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

Answer:

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

$f_r\times r=I\cdot \alpha$

where $I=moment\ of\ inertia$

$\alpha =angular\ acceleration$

$f_r\times r=\frac{2}{3}Mr^2\times \frac{a}{r}$

in pure rolling $a=\alpha r$

$f_r=\frac{2}{3}Ma$

$mg\sin \beta -\frac{2}{3}Ma=Ma$

$Mg\sin \beta =\frac{5}{3}Ma$

$a=\frac{3g\sin \beta }{5}$