Chapter 8, Problem 98P

To determine

Rank the order of the collection of objects.

Answer to Problem 98P

The objects reach the bottom in the following order from cube, solid sphere, solid cylinder, hollow sphere,and hollow cylinder.

Explanation of Solution

Write the equation of conservation of energy.

$K_{\text{i}}+U_{\text{i}}=K_{\text{tr,f}}+K_{\text{rot,f}}+U_{\text{f}}$ (I)

Here, $K_{\text{i}}$ is the initial kinetic energy of the object, $U_{\text{i}}$ is the initial potential energy of the object, $K_{\text{tr,f}}$ is the final translational kinetic energy of the object, $K_{\text{rot,f}}$ is the final rotational kinetic energy of the object, and $U_{\text{f}}$ is the final potential energy of the object.

Write the equation for initial potential energy of the object.

$U_{\text{i}}=mgh$ (II)

Here, $m$ is the mass of the object, $g$ is the gravitational acceleration, and $h$ is the height of the object rolling.

Write the equation for final translational kinetic energy of the object.

$K_{\text{tr,f}}=\frac{1}{2}m{v}^2$ (III)

Here, $v$ is the speed of the object.

Write the equation for final rotational kinetic energy of the object.

$K_{\text{rot,f}}=\frac{1}{2}I{\omega }^2$ (IV)

Here, $I$ is the moment of inertia and $\omega$ is the angular velocity.

Write the equation for moment of inertia of the solid sphere.

$I_{\text{ss}}=\frac{2}{5}m{r}^2$ (V)

Here, $r$ is the radius of the sphere.

Write the relatin between linear velocity and angular velocity.

$\omega =\frac{v}{r}$ (VI)

Write the equation for moment of inertia of the hollow sphere.

$I_{\text{hs}}=\frac{2}{3}m{r}^2$ (VII)

Write the equation for moment of inertia of the solid cylinder.

$I_{\text{sc}}=\frac{1}{2}m{r}^2$ (VIII)

Write the equation for moment of inertia of the hollow cylinder assumed as a thin wall like a hoop.

$I_{\text{hc}}=m{r}^2$ (IX)

Conclusion:

Solid sphere:

Substitute equation (II), (III), (IV), (V) and (VI) in equation (I).

$\begin{array}{c}0+mgh=\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2+0\\ mgh=\frac{1}{2}m{v}^2+\frac{1}{2}\left(\frac{2}{5}m{r}^2\right){\left(\frac{v}{r}\right)}^2\\ mgh=\frac{1}{2}m{v}^2+\frac{1}{5}m{r}^2\left(\frac{v^2}{r^2}\right)\end{array}$

Rewrite the above equation for final speed of the object.

$\begin{array}{c}mgh=\frac{1}{2}m{v}^2+\frac{1}{5}m{v}^2\\ mgh=\frac{7}{10}m{v}^2\\ v=\sqrt{\frac{10gh}{7}}\end{array}$

Hollow sphere:

Substitute equation (II), (III), (IV), (VII) and (VI) in equation (I).

$\begin{array}{c}mgh=\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2\\ mgh=\frac{1}{2}m{v}^2+\frac{1}{2}\left(\frac{2}{3}m{r}^2\right){\left(\frac{v}{r}\right)}^2\\ mgh=\frac{1}{2}m{v}^2+\frac{1}{3}m{v}^2\end{array}$

Rewrite the above equation for final speed of the object.

$\begin{array}{c}mgh=\frac{5}{6}m{v}^2\\ v^2=\frac{6gh}{5}\\ v=\sqrt{\frac{6gh}{5}}\end{array}$

Solid cylinder:

Substitute equation (II), (III), (IV), (VIII) and (VI) in equation (I).

$\begin{array}{c}mgh=\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2\\ mgh=\frac{1}{2}m{v}^2+\frac{1}{2}\left(\frac{1}{2}m{r}^2\right){\left(\frac{v}{r}\right)}^2\\ mgh=\frac{1}{2}m{v}^2+\frac{1}{4}m{v}^2\end{array}$

Rewrite the above equation for final speed of the object.

$\begin{array}{c}mgh=\frac{3}{4}m{v}^2\\ v^2=\frac{4gh}{3}\\ v=\sqrt{\frac{4gh}{3}}\end{array}$

Hollow cylinder:

Substitute equation (II), (III), (IV), (IX) and (VI) in equation (I).

$\begin{array}{c}mgh=\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2\\ mgh=\frac{1}{2}m{v}^2+\frac{1}{2}\left(m{r}^2\right){\left(\frac{v}{r}\right)}^2\\ mgh=\frac{1}{2}m{v}^2+\frac{1}{2}m{v}^2\end{array}$

Rewrite the above equation for final speed of the object.

$\begin{array}{c}mgh=m{v}^2\\ v^2=gh\\ v=\sqrt{gh}\end{array}$

Cube:

Substitute equation (III) and (II) in equation (I).

$\begin{array}{c}\frac{1}{2}m{v}^2=mgh\\ \frac{1}{2}{v}^2=gh\\ v^2=2gh\\ v=\sqrt{2gh}\end{array}$

Since $v_{\text{cube}}>v_{\text{solid sphere}}>v_{\text{solid cylinder}}>v_{\text{hollow sphere}}>v_{\text{hollow cylinder}}$.+

Therefore, the objects reach the bottom in the following order from cube, solid sphere, solid cylinder, hollow sphere,and hollow cylinder.