Answer to Question #102133 in Mechanics | Relativity for Jayme Johnston

Question

A regulation basketball has a 50 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 1.9 m down
an incline that makes an angle of 77.0
◦ with
the horizontal? The acceleration of gravity is
9.81 m/s
2
.
Answer in units of s.

Answer 1

The moment of inertia of the ball is

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

The initial height of the ball is

$$h=1.9\sin{77}$$

The initial potential energy is

$$mgh=mg(1.9\sin{77})$$

From the conservation of energy:

$$mgh=0.5mv^2 +0.5Iω^2 →mgh=0.5mv^2 +0.5(0.4mr^2 )$$

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

$$9.81(1.9\sin{77})=0.7v^2$$

$$v=5.094\frac{m}{s}$$

$$t=\frac{2s}{v}=\frac{2(1.9)}{5.094}=0.746\ s$$