Chapter 9, Problem 19SP

Convert (a) 50.0 rev to radians, (b) 48π rad to revolutions, (c) 72.0 rps to rad/s, (d) 1.50 × 103 rpm to rad/s, (e) 22.0 rad/s to rpm, (f) 2.000 rad/s to deg/s.

To determine

The value of $50.0\text{ rev}$ in radians.

Answer to Problem 19SP

Solution: $\text{314 radians}$

Explanation of Solution

Given data:

The number of revolutions is $50.0\text{ rev}$.

Formula used:

The number of radians corresponding to the revolution of a disk is,

$\theta =2\pi n$

Here, $n$ is the number of revolutions and $\theta$ is the number of radians corresponding to $n$ revolutions.

Explanation:

Consider the expression for number of radians covered.

$\theta =2\pi n$

Substitute $50$ for $n$.

$\begin{array}{c}\theta =2\pi \left(50\right)\\ =100\pi \\ =\text{314 radians}\end{array}$

Conclusion:

The value of $50.0\text{ rev}$ corresponds to $\text{314 radians}$.

To determine

The value of $48\pi \text{ radians}$ in revolutions.

Answer to Problem 19SP

Solution: $24\text{ revolutions}$

Explanation of Solution

Given data:

The value in radians is $48\pi \text{ radians}$.

Formula used:

The number of radians corresponding to revolutions is,

$\theta =2\pi n$

Here, $n$ is the number of revolutions and $\theta$ is the number of radians corresponding to $n$ revolutions.

Explanation:

Consider the expression for number of radians covered

$\theta =2\pi n$

Substitute $48\pi \text{ radians}$ for $\theta$

$\begin{array}{c}48\pi \text{ radians}=\left(2\pi \text{ radians/revolution}\right)n\\ 2\pi n=48\pi \\ n=\frac{48\pi }{2\pi }\\ n=24\text{ revolutions}\end{array}$

Conclusion:

The value of $48\pi \text{ radians}$ corresponds to $24\text{ revolutions}$.

To determine

The value of $72\text{ rps}$ in rad/s.

Answer to Problem 19SP

Solution: $452\text{ rad/s}$

Explanation of Solution

Given data:

The value in revolutions per second is $72\text{ rps}$.

Formula used:

The number of radians per second corresponding to revolutions per second is expressed as,

$\omega =2\pi N$

Here, $N$ is the revolutions per second and $\omega$ is the angular velocity in radians per second corresponding to $N$ revolutions per second.

Explanation:

Consider the expression for angular velocity.

$\omega =2\pi N$

Substitute $72.0\text{ rps}$ for $N$.

$\begin{array}{c}\omega =2\pi \left(\frac{\text{rad}}{\text{revolution}}\right)\left(72.0\frac{\text{revolution}}{\text{second}}\right)\\ =144\pi \text{ rad/s}\\ =452\text{ rad/s}\end{array}$

Conclusion:

The value of $72.0\text{ rps}$ corresponds to $452\text{ rad/s}$.

To determine

The value of $1.50\times {10}^3\text{ rpm}$ in rad/s.

Answer to Problem 19SP

Solution: $157\text{ rad/s}$

Explanation of Solution

Given data:

The value in revolutions per minute is $1.50\times {10}^3\text{ rpm}$.

Formula used:

The number of radians per minute corresponding to revolutions per minute is expressed as,

$\omega =2\pi N^{\prime }$

Here, $N^{\prime }$ is the revolutions per minute and $\omega$ is the angular velocity in radians per minute corresponding to $N^{\prime }$ revolutions per minute.

Explanation:

Consider the expression for angular velocity.

$\omega =2\pi N^{\prime }$

Substitute $1.50\times {10}^3\text{ rpm}$ for $N^{\prime }$.

$\begin{array}{c}\omega =2\pi \left(\frac{\text{rad}}{\text{revolution}}\right)\left(1.50\times {10}^3\frac{\text{revolution}}{\text{minute}}\right)\\ =3000\pi \text{ rad/min}\\ =3000\pi \left(\frac{\text{rad}}{\text{min}}\right)\left(\frac{1\text{ min}}{60\text{s}}\right)\\ =50\pi \text{ rad/s}\end{array}$

Further solve,

$\omega =157\text{ rad/s}$

Conclusion:

The value of $1.50\times {10}^3\text{ rpm}$ corresponds to $157\text{ rad/s}$.

To determine

The value of $22.0\text{ rad/s}$ in rpm.

Answer to Problem 19SP

Solution: $\text{210 rpm}$

Explanation of Solution

Given data:

The value in rad/s is $22.0\text{ rad/s}$.

Formula used:

The number of radians per second corresponding to revolutions per second is expressed as,

$\omega =2\pi N$

Here, $N$ is the revolutions per second and $\omega$ is the angular velocity in radians per second corresponding to $N$ revolutions per second.

Explanation:

Consider the expression for angular velocity.

$\omega =2\pi N$

Substitute $22.0\text{ rad/s}$ for $N$.

$\begin{array}{c}22.0\text{ rad/s}=2\pi N\\ 2\pi N=22\\ N=\frac{22}{2\pi }\\ N=\frac{11}{\pi }\text{ revolutions/second}\end{array}$

Further solve,

$\begin{array}{c}N=\left(\frac{11}{\pi }\text{ revolutions/second}\right)\left(\frac{60\text{ second}}{1\text{ minute}}\right)\\ =\frac{11}{\pi }\left(60\right)\\ =\frac{660}{\pi }\\ \approx \text{210 rpm}\end{array}$

Conclusion:

The value of $22.0\text{ rad/s}$ corresponds to $\text{210 rpm}$.

To determine

The value of $2.000\text{ rad/s}$ in $\text{deg/s}$.

Answer to Problem 19SP

Solution: $114.6\text{ deg/s}$

Explanation of Solution

Given data:

The value in rad/s is $2.000\text{ rad/s}$.

Formula used:

The number of radians per second corresponding to degree per second is expressed as,

${\omega }_D={\omega }_R\left(\frac{360}{2\pi }\right)$

Here, ${\omega }_R$ is the angular velocity in radians per second and ${\omega }_D$ is the angular velocity in degree per second.

Explanation:

Consider the expression for angular velocity in degree per second

${\omega }_D={\omega }_R\left(\frac{360}{2\pi }\right)$

Substitute $2.000\text{ rad/s}$ for ${\omega }_R$

$\begin{array}{l}{\omega }_D=\left(2.000\text{ rad/s}\right)\left(\frac{360}{2\pi }\right)\left(\frac{\text{deg}}{\text{rad}}\right)\\ {\omega }_D=114.6\text{ deg/s}\end{array}$

Conclusion:

The value of $2.000\text{ rad/s}$ corresponds to $114.6\text{ deg/s}$.