Question

A cooling fan is turned off when it is running at 850 rev/min. It turns 1500 revolutions before it comes to a stop. (a) What was the fan's angular acceleration, assumed constant? (b) How long did it take the fan to come to a complete stop?

Answer 1

The fan's angular acceleration assumed constant is -0.42rad/s²

The time it takes the fan to come to a complete stop is 211.90secs

In order to get the fan's angular acceleration, we will use the equation of rotation is expressed as:

ω² = ω₀² + 2 α θ where;

ω is the final angular velocity

ω₀ is the initial angular velocity

α  is the angular acceleration

θ is the angle

Since the body comes to rest, hence ω = 0

θ = 1500 x 2 x π

θ = 9424.78 rad

• Since the cooling fan revolution is 850rev/min, the final angular velocity will be 89 rad/s

Substitute the given parameters to get the angular acceleration

$0 = 89^2 + 2 \times \alpha \times 9424.78\\-7921=18,849.56\alpha\\\alpha=\frac{-7921}{18849.56}\\\alpha= -0.42rad/s^2$

b) Next is to get the time it takes for the fan to stop

$\omega = \omega_0+\alpha t\\0=89-0.42t\\0.42t=89\\t=\frac{89}{0.42}\\t= 211.90secs$

Hence the time it takes the fan to come to a complete stop is 211.90secs

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Answer 2

The angular acceleration of fan is $-16.58 \;\rm rad/s^{2}$.

The fan will take 5.36 seconds to stop completely.

Given data:

The number of rotation made by fan per minute is, $N = 850 \;\rm rpm$.

The number of revolutions is, $n = 1500$.

(a)

The angular displacement made by the fan is,

$\theta = \dfrac{n}{2 \pi} \\\theta = \dfrac{1500}{2 \pi}\\\theta = 238.73 \;\rm rad$

And, angular speed at initial is,

$\omega_{i}=\dfrac{2\pi N}{60} \\\omega_{i}=\dfrac{2\pi \times 850}{60}\\\omega_{i}=89 \;\rm rad/s$

Now, apply the third rotational equation of motion as,

$\omega^{2}_{f}=\omega^{2}_{i}+2 \alpha \theta\\0^{2}=89^{2}+2 \times \alpha \times 238.73\\\alpha =-16.58 \;\rm rad/s^{2}$

Thus, the angular acceleration of fan is $-16.58 \;\rm rad/s^{2}$.

(b)

Applying the first rotational equation of motion to obtain the time taken to stop as,

$\omega_{f}=\omega_{i}+\alpha t\\0= 89+(-16.58) t\\t = 5.36 \;\rm s$

Thus, the fan will take 5.36 seconds to stop completely.

Learn more about the rotational equation of motion here:

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