The system shown in the figure is in static equilibrium. The rod of length L and mass M is...

SKETCH:

The top of the rod is tied to a fixed vertical surface by a string, and a force is applied at the midpoint of the rod. It is shown in below figure.

RESEARCH:

The first condition for the static equilibrium:

The net force acting on the system must be zero when the system is in static equilibrium.

The second condition for the static equilibrium:

The net torque acting on the system about the pivot point must be zero when the system is in static equilibrium.

SIMPLIFY:

Let be the length of the rod.

Let be the mass of the rod.

When the rod is in static equilibrium, then the net force acting on the rod in vertical direction must be zero.

Therefore, the normal force exerted on the rod by horizontal surface can be expressed as

When the rod is in static equilibrium, then the net torque acting on the rod about the pivot point must be zero.

Therefore, the force acting on the rod can be expressed as

. . . . . . (1)

But, the frictional force exerted by the horizontal surface on the rod is

Here, the coefficient of static friction between the rod and the horizontal surface is.

Therefore, the equation (1) becomes

Hence, the maximum force applied on the rod is.