The blades of a fan running at low speed turn at 200 rpm. When the fan is switched to high speed, the rotation rate increases uniformly to 340 rpm in 5.92 s.

(a) What is the magnitude of the angular acceleration of the blades?

(b) How many revolutions do the blades go through while the fan is accelerating?

a. Vf - Vo = 340 - 200 = 140 rev/mi =

140 rev/60 s = 2.33 rev/s.

a = (Vf - Vo)/t = 2.33 rev/s / 5.92 s =
0.39 rev/s^2.

b. 2.33 rev/s * 5.92 s = 13.8 REVs.

For Part B, do (340+200)/2=270 rpm to find the average rpm during acceleration. Then divide this answer by 60 seconds to find the average revolutions per second during acceleration 270/60=4.5 rev/sec. Multiply this answer by the amount of seconds it takes for it to accelerate and you will have your answer 4.5*5.92=26.64 rev.

To solve this problem, we can use the formula for angular acceleration:

Angular acceleration (α) = (Final angular velocity (ωf) - Initial angular velocity (ωi)) / Time (t)

(a) To find the magnitude of the angular acceleration, we need to calculate the change in angular velocity and divide it by the time taken.

Given:
Initial angular velocity (ωi) = 200 rpm
Final angular velocity (ωf) = 340 rpm
Time (t) = 5.92 s

Angular acceleration (α) = (340 rpm - 200 rpm) / 5.92 s

First, let's convert the rpm values to radians per second (rad/s):
1 revolution = 2π radians

Initial angular velocity (ωi) = 200 rpm = (200 rpm) * (2π radians/1 revolution) * (1 minute/60 s) = 20π rad/s

Final angular velocity (ωf) = 340 rpm = (340 rpm) * (2π radians/1 revolution) * (1 minute/60 s) = 34π rad/s

Angular acceleration (α) = (34π rad/s - 20π rad/s) / 5.92 s

Simplifying,
α = 14π rad/s / 5.92 s

Now, let's calculate the value of α:
α = 14π rad/s ÷ 5.92 s ≈ 7.42 rad/s²

Therefore, the magnitude of the angular acceleration of the blades is approximately 7.42 rad/s².

(b) To find the number of revolutions the blades go through while the fan is accelerating, we can use the equation for angular displacement:

Angular displacement (θ) = (Initial angular velocity (ωi) + Final angular velocity (ωf)) / 2 * Time (t)

Given:
Initial angular velocity (ωi) = 200 rpm = 20π rad/s
Final angular velocity (ωf) = 340 rpm = 34π rad/s
Time (t) = 5.92 s

Angular displacement (θ) = (20π rad/s + 34π rad/s) / 2 * 5.92 s

Simplifying,
θ = 54π rad/s * 5.92 s / 2

θ = 319.68π rad

To find the number of revolutions, we divide the angular displacement by 2π:
Number of revolutions = 319.68π rad / (2π rad/revolution)

Simplifying,
Number of revolutions ≈ 319.68 / 2

Number of revolutions ≈ 159.84

Therefore, the blades go through approximately 159.84 revolutions while the fan is accelerating.

To answer these questions, we can use the formulas of rotational motion.

(a) To find the angular acceleration of the blades, we can use the formula:

ωf = ωi + αt

where ωf is the final angular velocity, ωi is the initial angular velocity, α is the angular acceleration, and t is the time.

Given:
ωi = 200 rpm
ωf = 340 rpm
t = 5.92 s

First, we need to convert the angular velocities from rpm to radians per second (rad/s). Since 1 revolution is equal to 2π radians, we can use the conversion factor:

1 rpm = (2π radians) / (60 seconds)

Therefore,
ωi = (200 rpm) * (2π radians / 60 seconds)
= (200 * 2π) / 60
= 20π / 3 rad/s

ωf = (340 rpm) * (2π radians / 60 seconds)
= (340 * 2π) / 60
=34π/3 rad/s

Now, we can substitute these values into the formula and solve for α:

34π/3 = (20π/3) + α(5.92)

Rearranging the equation to solve for α:

α(5.92) = 34π/3 - 20π/3
= (34π - 20π) / 3
= 14π / 3

α = (14π / 3) / 5.92
= (14/3)(π/5.92)
≈ 7.45 rad/s^2

Therefore, the magnitude of the angular acceleration of the blades is approximately 7.45 rad/s^2.

(b) To find the number of revolutions the blades go through while the fan is accelerating, we can use the formula for angular displacement:

Δθ = ωi*t + (1/2)α*t^2

where Δθ is the angular displacement.

However, we need to be careful because the given initial angular velocity is for low speed, and the acceleration occurs until the final angular velocity at high speed is reached. So, we need to find the angular displacement for the acceleration phase only.

Given:
ωi = 20π / 3 rad/s
α = 7.45 rad/s^2
t = 5.92 s

Plugging these values into the formula, we get:

Δθ = (20π / 3) * 5.92 + (1/2)(7.45)(5.92^2)

Δθ = (20π / 3) * 5.92 + 160.576

Δθ ≈ 98.926 radians

To convert this angular displacement to the number of revolutions, we can use the fact that 1 revolution is equal to 2π radians:

Number of revolutions = Δθ / 2π

Therefore,

Number of revolutions ≈ 98.926 / (2π)
≈ 15.740 revolutions

So, the number of revolutions the blades go through while the fan is accelerating is approximately 15.740 revolutions.