Chapter 8, Problem 19P

(a)

To determine

To find: The fan’s angular acceleration.

Answer to Problem 19P

The value of angular acceleration is $-0.42{\text{rad/s}}^{\text{2}}$ .

Explanation of Solution

Given information:

A cooling fan is turned off when it is running at $850\text{rev/min}$ . It turns $1500\text{revolutions}$ before it comes to a shop.

Formula used:

  ${\omega }_f^2={\omega }_0^2+2\alpha \theta$

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

  ${\omega }_0$ is the initial angular velocity,

  $\alpha$ is the angular acceleration,

  $\theta$ is the angular displacement.

Calculation:

Frequency of revolution of ceiling fan is $f=850\text{rev}$

  $\begin{array}{l}=\left(\frac{850}{60}\right)\text{rev/s}\\ =\frac{85}{6}\text{rev/s}\end{array}$

Angle described by the fan $\theta =1500\times 2\pi$

Final angular velocity ${\omega }_f=0$

Angular acceleration $\alpha$ is determined using

  $\begin{array}{l}{\omega }_f^2={\omega }_0^2+2\alpha \theta \\ \alpha =\left(\frac{{\omega }_f^2-{\omega }_0^2}{2\theta }\right)\\ =\left(\frac{0-{\left(2\pi \times \frac{85}{6}\right)}^2}{2\times 2\pi \times 1500}\right){\text{rad/s}}^{\text{2}}\\ =\left(-\frac{4{\pi }^2{\left(\frac{85}{6}\right)}^2}{\left(4\pi \times 1500\right)}\right){\text{rad/s}}^{\text{2}}\\ \alpha =\frac{-4{\pi }^2{\left(\frac{85}{6}\right)}^2}{4\pi \times 1500}\\ \alpha =-0.42{\text{rad/s}}^{\text{2}}\end{array}$

Angular acceleration $-0.42{\text{rad/s}}^{\text{2}}$

Conclusion:

The value of angular acceleration is $-0.42{\text{rad/s}}^{\text{2}}$ .

(b)

To determine

To find: The time that will take the fan to come to stop.

Answer to Problem 19P

The value is $t=212\text{s}$ .

Explanation of Solution

Given information:

A cooling fan is turned off when it is running at $850\text{rev/min}$ . It turns $1500\text{revolutions}$ before it comes to a shop.

Formula used:

  ${\omega }_f={\omega }_0+\alpha t$

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

  ${\omega }_0$ is the initial angular velocity,

  $\alpha$ is the angular acceleration,

  $t$ is the time.

Calculation:

The time taken by fan to stop,

  $\begin{array}{l}{\omega }_f={\omega }_0+\alpha t\\ t=\frac{0-\left(2\pi \times \frac{85}{6}\right)}{-0.42}\text{s}\\ t=212\text{s}\end{array}$

Time $t=212\text{s}$

Conclusion:

Hence, the value is $t=212\text{s}$