Question

A solid sphere is set into motion on a rough horizontal surface with a linear speed ν in the forward direction and an angular speed ν/R in the anticlockwise directions as shown in figure (10-E16). Find the linear speed of the sphere (a) when it stops rotating and (b) when slipping finally ceases and pure rolling starts.
Figure

Answer 1

(a)The linear speed of the sphere when it stops rotating is $\frac{3v}{5}$

(b)The linear speed when slipping finally ceases and pure rolling starts is $\frac{3v}{7}$

Explanation:

Given Data

A solid sphere is in motion with speed = v

Angular speed of the solid sphere = $\frac{v}{R}$

Find the Linear sphere of the sphere (a) when it stops rotating (b) when slipping finally ceases and pure rolling

$m v_{0} R+I \omega_{0}=m V R+I \omega$

$m v R-\frac{2}{5} m R^{2} \frac{v}{R}=m V R+\frac{2}{5} m R^{2} \frac{V}{R}$

$\frac{3}{5} m v R=\frac{7}{5} m V R$

Eliminate 'R' and 'm' on both sides of the above relation, we get

$\frac{3}{5} v =\frac{7}{5} V$

In the above relation Initial linear speed is equate with the final linear speed.

Where the linear speed of the sphere when it stops rotating is $\frac{3v}{5}$ and the linear speed when slipping finally ceases and pure rolling starts is  $\frac{3v}{7}$