Question

A ceiling fan with 80-cm-diameter blades is turning at 60 rpm.Suppose the fan coasts to a stop 25s after being turned off.

a.) What is the speed of the trip of a blade 10 s after the fan is turned off?

b.) Through how many revolutions does the fan turn while stopping?

Answer 1

Concepts and reason

The required concepts to solve these questions are rotational kinematic equations, angular velocity, angular acceleration and angular displacement.

Firstly, calculate the angular acceleration by using angular velocity equation. Calculate the linear speed of the trip of a blade 10 s after the fan is turned off by using angular velocity equation. Calculate the number of revolutions by the fan while stopping by using angular displacement equation.

Fundamentals

The expression for angular velocity is expresses as follows,

$\omega = {\omega _0} + \alpha t$

Here, $\alpha$is the angular acceleration, $t$ is the time and ${\omega _0}$is the initial angular velocity.

The expression for angular displacement as a function of time is expresses as follows,

$\theta = \frac{{{\omega ^2} - {\omega _0}^2}}{{2\alpha }}$

Here, ${\omega _0}$is the initial angular velocity, $\omega$is final angular velocity. and $\alpha$is the angular acceleration.

The expression for linear speed in term of angular velocity is expresses as follows,

$v = r\omega$

Here, $r$is the radius and $\omega$is the angular velocity.

The expression for radius is expresses as follows,

$r = \frac{d}{2}$

Here, $d$ is the diameter.

(a)

The expression for angular velocity is expresses as follows,

$\omega = {\omega _0} + \alpha t$ …… (1)

Rearrange the equation for angular acceleration.

$\alpha = \frac{{\omega - {\omega _0}}}{t}$ …… (2)

Convert the unit of $\left( {{\omega _0}} \right)$initial angular velocity $\left( {60{\rm{ rpm}}} \right)$to${\rm{rad}} \cdot {{\rm{s}}^{ - 1}}$.

$\begin{array}{c}\\60{\rm{ rpm}} = \left( {60{\rm{ rpm}}} \right)\left( {\frac{{0.104{\rm{ rad}} \cdot {{\rm{s}}^{ - 1}}}}{{1{\rm{ rpm}}}}} \right)\\\\ = 6.24{\rm{ rad}} \cdot {{\rm{s}}^{ - 1}}\\\end{array}$

Substitute $0$ for $\omega$ , $6.24{\rm{ rad}} \cdot {{\rm{s}}^{ - 1}}$ for ${\omega _0}$ and$25{\rm{ s}}$ for $t$in the equation (2).

$\begin{array}{c}\\\alpha = \frac{{0 - \left( {6.24{\rm{ rad}} \cdot {{\rm{s}}^{ - 1}}} \right)}}{{\left( {25{\rm{ s}}} \right)}}\\\\ = - 0.24{\rm{ rad}} \cdot {{\rm{s}}^{ - 2}}\\\end{array}$

Substitute $6.24{\rm{ rad}} \cdot {{\rm{s}}^{ - 1}}$ for ${\omega _0}$ , $- 0.24{\rm{ rad}} \cdot {{\rm{s}}^{ - 2}}$ for $\alpha$and$10{\rm{ s}}$ for $t$in the equation (1).

$\begin{array}{c}\\\omega = 6.24{\rm{ rad}} \cdot {{\rm{s}}^{ - 1}} + \left( { - 0.24{\rm{ rad}} \cdot {{\rm{s}}^{ - 2}}} \right)\left( {10{\rm{ s}}} \right)\\\\ = 3.84{\rm{ rad}} \cdot {{\rm{s}}^{ - 1}}\\\end{array}$

The expression for radius is,

$r = \frac{d}{2}$

Substitute $80{\rm{ cm}}$ for $d$in the above equation.

$\begin{array}{c}\\r = \frac{{80{\rm{ cm}}}}{2}\\\\ = 40{\rm{ cm}}\left( {\frac{{1{\rm{ m}}}}{{{{10}^2}\,{\rm{cm}}}}} \right)\\\\ = 40 \times {10^{ - 2}}\;{\rm{m}}\\\end{array}$

The expression for linear speed of the trip of a blade 10 s after the fan is turned off,

$v = r\omega$

Substitute $3.84{\rm{ rad}} \cdot {{\rm{s}}^{ - 1}}$ for $\omega$ and $40 \times {10^{ - 2}}\;{\rm{m}}$for $r$in the above equation.

$\begin{array}{c}\\v = \left( {3.84{\rm{ rad}} \cdot {{\rm{s}}^{ - 1}}} \right)\left( {40 \times {{10}^{ - 2}}\;{\rm{m}}} \right)\\\\ = 1.536{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}\\\end{array}$

(b)

The expression for angular displacement as a function of time is,

$\theta = \frac{{{\omega ^2} - {\omega _0}^2}}{{2\alpha }}$

Substitute $0$ for $\omega$ , $6.24{\rm{ rad}} \cdot {{\rm{s}}^{ - 1}}$ for ${\omega _0}$ , $- 0.24{\rm{ rad}} \cdot {{\rm{s}}^{ - 2}}$ for $\alpha$ in the above equation.

$\begin{array}{c}\\\theta = \frac{{0 - {{\left( {6.24{\rm{ rad}} \cdot {{\rm{s}}^{ - 1}}} \right)}^2}}}{{2\left( { - 0.24{\rm{ rad}} \cdot {{\rm{s}}^{ - 2}}} \right)}}\\\\ = 81.12{\rm{ rad}}\left( {\frac{{\;1{\rm{ rev}}}}{{2\pi {\rm{ rad}}}}{\rm{ }}} \right)\\\\ = 12.91{\rm{ rev}}\\\end{array}$

Ans: Part a

The speed of the trip of a blade 10 s after the fan is turned off is $1.536{\rm{ m}} \cdot {{\rm{s}}^{ - 1}}$.

Part b

The number of revolutions by the fan while stopping is $12.91{\rm{ rev}}$.