A desk fan has three (approximately) rectangular sheets of 15.8 cm long and 5 cm wide, each with a mass of 140 g. The fan is rotating at a constant angular speed of 19.8 rad / s when the power is turned off. Due to friction the angular velocity of the fan slows down in a uniform manner and after 5.2 s it still runs at 4.30 rad / s.

How many revolutions has the fan performed at this time?

{[(19.8 + 4.30) / 2] * 5.2} / π

Thx

To determine the number of revolutions the fan has performed, we need to find the initial angular velocity and final angular velocity, and then calculate the total change in angular displacement.

First, let's convert the angular velocities from rad/s to rev/s. Since one revolution is equal to 2π radians, we can do the following conversions:

Initial angular velocity (ω₁) = 19.8 rad/s * (1 rev / 2π rad) ≈ 3.15 rev/s
Final angular velocity (ω₂) = 4.30 rad/s * (1 rev / 2π rad) ≈ 0.68 rev/s

Next, we can calculate the total change in angular displacement (Δθ) using the formula:

Δθ = ω₂ - ω₁

Δθ = 0.68 rev/s - 3.15 rev/s ≈ -2.47 rev/s

Since the final angular velocity is slower than the initial angular velocity, the total change in angular displacement is negative (-2.47 rev/s).

To determine the number of revolutions, we can multiply the change in angular displacement by the time elapsed (t):

Number of revolutions = Δθ * t

Number of revolutions = -2.47 rev/s * 5.2 s ≈ -12.84 revolutions

Since negative revolutions don't have physical meaning, we can take the absolute value to get the positive number of revolutions.

Therefore, the fan has performed approximately 12.84 revolutions at this time.