Question 745: Centre of Mass and Rotational Motion

Question

A solid sphere of mass $1 \mathrm{~kg}$ and radius $1 \mathrm{~m}$ rolls without slipping on a fixed inclined plane with an angle of inclination $	heta=30^{\circ}$ from the horizontal. Two forces of magnitude $1 \mathrm{~N}$ each, parallel to the incline, act on the sphere, both at distance $r=0.5 \mathrm{~m}$ from the center of the sphere, as shown in the figure. The acceleration of the sphere down the plane is _________ $m \,s^{-2} .\left(ight.$ Take $g=10\, m s^{-2}$)

Accepted Answer

Taking torque about contact point.
$\vec{	au}=m g R \sin 30^{\otimes}+1 	imes 1^{\odot}$
taking $\otimes$ is positive
$$
\begin{aligned}
& 5-1=\frac{7}{5} m R^2 \alpha \\
& \Rightarrow \quad \alpha=\frac{20}{7} \mathrm{rad} / \mathrm{s}^2
\end{aligned}
$$
So, $a_{\mathrm{cm}}=\alpha R=\frac{20}{7} \mathrm{~m} / \mathrm{s}^2$
$$
a_{\mathrm{cm}}=2.86 \mathrm{~m} / \mathrm{s}^2
$$

Centre of Mass and Rotational Motion

Integer / Numeric Answer Type