is the work done by friction. The rod's left end moves from $s=0$ to $s=L$. At any instant, when the left end is at $s$, the portion of the rod from $s$ to $L$ is in contact with the rough surface. For an element $dx'$ at $x'$ on the surface, the friction force is $dF_f = (kx') (M/L)g dx'$. Integrating this from $s$ to $L$ gives the total friction force $F_f(s) = \frac{kMg}{2L}(L^2 - s^2)$. The total heat generated is the integral of $F_f(s)$ over the displacement $s$ from $0$ to $L$, which evaluates to $\frac{kMgL^2}{3}$.