The uniform ball of radius ρ and mass m is gently placed in the bowl B with inner radius R...

The uniform ball of radius ρ and mass m is gently placed in the bowl B with inner radius R and is released. The angle ϕ measures the position of the center of the ball at G with respect to a vertical line, and the angle θ measures the rotation of the ball with respect to a vertical line. Assume that the system lies in the vertical plane. Hint: In working the following problems, we recommend using the rϕ coordinate system shown.

Figure P7.61

Assume that friction is sufficient to prevent slipping.

(a) Derive the equation(s) of motion of the ball in terms of the angle ϕ.

(b) Determine the friction force as a function of ϕ.

(c) Letting ϕ(0) = ϕ0 and = 0, with 0° < ϕ0 < 90° integrate the equation(s) of motion to determine the normal force as a function of ϕ.

(d) Using the results of Parts (b) and (c), given a value for μs, determine the maximum value of ϕ(0) = ϕ0 so that the ball does not slip.