Chapter 8, Problem 57A

To determine

The angular acceleration of the propeller.

Answer to Problem 57A

  $-7.54{\text{rad/s}}^{\text{2}}$

Explanation of Solution

Given:

Initial angular velocity of the propeller, ${\omega }_i=475\text{rev/min}$ .

Final angular velocity of the propeller, ${\omega }_f=187\text{rev/min}$ .

Time taken, $t=4.00\text{s}$

Formula used:

The angular acceleration is:

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

Where, ${\omega }_f$ is the final angular velocity, ${\omega }_i$ is the initial angular velocity, and the $\Delta t$ is the time taken.

Calculation:

Initial angular velocity of the propeller is

  $\begin{array}{c}{\omega }_i=475\text{rev/min}\\ \text{= }475\text{rev/min}\times \frac{2\pi \text{rad}}{1\text{rev}}\times \frac{1\text{min}}{60\text{s}}\\ =49.72\text{rad/s}\end{array}$

Final angular velocity of the propeller is

  $\begin{array}{c}{\omega }_f=187\text{rev/min}\\ \text{= }187\text{rev/min}\times \frac{2\pi \text{rad}}{1\text{rev}}\times \frac{1\text{min}}{60\text{s}}\\ =19.57\text{rad/s}\end{array}$

Time taken to stop is, $\Delta t=4.0\text{s}$

Then, the angular acceleration will be,

  $\begin{array}{c}\alpha =\frac{{\omega }_f-{\omega }_i}{\Delta t}\\ =\frac{19.57-49.72}{4}\\ =-7.54{\text{rad/s}}^{\text{2}}\end{array}$

Negative sign shows that the angular velocity decreases with time.

Conclusion:

Thus, angular acceleration of the propeller is $-7.54{\text{rad/s}}^{\text{2}}$ .