Question

Problem 4. A solid sphere of mass m and radius rrolls without slipping along the track shown below. It starts from rest with the lowest point of the sphere at height 3R above the bottom of the loop of radius R, much larger than r. Point P is on the track and it is R above the bottom of the loop. The moment of inertia of the ball about an axis through its center is I-2/5 mr. The ball should not be treated as a point mass. For the following parts, you can express your answers in terms of m, g, r, and R. Find (a) the center of mass speed of the ball when it is at point P (b) the angular speed of the ball when it is at point P; (c) the angular acceleration of the ball when it is at point P; (d) the tangential acceleration ofthe ball when it is at point P; (e) the static frictional force acting on the ball when it is at point P Solid sphere of mass m and radius <<R.

Answer 1

Using conservation of Gauss initial energy = Final energy I_2 mv^2 + mgh = 1 /2 m v^2_cm + 1 /2 I_c W^2 + m gh 3 mgh = 1/2 m v^2_cm + V/2 mv^2 + n SR 2 mg R = 1/2 mv^2_cm + 1 /5 m v^2 times v^2_cm/r^2 2 mg H = in x^2_m [1 /2 + 1/5] 2 g h = V^2_cm [7/10] V^2_cm = 20 gR/7 V_cm = squareroot 20 g R/7 W = V- cm/r = squareroot 20 g R/7 1/r W = squareroot 20 g h/7 r^2_cm [1 /2 + 1/5] 2 g h = V^2_cm [7/10] V^2_cm = 20 g R/7