A spherical shell of mass M radius R and a ball of Mass M ra

A spherical shell of mass M radius R) and a ball of Mass M (radius R) start from rest at a height of 0.5 m and rolls down simultaneously a 20 degree slope as he Fig. above. Who reach to bottom first? Who reach to bottom second? (I = 2/5 MR^2 for ball; I = 2/3 MR^2 for spherical shell); who rotate fast? Who rotate slow? v_shell = ____ v_ball = _____ omega_shell = _____ omega_ball = ______

Solution

Appplying energy conservation,

m g h = m v^2 /2 + I w^2 /2

for rolling without slipping,

w= v /R

for shell:

m g h = m v^2 /2 + m v^2 / 3

v = sqrt(6 g h / 5)

for Ball : m g h = m v^2 /2 + m v^2 / 5

v = sqrt(10 g h / 7 )

putting h = 0.5 m

(A) v_shell = 2.43 m/s

(B) v_ball = 2.65 m/s

(C) w_shell = V_shell / R = 24.3 rad/s

(D) w_ball = 26.5 rad/s