Question

A thin-walled, hollow sphere of mass M rolls without slipping down a ramp that is inclined at an angle β to the horizontal. Find the magnitude of the acceleration of the sphere along the ramp. Express your answer in terms of β and acceleration due to gravity g.

Answer 1

Step-by-step explanation:

Given

inclination
`=\beta`

Assuming radius of sphere is r

Now from Free Body Diagram

`Mg\sin \theta -f_r=Ma`

where
`f_r=friction\ force`

`a=acceleration\ of\ system`

Now friction will Provide the Torque

`f_r* r=I\cdot \alpha`

where
`I=moment\ of\ inertia`

`\alpha =angular\ acceleration`

`f_r* r=(2)/(3)Mr^2* (a)/(r)`

in pure rolling
`a=\alpha r`

`f_r=(2)/(3)Ma`

`mg\sin \beta -(2)/(3)Ma=Ma`

`Mg\sin \beta =(5)/(3)Ma`

`a=(3g\sin \beta )/(5)`