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

Question #102133
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.
1
Expert's answer
2020-02-03T13:59:37-0500

The moment of inertia of the ball is


I=25mr2I=\frac{2}{5}mr^2

The initial height of the ball is


h=1.9sin77h=1.9\sin{77}

The initial potential energy is


mgh=mg(1.9sin77)mgh=mg(1.9\sin{77})

From the conservation of energy:


mgh=0.5mv2+0.5Iω2mgh=0.5mv2+0.5(0.4mr2)mgh=0.5mv^2 +0.5Iω^2 →mgh=0.5mv^2 +0.5(0.4mr^2 )

gh=0.5v2+0.5(0.4v2)=0.7v2gh=0.5v ^ 2 +0.5(0.4v^ 2 )=0.7v^ 2

9.81(1.9sin77)=0.7v29.81(1.9\sin{77})=0.7v^2

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

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


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