an electric fan blade, finding angular speed and acceleration

Question

An electric fan blade is rotating at a rate of 210 rev/min when it is subsequently switched off. The blade then takes 35.5 seconds to come to rest.

A. What is the initial angular speed of the fan blade?

𝜔₀ =   rad/s

B. What is the angular acceleration of the fan blade?

𝛼 =  rad/s²

C. How many revolutions does the fan blade go through after the fan is switched off?

 revolutions

Answer 1

A. Formula for angular speed of the fan

ω₀ = 2πn,

where n = 3.5 1/s - frequency of revolution. I converted it to s^-1 by dividing number of revolution per minute by 60.

So

ω₀ = 2π x 3.5 s^-1 = 7.0 π rad/s

If you want, you can put value of π = 3.14 and get an answer not in π form.

B. As we know

ω = ω₀ - αt

Because the fan stopped, I took minus sign near angular acceleration. At a moment when fan comes to rest ω = 0, hence

0 = ω₀ - αt

From this formula

α = ω₀ / t

Let's find the numeric value.

α = 7.0 ·3.14 rad/s / 35.5s = 0.61 rad/s² or 0.19 π rad/s²

C. To find the number of revolution the fan made before stopping, we need at first find the angle θ and then divide it by 2π.

The easiest way to find angle is by using the formula

ω₀² = 2αΘ

Hence,

θ = ω₀² / 2α

Or

Θ = (7.0 π)² / (2 x 0.19 π) = 129π rad

Finally, the number of revolution

N = 129 π / 2π = 64.5 revolutions