Answer to Question #101398 in Physics for Nyx

Question

A regulation basketball has a 25 cm diameter
and may be approximated as a thin spherical
shell.
How long will it take a basketball starting
from rest to roll without slipping 4.6 m down
an incline that makes an angle of 52.6◦ with the horizontal? The acceleration of gravity is
9.81 m/s2. Answer in units of s. Don't round answer.

Answer 1

The moment of inertia of the ball is

$$I=\frac{2}{5}mr^2$$

The initial height of the ball is

$$h=4.6\sin{52.6\degree}$$

The initial potential energy is

$$mgh=mg(4.6\sin{52.6\degree})$$

From the conservation of energy:

$$mgh=0.5mv^2+0.5I\omega^2\to mgh=0.5mv^2+0.5(0.4mr^2)\frac{v^2}{r^2}$$

$$gh=0.5v^2+0.5(0.4v^2)=0.7v^2$$

$$(9.81)(4.6\sin{52.6\degree})=0.7v^2$$

$$v=7.156\frac{m}{s}$$

$$t=\frac{2s}{v}=\frac{2(4.6)}{7.156}=1.29\ s$$