Chapter 8, Problem 20P

(a)

To determine

To Find: The angular acceleration of the pottery wheel.

Answer to Problem 20P

The value of the acceleration is ${\alpha }_2=0.576{\text{rad/s}}^{\text{2}}$ .

Explanation of Solution

Given information:

The small wheel has a radius of $2.0\text{cm}$ and accelerates at the rate of $7.2{\text{rad/s}}^{\text{2}}$ , and it is in contact with the pottery wheel (radius $25.0\text{cm}$ ) without slipping.

Formula used:

  ${\alpha }_{\tan }=r_1{\alpha }_1=r_2{\alpha }_2$

Here,

  ${\alpha }_{\tan }$ is the tangential acceleration.

  ${\alpha }_1,{\alpha }_2$ is the angular acceleration.

  $r_1,r_2$ is the radius.

Calculation:

The magnitude of tangential component of linear acceleration $=a_{\tan }$

The tangential component of the linear acceleration of each wheel must be the same.

  ${\alpha }_{\tan }=r_1{\alpha }_1=r_2{\alpha }_2$

As they are in contact with the circular edge

For small rubber wheel

  $\begin{array}{l}{a}_{\tan }=\left(0.02\times 7.2\right){\text{rad/s}}^{\text{2}}\\ a_{\tan }=0.144{\text{rad/s}}^{\text{2}}\end{array}$

  $\begin{array}{l}{\alpha }_2=\frac{0.144}{r_2}\\ =\frac{0.144}{0.25}\\ {\alpha }_2=0.576{\text{rad/s}}^{\text{2}}\end{array}$

Conclusion:

The value of the acceleration is ${\alpha }_2=0.576{\text{rad/s}}^{\text{2}}$

(b)

To determine

To Find: The time that is required for the pottery wheel.

Answer to Problem 20P

The time taken by the pottery wheel to reach a speed of $65\text{rpm}$ will be $11.806\text{s}$ .

Explanation of Solution

Given information:

The small wheel has a radius of $2.0\text{cm}$ and accelerates at the rate of $7.2{\text{rad/s}}^{\text{2}}$ , and it is in contact with the pottery wheel (radius $25.0\text{cm}$ ) without slipping.

Formula used:

  $t=\frac{{\omega }_f-{\omega }_i}{\alpha }$

Here, ${\omega }_f$ is the final angular velocity,

  ${\omega }_i$ is the initial angular velocity,

  $\alpha$ is the angular acceleration,

  $t$ is the time.

Calculation:

Time taken by pottery wheel to reach its required speed of $65\text{rpm}$

  $\begin{array}{l}65\text{rev/min}=\frac{65}{60}\text{rev/s}\\ =\frac{\text{65}}{\text{60}}\times 2\pi \text{rad/s}\\ =\text{6}\text{.8}\text{rad/s}\end{array}$

Hence if initial angular velocity ${\omega }_i=0$

And final angular velocity ${\omega }_f=6.8\text{rad/s}$

As the time $t=\frac{{\omega }_f-{\omega }_i}{\alpha }$

Where, $\alpha =0.576{\text{rad/s}}^{\text{2}}$

  $\begin{array}{l}t=\left(\frac{6.8-0}{0.576}\right)\text{s}\\ t=11.806\text{s}\end{array}$

Conclusion:

The time taken by the pottery wheel to reach a speed of $65\text{rpm}$ will be $11.806\text{s}$ .