Question

When a ceiling fan is switched on, it makes 10 revolutions in the first 3 seconds. Assuming a uniform angular acceleration, how many rotations will it make in the next 3 seconds?

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Answer

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Let's determine how many rotations the ceiling fan will make in the next 3 seconds, given that it made 10 revolutions in the first 3 seconds under uniform angular acceleration.

To begin with: The fan makes 10 revolutions in the first 3 seconds.

We need to convert these revolutions to radians: $$ \theta_{1} = 2 \pi \times 10 = 20 \pi \text{ radians} $$

Using the equation of motion for circular motion: $$ \theta = \omega_{0} t + \frac{1}{2} \alpha t^{2} $$

where:

  • $\theta$ is the angular displacement
  • $\omega_{0}$ is the initial angular velocity (which is 0 since the fan starts from rest)
  • $\alpha$ is the angular acceleration
  • $t$ is the time

Applying the given values: $$ 20 \pi = \frac{1}{2} \alpha \times 3^2 $$

Solving for $\alpha$: $$ 20 \pi = \frac{1}{2} \alpha \times 9 $$ $$ \alpha = \frac{40 \pi}{9} \text{ radians/s}^2 $$

Next, we calculate the angular velocity at the end of the first 3 seconds: $$ \omega_{1} = \omega_{0} + \alpha t = 0 + \frac{40 \pi}{9} \times 3 = \frac{40 \pi}{3} \text{ radians/s} $$

Now, let's determine $\theta_{2}$, the angular displacement from $3 \text{ s}$ to $6 \text{ s}$: $$ \theta_{2} = \omega_{1} t + \frac{1}{2} \alpha t^{2} $$ $$ \theta_{2} = \frac{40 \pi}{3} \times 3 + \frac{1}{2} \times \frac{40 \pi}{9} \times 9 $$ $$ \theta_{2} = 40 \pi + 20 \pi = 60 \pi \text{ radians} $$

Finally, converting radians back to rotations: $$ \text{Number of rotations} = \frac{60 \pi}{2 \pi} = 30 $$

Therefore, the ceiling fan will make 30 rotations in the next 3 seconds.


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