A thin steel hoop of weight W and radius r starts from rest at A and rolls down

along a circular cylindrical surface of radius a as shown in Fig. G. Determine the
angle defining the position of point B where the hoop will begin to slip if the
coefficient of friction at the point of contact is  = 1/3

To determine the angle defining the position of point B where the hoop will begin to slip, we need to consider the forces acting on the hoop.

1. Weight (W): The hoop experiences a downward force due to its weight, which acts at its center of mass. The magnitude of the weight is given as W.

2. Normal force (N): The normal force acts perpendicular to the surface of contact and counterbalances the weight. Its magnitude is equal to W.

3. Frictional force (f): The frictional force opposes the motion of the hoop and acts tangentially to the circular cylindrical surface of radius a. Its magnitude is given by f = μN, where μ is the coefficient of friction at the point of contact.

Now, let's analyze the forces acting on the hoop when it reaches the point B and is about to slip:

1. Weight (W): The weight acts vertically downward, passing through the center of mass of the hoop. It does not contribute to the force causing slipping.

2. Normal force (N): The normal force acts perpendicular to the surface of contact. At point B, it acts towards the center of the circular cylindrical surface.

3. Frictional force (f): The frictional force acts tangentially to the circular cylindrical surface at point B. It is directed in the opposite direction of the motion.

For the hoop to slip, the maximum static frictional force should be overcome by the force causing the slipping. The moment the static frictional force reaches its maximum value, the hoop begins to slip.

The maximum static frictional force is given by f_max = μN. In this case, f_max = (1/3)N.

The force causing the slipping is the component of the weight acting tangentially to the surface of the cylindrical surface. It is given by Wsinθ, where θ is the angle defining the position of point B.

For the hoop to begin to slip, the magnitude of the frictional force should be equal to the force causing the slipping. Therefore, we have the equation:

f_max = Wsinθ

Substituting the values for f_max and solving for θ:

(1/3)N = Wsinθ

Since N = W (normal force equals weight), we have:

(1/3)W = Wsinθ

Simplifying the equation:

sinθ = (1/3)

To find the angle θ, we can take the inverse sine (sin^(-1)) of both sides of the equation:

θ = sin^(-1)(1/3)

Therefore, the angle defining the position of point B where the hoop will begin to slip is θ = sin^(-1)(1/3).