Question

A regulation basketball has a 29 cm diameter and may be approximated as a thin spherical t shell. How long will it take a basketball starting from rest to roll without slipping 4.5 m down P an incline that makes an angle of 41.7 witht the horizontal? The acceleration of gravity is 9.81 m/s2

016 10.0 points A regulation basketball has a 29 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 4.5 m down an incline that makes an angle of 41.7 with the horizontal? The acceleration of gravity is 9.81 m/s2 Answer in units of s.

Answer 1

Solution:

This question is based on the concepts of:

1. Moment of inertia

2. Conservation of energy

3. Newton’s equations of motion

The detailed solution is explained below:

Let us go to the basics first.

The moment of inertia of the ball is 2/3*mr^2.
The initial height of the ball is h = 4.5*sin(41.7)
The initial potential energy is mgh = mg*4.5*sin(41.7)
The potential energy converts into kinetic energy = 1/2*mv^2 and rotation energy = 1/2*I*ω^2
with ω = v/r.

4.5mgsin41.7 = 1/2*mv^2 + 1/2*(2/3*mr^2)*v^2/r^2
4.5mgsin41.7 = 1/2*mv^2 + 1/3*mv^2
4.5mgsin41.7 = 5/6*mv^2
4.5gsin41.7 = 5/6*v^2
v^2 = 27gsin41.7/5
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find the acceleration:
v^2 = 2as = 2*4.5*a
a = v^2/9
a = 27gsin41.7/(5*9)
a = 3.915 m/s^2
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find the time by
s = 1/2*at^2
t^2 = 2s/a
t^2 = 2*4.5/3.915
t = 1.516 s ---> answer