A ball is held up by a string with the string tangent to the ball. If the angle between the string and the wall is , what is the minimum coefficient of static friction between the ball and the wall that keeps the ball from falling?
Problem Source : Introduction to Classical Mechanics
First, we identify the forces exerted on the ball :
To achieve a static state, we should have a balanced forces and balanced torque.
For balancing forces, we have
Noticed that we used larger or equal to instead of equal to. This is because the gravitational force may not overcome the sum of the friction force and tension force.
For balancing torques, we take the point where the ball is in contact with the wall as fulcrum.
Substitute into the previous equation,