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

Answer 1

Answer: (D)

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