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Expert-verified Found in: Page 335 ### Physics For Scientists & Engineers

Book edition 9th Edition
Author(s) Raymond A. Serway, John W. Jewett
Pages 1624 pages
ISBN 9781133947271 # Question: A rotating wheel requires rotating through 37.0 revolutions. Its angular speed at the end of the interval is. What is the constant angular acceleration of the wheel?

The solution of the constant angular acceleration is $\alpha =13.7\text{rad}/{\text{s}}^{2}$.

See the step by step solution

## Step 1: Converting the given units and deriving angular speed

First converting units:

Multiply the angular speed $\left({\omega }_{i}\right)$ by a conversion factor to convert its units from

$\theta =37\left(rev\right)\left(\frac{2\pi \text{rad}}{1 \text{rev}}\right)=232.47 \text{rad}$

Second, solving the problem:

Model the wheel as a rigid object under constant angular acceleration and use the following equation to find the initial angular speed.

${\omega }_{f}={\omega }_{i}+\alpha t$

Solve for ${\omega }_{i}$

${\omega }_{i}={\omega }_{f}-\alpha t$

Substitute the known numerical values.

${\omega }_{i}=98-3\alpha$

## Step 2: Deriving the equation and finding angular acceleration

Use the following equation to find the angular acceleration of the wheel:

$\theta ={\omega }_{i}t+\frac{1}{2}\alpha {t}^{2}$

Solve for $\left(\alpha \right)$.

$\alpha =\frac{2\left(\theta -{\omega }_{i}t\right)}{{t}^{2}}$

Substitute for $\left({\omega }_{i}\right)$ from Equation (1) in Equation (2).

$\begin{array}{rcl}\alpha & =& \frac{2\left[\theta -\left(98-3\alpha t\right)\right]}{{t}^{2}}\\ \alpha & =& \frac{2\theta -196t+6\alpha t}{{t}^{2}}\\ \alpha {t}^{2}& =& 2\theta -196t+6\alpha t\\ \alpha \left({t}^{2}-6t\right)& =& 2\theta -196t\end{array}$

Solve for $\left(\alpha \right)$

$\alpha =\frac{2\theta -196t}{{t}^{2}-6{t}^{2}}$

Substitute numerical values.

$\begin{array}{rcl}\alpha & =& \frac{2\left(232.47\right)-196\left(3\right)}{{3}^{2}-6\left(3\right)}\\ & =& 13.7 \text{rad}/{\text{s}}^{2}\end{array}$

Hence, the answer is $13.7 \text{rad}/{\text{s}}^{2}$. ### Want to see more solutions like these? 