Question

A hollow, thin-walled sphere of mass 12.0 kg and diameter 0.50 m is rotating about an axle through its center. The angle (in radians) through which it turns as a function of time (in seconds) is given by f(t) = At ² + Bt ⁴, where the magnitude of A =1.50 and B has numerical value 1.10. (a) What are the units of the constants A and B? (b) for time t=4s find (i) the angular momentum of the sphere and (ii) the net torque on the sphere.

Answer 1

a) units of A and B

Units of A = rad /s^2

Units of B = rad /s^4

b) the angular speed is

$\omega = \frac{d \theta}{dt} = 2At + 4 B t^3$

At t = 4 s angular speed is

$\omega = 2A(4)+ 4 B(4)^3 = 291.4 rad/s$

moment of inertia I = (2/3)mr^2 = (2/3)(12)(0.5)^2 = 2 kgm^2

Angular momentum

$L = I \omega = (2)(291.4) = 582.8 kgm^2/s$

b)

taking the derivative of angular speed for angular acceleration

$\alpha = \frac{d\omega}{dt} = 2A + 12 B t^2$

At t = 4 s angular speed is

$\omega = 2A+ 12 B(4)^2 = 214.2 rad/s$

net torque

$\tau = I \alpha = (2) 214.2 rad/s = 428.4 Nm$