Problem 48

We are give that:-

The coefficient of static friction,  .    

The coefficient of kinetic friction,  . 

The magnitude of the applied force,  .  

The angle made by the force to the horizontal,  .

Step 1.

The box of mass is held in place when the net force on the box is zero. It starts to slide when the force acting downwards is more than the force acting in the upward direction.

 

The free-body diagram of the box.

 

 

The box is pushed against the wall by a force F acting at an angle to the horizontal. The weight   of the box acts downwards. Since the box tends to slide down, the force of friction acts in the upward direction and the normal force  acts perpendicular to the wall.

 

Resolve the force into two components , as shown in the figure.

For horizontal equilibrium,

……(1)

 

For vertical equilibrium,

……(2)

 

The force of friction  and normal force   are related as follows.

.……(3)

 

Substitute equations (1) and (3) in equation (2).

.  

 

Simplify and write the expression for  .