First slide

Circular motion

Question

When a ceiling fan is switched off its angular velocity reduces to 50% while it makes 36 rotations. How many more rotation will it make before coming to rest (Assume uniform angular retardation)?

Moderate

A

18
B

12
C

36
D

48

Solution

$By using equation ω^{2} = ω_{0}^{2} - 2 α θ$
${(\frac{ω_{0}}{2})}^{2} = ω_{0}^{2} - 2 α (2 π n) \Rightarrow α = \frac{3}{4} \frac{ω_{0}^{2}}{4 π \times 36}, (n = 36) . . . . . . . (i)$
Now let fan completes total n' revolution from the starting to come to rest
$0 = ω_{0}^{2} - 2 α (2 π n^{'}) \Rightarrow n^{'} = \frac{ω_{0}^{2}}{4 α π}$
Substituting the value of $α$ from equation (i)
$n^{'} = \frac{ω_{0}^{2}}{4 π} \frac{4 \times 4 π \times 36}{3 ω_{0}^{2}} = 48 revolutions$
Number of rotation = 48 - 36 = 12

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