Question

A wooden block of mass M lies on a rough floor. Another wooden block of the same mass is hanging from the point O through strings as shown in the figure. To achieve equilibrium, the coefficient of static friction between the block on the floor and the floor itself is 

• A. \( \mu = \tan \theta \)
• B. \( \mu = \cos \theta \)
• C. \( \mu = \sin \theta \)
• D. \( \mu = \tan \theta \)

Hint:

In problems involving static friction, the frictional force can be found by equating the horizontal force to the frictional force and using the relationship \( f = \mu N \), where \( N \) is the normal force.

Answer 1

The Correct Option is A

In this case, the block on the floor is being pulled horizontally by the block hanging vertically. The force of static friction is what prevents the block from moving. The frictional force must balance the horizontal force exerted by the tension in the string, which is equal to the weight of the hanging block. From the equilibrium condition: \[ T = M g \sin \theta \] The frictional force \( f \) is given by: \[ f = \mu M g \cos \theta \] At equilibrium, \( f = T \), so: \[ \mu M g \cos \theta = M g \sin \theta \] Canceling out \( M g \) from both sides: \[ \mu \cos \theta = \sin \theta \] Thus, the coefficient of static friction is: \[ \mu = \tan \theta \]