Question

A plate of mass $m$ is placed on a solid and hollow spheres each of mass $m$ . If the speed of the plate is $v$ , assuming pure rolling of the spheres with all contacting surfaces, the kinetic energy of the system (spheres $+$ plate) will be,

• A. $\frac{43}{50}m{v}^2$
• B. $\frac{33}{50}m{v}^2$
• C. $\frac{43}{60}m{v}^2$
• D. $\frac{53}{60}m{v}^2$

Answer 1

Answer: (D)

The total $KE$ is $K=K_{rod}+K_{hollowsphere}+K_{solidsphere}$
Where $K_{rod}=\frac{1}{2}m{v}^2$
Since the $CM$ of each sphere moves with a velocity
$v_c=\frac{v^{\prime }}{2}$
$K_{hollowsphere}=\frac{1}{2}m{v}_C^2\left(1+\frac{K^2}{R^2}\right)=\frac{5m{V}^2}{24}$
$K_{solidsphere}=\frac{1}{2}m{\left(\frac{v}{2}\right)}^2\left(1+\frac{2}{5}\right)=\frac{7}{40}{\left(mv\right)}^2$
Using the above four equations,
$K=\frac{1}{2}m{v}^2+\frac{5}{24}m{v}^2+\frac{7}{40}m{v}^2=\frac{53}{60}m{v}^2$