Question

A 500g uniform sphere of 7.0cm radius spins at 30 rev/s on an axis through it's centre. Find it's, a) rotational kinetic energy, b)angular momentum, C) radius of gyration

Answer 1

Given:

$m=0.5\:\rm kg$

$n=30\:\rm rev/s$

$R=0.07\:\rm m$

(a)

$$E_k=\frac{I\omega^2}{2}=\frac{2/5mR^2*(2\pi n)^2}{2}=4/5\pi^2mR^2n^2$$

$$=4/5\pi^2*0.5*0.07^2*30^2=17.4\:\rm J$$

(b)

$$L=I\omega=2/5mR^2*2\pi n=4/5\pi mR^2n$$

$$=4/5\pi*0.5*0.07^2*30=0.37\:\rm kg\cdot m^2/s$$

(c)

$$r=\sqrt{\frac{I}{m}}=\sqrt{\frac{2R^2}{5}}$$

$$=\sqrt{2/5}*0.07=0.044\:\rm m=4.4\:\rm cm$$