Question

A bar AB and a block are placed on a frictionless horizontal floor with their adjacent vertical faces in contact as shown by the top view in the figure. The bar is made to move with a constant velocity $\dot{s}$ along its length and with another constant velocity u perpendicular to tits length. Coefficient of friction between the block and the floor is $\mu_1$ and between the block and the bar is $\mu_2$. Find work done by the bar on the block during its transverse displacement s.

Answer 1

$\mu_1 mgs$

Answer 2

The bar exerts a normal force on the block in the transverse direction (perpendicular to its length) and a friction force along its length. The block moves with the bar's transverse velocity $u$. Due to friction from the floor, a normal force $N_{bar} = \mu_1 mg$ is required from the bar to maintain the block's transverse motion. The work done by the bar on the block during transverse displacement 's' is solely due to this normal force, as the friction force from the bar is perpendicular to the transverse displacement. Thus, work done is $N_{bar} \times s = \mu_1 mg s$.