Question

A solid sphere of a 0.72 m diameter can be rotated about an axis through its center by a torque, which accelerates it uniformly from rest through a total of 160 revolutions in 15.0 s. What is the mass of the sphere?

Short answer

The mass of the solid sphere is \(23.33\;{\rm{kg}}\).

Step-by-step solution

Concepts

The torque is the product of the moment of inertia and the square of the distance.For this problem, first, find the angular acceleration of the solid sphere and then find the torque.

Given data

The initial angular velocity of the solid sphere is \({\omega _1} = 0\).

The diameter of the solid sphere is \(d = 0.72\;{\rm{m}}\).

The time taken by the solid sphere is \(t = 15.0\;{\rm{s}}\).

The torque on the sphere is \(\tau = 10.8\;{\rm{m}} \cdot {\rm{N}}\).

The angular displacement is \(\theta = 160\;{\rm{revolution}} = \left( {160 \times 2\pi } \right)\;{\rm{rad}}\).

Let m be the mass of the sphere and \(\alpha \) be the angular acceleration of the sphere.

Calculation of torque

The radius of the solid sphere is \(r = \frac{d}{2}\).

The moment of inertia of the sphere is

\(\begin{align}I &= \frac{2}{5}m{r^2}\\ &= \frac{2}{5}m{\left( {\frac{d}{2}} \right)^2}\end{align}\)

Now, for the rotational motion of the sphere,

\(\begin{align}\theta &= {\omega _1}t + \frac{1}{2}\alpha {t^2}\\\theta &= \frac{1}{2}\alpha {t^2}\;{\rm{as}}\;{\omega _1} &= 0\\\alpha &= \frac{{2\theta }}{{{t^2}}}\end{align}\).

Now, the torque is

\(\begin{align}\tau &= I\alpha \\\tau &= \frac{2}{5}m{\left( {\frac{d}{2}} \right)^2}\frac{{2\theta }}{{{t^2}}}\\\tau &= m \times \frac{2}{5}\frac{{{d^2}}}{4}\frac{{2\theta }}{{{t^2}}}\\m &= \frac{{5\tau {t^2}}}{{{d^2}\theta }}\end{align}\)

Now, substituting the values in the above equation,

\(\begin{align}m &= \frac{{5 \times \left( {10.8\;{\rm{m}} \cdot {\rm{N}}} \right) \times {{\left( {15.0\;{\rm{s}}} \right)}^2}}}{{{{\left( {0.72\;{\rm{m}}} \right)}^2} \times \left( {160 \times 2\pi } \right)\;{\rm{rad}}}}\\ &= 23.33\;{\rm{kg}}\end{align}\).

Hence, the mass of the solid sphere is \(23.33\;{\rm{kg}}\).