StudySmarter AI is coming soon!

- :00Days
- :00Hours
- :00Mins
- 00Seconds

A new era for learning is coming soonSign up for free

Suggested languages for you:

Americas

Europe

11 P

Expert-verifiedFound in: Page 335

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 $\alpha =13.7\text{rad}/{\text{s}}^{2}$.

First converting units:

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

(rev/min) to (rad/sec)

$\theta =37\left(rev\right)\left(\frac{2\pi \u200a\text{rad}}{1\u200a\text{rev}}\right)=232.47\u200a\text{rad}$

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 $

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).

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

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

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

Substitute numerical values.

$\begin{array}{rcl}\alpha & =& \frac{2(232.47)-196\left(3\right)}{{3}^{2}-6\left(3\right)}\\ & =& 13.7\u200a\text{rad}/{\text{s}}^{2}\end{array}$

Hence, the answer is $13.7\u200a\text{rad}/{\text{s}}^{2}$.

94% of StudySmarter users get better grades.

Sign up for free