Question

A solid sphere of mass ‘m’ and radius ‘R’ is kept on a rough surface. The velocities of air (density) around the sphere area as shown in the figure. Assuming ‘R’ to be small and $m=\frac{4\pi \rho R^2}{g}$ kg, the minimum value of coefficient of friction so that the sphere starts pure rolling is $\frac{1}{\beta }$. (Assuming force due to pressure difference is acting on the centre of mass of the sphere), determine the value of $‘{\beta }^{\prime }.$___

Answer 1

Force due to pressure difference is $F=P\times A$
$=\frac{1}{2}\rho (V_2^2-V_1^2)\times \pi R^2,F=\frac{1}{2}\rho (14-7)\times \pi R^2=\frac{7\pi \rho R^2}{2}$
For translational motion F – f = ma . . . . .(1)
For rotational motion
$f\times R=I\alpha =\frac{Ia}{R}$
$f=\frac{I_a}{R^2}=\frac{2}{5}m{R}^2\times \frac{a}{R^2}=\frac{2}{5}maF=ma+f=ma+\frac{2}{5}ma=\frac{7}{5}ma,ma=\frac{5}{7}Ff=\frac{2}{5}\times \frac{5}{7}F=\frac{2F}{7}=\frac{2}{7}\times \frac{7\pi \rho R^2}{2}f=\pi \rho R^2\mu mg=\pi \rho R^2\mu =\frac{\pi \rho R^2}{mg}=\frac{1}{4}\Rightarrow \mu =\frac{1}{4}=\frac{1}{\beta }\Rightarrow \beta =4$