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Question: The frequency of rotation of a ceiling fan increases from \(3\,r.p.s\) to \(4\,r.p.s.\) in \(6.28\) ...

The frequency of rotation of a ceiling fan increases from 3 r.p.s3\,r.p.s to 4 r.p.s.4\,r.p.s. in 6.286.28 seconds. Its average angular acceleration is
(A) 1 rads−21\,rad{s^{ - 2}}
(B) 0.1 rads−20.1\,rad{s^{ - 2}}
(C) 10 rads−210\,rad{s^{ - 2}}
(D) 100 rads−2100\,rad{s^{ - 2}}

Explanation

Solution

Hint Use the formula of the angular acceleration given below and substitute the calculated value of the difference in the frequency of the rotation and the time taken in it and the simplification of this equation, to find the value of the acceleration from it.

Useful formula
The formula of the angular acceleration is given by
α=Δωt\alpha = \dfrac{{\Delta \omega }}{t}
Where α\alpha is the angular acceleration of the rotating ceiling fan, Δω\Delta \omega is the difference in the frequency of rotation and tt is the time taken.

Complete step by step solution
It is given that the
At first, frequency of the ceiling fan, ωi=3 r.p.s.=3×2π=6π s−1{\omega _i} = 3\,r.p.s. = 3 \times 2\pi = 6\pi \,{s^{ - 1}}
Then, frequency of the ceiling fan, ωf=4 r.p.s.=4×2π=8π s−1{\omega _f} = 4\,r.p.s. = 4 \times 2\pi = 8\pi \,{s^{ - 1}}
Time taken for the rotation of the ceiling fan, t=6.28 st = 6.28\,s
The difference in the angular acceleration is calculated as Δω=ω1−ω2\Delta \omega = {\omega _1} - {\omega _2}
Substitute the known values in it,
Δω=8π−6π\Delta \omega = 8\pi - 6\pi
Simplifying the above equation, we get
Δω=2π s−1\Delta \omega = 2\pi \,{s^{ - 1}}
The difference in the value of the frequency is obtained as 2π s−12\pi \,{s^{ - 1}} .
By using the formula of the angular acceleration,
α=Δωt\alpha = \dfrac{{\Delta \omega }}{t}
Substituting the known values in the above equation,
α=2π6.28\alpha = \dfrac{{2\pi }}{{6.28}}
By performing the division in the above equation, we get
α=1 rads−2\alpha = 1\,rad{s^{ - 2}}
Hence the value of the angular acceleration of the rotating ceiling fan is obtained as 1 rads−21\,rad{s^{ - 2}} .

Thus the option (A) is correct.

Note The ceiling fan rotates in the circular motion, hence the velocity considered is angular velocity and the acceleration is the angular acceleration. The frequency considered in the above solution is the same as that of the angular velocity.