Chapter 10, Problem 10.1QQ

A rigid object rotates in a counterclockwise sense around a fixed axis. Each of the following pairs of quantities represents an initial angular position and a final angular position of the rigid object. (i) Which of the sets can only occur if the rigid object rotates through more than 180°? (a) 3 rad, 6 rad (b) −1 rad, 1 rad (c) 1 rad, 5 rad (ii) Suppose the change in angular position for each of these pairs of values occurs in 1 s. Which choice represents the lowest average angular speed?

To determine

The pair of given quantities from the given sets that can only occur if the objects rotates through more than $180°$.

Answer to Problem 10.1QQ

Option (c) $1\text{rad}$, $5\text{rad}$.

Explanation of Solution

For option (a), the initial and final angular positions of the object are $3\text{rad}$ and $6\text{rad}$.

Formula to calculate the change in position of the object due to rotation is,

    $\Delta {\theta }_1={\theta }_{\text{f}}{}_1-{\theta }_{\text{i1}}$

Here, $\Delta {\theta }_1$ is the change in position of the object due to rotation for set $1$ quantities, ${\theta }_{\text{f1}}$ is the final angular position of the object for set $1$ quantities, and ${\theta }_{\text{i1}}$ is the initial angular position of the object for set $1$ quantities.

Substitute $3\text{rad}$ for ${\theta }_{\text{i1}}$ and $6\text{rad}$ for ${\theta }_{\text{f1}}$ in above equation.

    $\begin{array}{c}\Delta {\theta }_1=\left(6\text{rad}-3\text{rad}\right)\times \left(\frac{180°}{\pi \text{rad}}\right)\\ =171.88°\end{array}$

Thus, for set $1$ the object does not rotate more than $180°$.

For option (b), the initial and final angular positions of the object are $-1\text{rad}$ and $1\text{rad}$.

Formula to calculate the change in position of the object due to rotation is,

    $\Delta {\theta }_2={\theta }_{\text{f}}{}_2-{\theta }_{\text{i2}}$

Here, $\Delta {\theta }_2$ is the change in position of the object due to rotation for set $2$ quantities, ${\theta }_{\text{f2}}$ is the final angular position of the object for set $2$ quantities and ${\theta }_{\text{i2}}$ is the initial angular position of the object for set $2$ quantities.

Substitute $-1\text{rad}$ for ${\theta }_{\text{i2}}$ and $1\text{rad}$ for ${\theta }_{\text{f2}}$ in above equation.

    $\begin{array}{c}\Delta {\theta }_2=\left(1\text{rad}-\left(-1\text{rad}\right)\right)\times \frac{180°}{\pi \text{rad}}\\ =114.59°\end{array}$

Thus, for set $2$ the object does not rotate more than $180°$.

For option (c), the initial and final angular positions of the object are $1\text{rad}$ and $5\text{rad}$.

Formula to calculate the change in position of the object due to rotation is,

    $\Delta {\theta }_3={\theta }_{\text{f}}{}_3-{\theta }_{\text{i3}}$

Here, $\Delta {\theta }_3$ is the change in position of the object due to rotation for set $3$ quantities, ${\theta }_{\text{f3}}$ is the final angular position of the object for set $3$ quantities and ${\theta }_{\text{i3}}$ is the initial angular position of the object for set $3$ quantities.

Substitute $1\text{rad}$ for ${\theta }_{\text{i3}}$ and $5\text{rad}$ for ${\theta }_{\text{f3}}$ in above equation.

    $\begin{array}{c}\Delta {\theta }_3=\left(5\text{rad}-1\text{rad}\right)\times \frac{180°}{\pi \text{rad}}\\ =229.18°\end{array}$

Thus, for set $3$ the object rotate more than $180°$.

Conclusion:

The object rotates more than $180°$ for which the initial and final angular speed are $1\text{rad}$ and $5\text{rad}$ and the option (c) is $1\text{rad}$, $5\text{rad}$. Thus option (c) is correct.

The object rotates more than $180°$ for which the initial and final angular speed are $1\text{rad}$ and $5\text{rad}$ but the option (a) is $3\text{rad}$, $6\text{rad}$. Thus option (a) is incorrect.

The object rotates more than $180°$ for which the initial and final angular speed are $1\text{rad}$ and $5\text{rad}$ but the option (b) is $-1\text{rad}$, $1\text{rad}$. Thus option (b) is incorrect.

To determine

The pair of given quantities that represents the lowest average angular speed.

Answer to Problem 10.1QQ

Option (b) $-1\text{rad}$, $1\text{rad}$.

Explanation of Solution

The objects rotate in counterclockwise direction.

For option (a), the initial and final angular positions of the object are $3\text{rad}$ and $6\text{rad}$.

Formula to calculate the average angular speed of the object due to rotation is,

    ${\omega }_1=\frac{\left(\frac{{\theta }_{\text{f}}{}_1+{\theta }_{\text{i1}}}{\Delta t}\right)}{2}$

Substitute $3\text{rad}$ for ${\theta }_{\text{i1}}$ and $6\text{rad}$ for ${\theta }_{\text{f1}}$ in above equation.

    $\begin{array}{c}{\omega }_1=\left(\frac{\frac{6\text{rad}+3\text{rad}}{1\text{s}}}{2}\right)\\ =4.5\text{rad}/\text{s}\end{array}$

Thus, for set $1$ the average angular speed of the object is $4.5\text{rad}/\text{s}$.

For option (b), the initial and final angular positions of the object are $-1\text{rad}$ and $1\text{rad}$.

Formula to calculate the average angular speed of the object due to rotation is,

    ${\omega }_2=\frac{\left(\frac{{\theta }_{\text{f}}{}_2+{\theta }_{\text{i2}}}{\Delta t}\right)}{2}$

Substitute $-1\text{rad}$ for ${\theta }_{\text{i2}}$ and $1\text{rad}$ for ${\theta }_{\text{f2}}$ in above equation.

    $\begin{array}{c}\Delta {\theta }_2=\left[\frac{\frac{1\text{rad}+\left(-1\text{rad}\right)}{1\text{s}}}{2}\right]\\ =0\end{array}$

Thus, for set $2$ the average angular speed of the object is $0$.

For option (c), the initial and final angular positions of the object are $1\text{rad}$ and $5\text{rad}$.

Formula to calculate the average angular speed of the object due to rotation is,

    ${\omega }_3=\frac{\left(\frac{{\theta }_{\text{f}}{}_3+{\theta }_{\text{i3}}}{\Delta t}\right)}{2}$

Substitute $1\text{rad}$ for ${\theta }_{\text{i3}}$ and $5\text{rad}$ for ${\theta }_{\text{f3}}$ in above equation.

    $\begin{array}{c}\Delta {\theta }_3=\left[\frac{\frac{5\text{rad}+1\text{rad}}{1\text{s}}}{2}\right]\\ =3\text{rad}/\text{s}\end{array}$

Thus, for set $3$ the average angular speed of the object is $3\text{rad}/\text{s}$.

Conclusion:

The lowest average angular speed of the object is $0$ for which the initial and final angular speed are $-1\text{rad}$ and $1\text{rad}$ but the option (b) is $-1\text{rad}$, $1\text{rad}$. Thus option (b) is correct.

The lowest average angular speed of the object is $0$ for which the initial and final angular speed are $-1\text{rad}$ and $1\text{rad}$ but the option (a) is $3\text{rad}$, $6\text{rad}$. Thus option (a) is incorrect.

The lowest average angular speed of the object is $0$ for which the initial and final angular speed are $-1\text{rad}$ and $1\text{rad}$ but the option (c) is $1\text{rad}$, $5\text{rad}$. Thus option (c) is incorrect.