## Statement

difficulty

A fan whose blades measure 30cm is located in the roof spinning at 140 r.p.m. A blackout makes the fan stops, after 25 seconds. Calculate:

1. Angular acceleration
2. Distance traveled by the end of a blade until it stops, and the number of rotations completed
3. The value of the linear velocity, tangential, normal and full acceleration at 15 seconds after the power outage

## Solution

Data

• Length of the blade: R = 30 cm = 0.3 m
• Number of revolutions per minute: 140
• Time to full stop tf = 25 s
• Another time considered: t2=15 s

Preliminary consideratioins

We are given as data the number of revolutions per minute, that is the initial angular velocity. We need to make the conversion of revolutions per minute to radians per second knowing that a revolution is 2·π radians and one minute is 60 seconds:

${\omega }_{0}=\frac{2·\pi }{60}·140=14.66rad/s$

Resolution

Section 1

We know that the final angular velocity is 0, so:

$\omega ={\omega }_{0}+\alpha ·t⇒0=14.66+\alpha ·25⇒\alpha =-0.58rad/{s}^{2}$

Section 2

We are being asked for the final angular position (as number of revolutions):

$\phi ={\phi }_{0}+{\omega }_{0}·t+\frac{1}{2}·\alpha ·{t}^{2}⇒\phi =\overline{){\phi }_{0}}+14.66·25+\frac{1}{2}·\left(-0.58\right)·{25}^{2}=185.25rad$

We make the conversion by dividing the revolutions by 2·π:

$\frac{185.25}{2·\pi }=29.48$

Section 3

${a}_{t}=\alpha ·R=-0.58·0.3=-0.174m/{s}^{2}$

${a}_{n}={\omega }^{2}·R={5.96}^{2}·0.3=10.65m/{s}^{2}$

Finally, we can determine the value of the total acceleration (scalar), according to:

$a=\sqrt{{{a}_{n}}^{2}+{{a}_{t}}^{2}}=\sqrt{{10.65}^{2}+{\left(-0.174\right)}^{2}}=10.651m/{s}^{2}$

## Formulas worksheet

These are the main formulas that you must know to solve this exercise. If you are not clear about their meaning, we recommend you to check the theory in the corresponding sections. Furthermore, you will find in them, under the Formulas tab, the codes that will allow you to integrate these equations in external programs like Word or Mathematica.

Formulas
Related sections
$\omega ={\omega }_{0}+\alpha ·t$
$s=\phi ·R$
$\phi ={\phi }_{0}+\omega ·t+\frac{1}{2}·\alpha ·{t}^{2}$
$v=\omega ·R$
$\alpha =\text{constant}$
${a}_{t}=\alpha ·R$