Question

Two spheres are each rotating at an angular speed of 24.6 rad/s about axes that pass through their centers. Each has a radius of 0.440 m and a mass of 1.65 kg. However, one is solid and the other is a thin-walled spherical shell. Suddenly, a net external torque due to friction (magnitude = 0.230 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

a)

for solid sphere

I = moment of inertia = (0.4) m r² = (0.4) (1.65) (0.44)² = 0.128 kgm²

= angular acceleration

Net external torque = 0.230 Nm

Torque is given as

= I

-0.230 = 0.128

= -1.8 rad/s²

w_i = initial angular speed = 24.6 rad/s

w_f = final angular speed = 0 rad/s

t = time taken = ?

Using the equation

= (w_f - w_i )/t

-1.8 = (0 - 24.6)/t

t = 13.7 sec

b)

for hollow sphere :

I = moment of inertia = (0.67) m r² = (0.67) (1.65) (0.44)² = 0.214 kgm²

= angular acceleration

Net external torque = 0.230 Nm

Torque is given as

= I

-0.230 = 0.214

= -1.1 rad/s²

w_i = initial angular speed = 24.6 rad/s

w_f = final angular speed = 0 rad/s

t = time taken = ?

Using the equation

= (w_f - w_i )/t

-1.1 = (0 - 24.6)/t

t = 22.4 sec