Chapter 10, Problem 10.1CYU

Check Your Understanding The fan blades on a turbofan jet engine (shown below) accelerate from rest up to a rotation rate of 40.0 rev/s in 20 s. The increase in angular velocity of the fan is constant in time. (The GE90-11OB1 turbofan engine mounted on a Boeing 777, as shown, is currently the largest turbofan engine in the world, capable of thrusts of 330-510 kN.)

(a) What is the average angular acceleration?

(b) What is the instantaneous angular acceleration at any time during the first 20 s?

To determine

(a)

The average angular acceleration of fan blades.

Answer to Problem 10.1CYU

The average angular acceleration of fan blades is $4.0\pi \text{rad}/{\text{s}}^2$.

Explanation of Solution

Given:

The rate of rotation of wheel is, $\Delta \omega =40.0\text{rev}/\text{s}$.

The time taken to achieve the rotation rate is, $\Delta t=20\text{ s}$.

Formula used:

The average angular acceleration is given by,

$\overline{\alpha }=\frac{\Delta \omega }{\Delta t}$

Calculation:

The average angular acceleration is calculated as,

$\begin{array}{c}\overline{\alpha }=\frac{\Delta \omega }{\Delta t}\\ =\frac{\left(40.0\text{rev}/\text{s}\times \frac{\text{2}\pi \text{rad}/\text{s}}{1\text{rev}/\text{s}}\right)}{20.0\text{ s}}\\ =\frac{80.0\pi \text{rad}/\text{s}}{20.0\text{ s}}\\ =4.0\pi \text{rad}/{\text{s}}^2\end{array}$

Conclusion:

The average angular acceleration of fan blades is $4.0\pi \text{rad}/{\text{s}}^2$.

To determine

(b)

The instantaneous angular acceleration at any time during the first $20\text{ s}$.

Answer to Problem 10.1CYU

The instantaneous angular acceleration at any time during the first $20\text{ s}$ is $4.0\pi \text{rad}/{\text{s}}^2$.

Explanation of Solution

Formula used:

The instantaneous angular acceleration when the increase in the angular velocity is constant is given by,

$\alpha =\overline{\alpha }$

Calculation:

The instantaneous angular acceleration is calculated as,

$\begin{array}{c}\alpha =\overline{\alpha }\\ =4.0\pi \text{rad}/{\text{s}}^2\end{array}$

Conclusion:

The instantaneous angular acceleration at any time during the first $20\text{ s}$ is $4.0\pi \text{rad}/{\text{s}}^2$.