Question

A ceiling fan is turned on and reaches an angular speed of 120 rev/min in 20.0 s. It is then turned off and coasts to a stop in 40,0 s. In the 60.0 s of rotation, through how many revolutions did the fan turn?

Answer 1

Initial angular speed of the fan = $\omega$₁ = 0 rad/s

Maximum angular speed of the fan = $\omega$₂ = 120 rev/min = 120 x (2$\pi$/60) rad/s = 12.566 rad/s

Time taken by the fan to reach maximum speed = T₁ = 20 sec

Angular acceleration of the fan while speeding up = $\alpha$₁

$\omega$₂ = $\omega$₁ + $\alpha$₁T₁

12.566 = 0 + $\alpha$₁(20)

$\alpha$₁ = 0.6283 rad/s²

Angle through which the fan spins through while speeding up = $\theta$₁

$\theta$₁ = $\omega$₁T₁ + $\alpha$₁T₁²/2

$\theta$₁ = (0)(20) + (0.6283)(20)²/2

$\theta$₁ = 125.66 rad

Final angular speed of the fan = $\omega$₃ = 0 rad/s

Time taken by the fan to come to stop = T₂ = 40 sec

Angular acceleration of the fan while slowing down = $\alpha$₂

$\omega$₃ = $\omega$₂ + $\alpha$₂T₂

0 = 12.566 + $\alpha$₂(40)

$\alpha$₂ = -0.31415 rad/s²

Angle through which the fan rotates while slowing down = $\theta$₂

$\theta$₂ = $\omega$₂T₂ + $\alpha$₂T₂²/2

$\theta$₂ = (12.566)(40) + (-0.31415)(40)²/2

$\theta$₂ = 251.32 rad

Total angle through which the fan rotates = $\theta$

$\theta$ = $\theta$₁ + $\theta$₂

$\theta$ = 125.66 + 251.32

$\theta$ = 376.98 rad

Total number of revolutions made by the fan = n

$n = \frac{\theta }{2\pi }$

376.98 n = 275

n = 60 rev

Total number of revolutions made by the fan in the 60 sec = 60 rev

Answer 2

Initial angular speed of the fan = $\omega$₁ = 0 rad/s

Maximum angular speed of the fan = $\omega$₂ = 120 rev/min = 120 x (2$\pi$/60) rad/s = 12.566 rad/s

Time taken by the fan to reach maximum speed = T₁ = 20 sec

Angular acceleration of the fan while speeding up = $\alpha$₁

$\omega$₂ = $\omega$₁ + $\alpha$₁T₁

12.566 = 0 + $\alpha$₁(20)

$\alpha$₁ = 0.6283 rad/s²

Angle through which the fan spins through while speeding up = $\theta$₁

$\theta$₁ = $\omega$₁T₁ + $\alpha$₁T₁²/2

$\theta$₁ = (0)(20) + (0.6283)(20)²/2

$\theta$₁ = 125.66 rad

Final angular speed of the fan = $\omega$₃ = 0 rad/s

Time taken by the fan to come to stop = T₂ = 40 sec

Angular acceleration of the fan while slowing down = $\alpha$₂

$\omega$₃ = $\omega$₂ + $\alpha$₂T₂

0 = 12.566 + $\alpha$₂(40)

$\alpha$₂ = -0.31415 rad/s²

Angle through which the fan rotates while slowing down = $\theta$₂

$\theta$₂ = $\omega$₂T₂ + $\alpha$₂T₂²/2

$\theta$₂ = (12.566)(40) + (-0.31415)(40)²/2

$\theta$₂ = 251.32 rad

Total angle through which the fan rotates = $\theta$

$\theta$ = $\theta$₁ + $\theta$₂

$\theta$ = 125.66 + 251.32

$\theta$ = 376.98 rad

Total number of revolutions made by the fan = n

$n = \frac{\theta }{2\pi }$

376.98 n = 275

n = 60 rev

Total number of revolutions made by the fan in the 60 sec = 60 rev