Question

Two spheres are each rotating at an angular speed of 23.7 rad/s about axes that pass through their centers. Each has a radius of 0.360 m and a mass of 1.68 kg. However, as the figure shows, one is solid and the other is a thin-walled spherical shell. Suddenly, a net external torque due to friction (magnitude = 0.400 N · m) begins to act on each sphere and slows the motion down. How long does it take (a) the solid sphere and (b) the thin-walled sphere to come to a halt?

Answer 1

here,

a)

for solid sphere

the initial angular speed , w0 = 23.7 rad/s

radius , r = 0.36 m

mass ,m = 1.68 kg

the net external torque , T = 0.4 N.m

the magnitude of angular acceleration , alpha = T /I

alpha = T /(2/5 * m * r^2)

alpha = 0.4 /(0.4 * 1.68 * 0.36^2) rad/s^2

alpha = 4.59 rad/s^2

the time taken to stop , t = w0 /alpha

t = 23.7 /4.59 s = 5.16 s

the time taken is 5.16 s

b)

for Hollow sphere

the initial angular speed , w0 = 23.7 rad/s

radius , r = 0.36 m

mass ,m = 1.68 kg

the net external torque , T = 0.4 N.m

the magnitude of angular acceleration , alpha = T /I

alpha = T /(2/3 * m * r^2)

alpha = 0.4 /(0.67 * 1.68 * 0.36^2) rad/s^2

alpha = 2.74 rad/s^2

the time taken to stop , t = w0 /alpha

t = 23.7 /2.74 s = 8.64 s

the time taken is 8.64 s