Chapter 11, Problem 11.1CYU

Check Your Understanding A hollow cylinder is on an incline at an angle of $\text{60}°$ . The coefficient of static friction on the surface is ${\mu }_s=0.6$ . (a) Does the cylinder roll without slipping? (b) Will a solid cylinder roll without slipping?

To determine

(a)

Whether the cylinder will roll without slipping.

Answer to Problem 11.1CYU

The cylinder will roll with slipping.

Explanation of Solution

Given:

The inclination of hollow cylinder with the horizontal is $\theta =60°$.

The value of static friction of the surface is ${\mu }_{\text{s}}=0.6$.

Formula used:

The expression for the checking the slippage condition is given by,

${\mu }_{\text{s}}>\frac{\tan \theta }{1+\left(\frac{m{r}^2}{I_{\text{cm}}}\right)}$

Here, $m$ is the mass of the cylinder, $r$ is the radius of the cylinder, $\theta$ is the angle of inclination and $I_{\text{cm}}$ is the moment of inertia of the body.

Calculation:

The moment of inertia of the hollow cylinder is $I_{\text{cm}}=m{r}^2$.

The slipping condition is checked as,

$\begin{array}{c}{\mu }_{\text{s}}>\frac{\tan \theta }{1+\left(\frac{m{r}^2}{I_{\text{cm}}}\right)}\\ 0.6>\frac{\tan 60°}{1+\left(\frac{m{r}^2}{m{r}^2}\right)}\\ 0.6>\frac{\tan 60°}{2}\\ 0.6<0.87\end{array}$

Conclusion:

Therefore, as ${\mu }_{\text{s}}$ is less than $\frac{\tan \theta }{1+\left(\frac{m{r}^2}{I_{\text{cm}}}\right)}$, the cylinder will roll with slipping.

To determine

(b)

Whether the solid cylinder will roll without slipping.

Answer to Problem 11.1CYU

The solid cylinder will roll without slipping.

Explanation of Solution

Given:

The inclination of hollow cylinder with the horizontal is $\theta =60°$.

The value of static friction of the surface is ${\mu }_{\text{s}}=0.6$.

Formula used:

The expression for the checking the slippage condition is given by,

${\mu }_{\text{s}}>\frac{\tan \theta }{1+\left(\frac{m{r}^2}{I_{\text{cm}}}\right)}$

Here, $m$ is the mass of the cylinder, $r$ is the radius of the cylinder, $\theta$ is the angle of inclination and $I_{\text{cm}}$ is the moment of inertia of the body..

Calculation:

The moment of inertia of the solid cylinder is $I_{\text{cm}}=\frac{m{r}^2}{2}$.

The slipping condition is checked as,

$\begin{array}{c}{\mu }_{\text{s}}>\frac{\tan \theta }{1+\left(\frac{m{r}^2}{I_{\text{cm}}}\right)}\\ 0.6>\frac{\tan 60°}{1+\left[\frac{m{r}^2}{\left(\frac{m{r}^2}{2}\right)}\right]}\\ 0.6>\frac{\tan 60°}{3}\\ 0.6>0.57\end{array}$

Conclusion:

Therefore, as ${\mu }_{\text{s}}$ is greater than $\frac{\tan \theta }{1+\left(\frac{m{r}^2}{I_{\text{cm}}}\right)}$, the solid cylinder will roll without slipping.