The blades of a fan running at low speed turn at 230rpm . When the fan is switched to high speed, the rotation rate increases uniformly to 370rpm in 5.80s .

Part A
What is the magnitude of the angular acceleration of the blades?

Part B
How many revolutions do the blades go through while the fan is accelerating?

angular acceleration=(wf-wi)/time

for here, you have to choose your units. I think I would change time to minutes, so angular acceleration would be in rev/min^2

acceleration= (370-230)/(5.8/60)
= 140*60/5.8 work that our rpm^2

rev=wi*time+1/2 acceleration*time^2
= 230*5.8/60 + 1/2 above acc*(5.8/60)^2
or you could have done it with avg velocity
rev=avgvelocity*time=(wf+wi)/2 * 5.8/60

Part A:

To find the magnitude of the angular acceleration, we can use the formula:

Angular acceleration (α) = (final angular velocity - initial angular velocity) / time

Given:
Initial angular velocity (ωi) = 230 rpm
Final angular velocity (ωf) = 370 rpm
Time (t) = 5.80 s

Converting the angular velocities from rpm to rad/s:

Initial angular velocity (ωi) = 230 rpm * 2π rad/1 min * 1 min/60 s = 24.2 rad/s
Final angular velocity (ωf) = 370 rpm * 2π rad/1 min * 1 min/60 s = 38.8 rad/s

Substituting these values into the formula:

Angular acceleration (α) = (38.8 rad/s - 24.2 rad/s) / 5.80 s
= 14.6 rad/s / 5.80 s
≈ 2.5172 rad/s^2

Therefore, the magnitude of the angular acceleration of the blades is approximately 2.5172 rad/s^2.

Part B:

To find the number of revolutions the blades go through while the fan is accelerating, we need to find the angular displacement (θ) during the acceleration phase.

The formula to calculate angular displacement is:

Angular displacement (θ) = (initial angular velocity + final angular velocity) / 2 * time

Substituting the given values:

Angular displacement (θ) = (24.2 rad/s + 38.8 rad/s) / 2 * 5.80 s
= 63 rad/s / 2 * 5.80 s
≈ 10.8621 rad

Each revolution is equal to 2π rad, so we can convert the angular displacement to revolutions:

Number of revolutions = Angular displacement (θ) / 2π
≈ 10.8621 rad / 2π
≈ 1.7253 revolutions

Therefore, the blades go through approximately 1.7253 revolutions while the fan is accelerating.

Part A:

To find the magnitude of the angular acceleration, we can use the formula:

angular acceleration (α) = (final angular velocity - initial angular velocity) / time

Given:
Initial angular velocity (ω1) = 230 rpm
Final angular velocity (ω2) = 370 rpm
Time (t) = 5.80 s

First, we need to convert the angular velocities from rpm to radians per second (rad/s). To do this, we need to multiply them by 2π/60 (since 1 revolution = 2π radians and 1 minute = 60 seconds):

ω1 = 230 rpm * (2π/60) rad/s ≈ 24.13 rad/s
ω2 = 370 rpm * (2π/60) rad/s ≈ 38.83 rad/s

Now we can substitute these values into the formula:

α = (ω2 - ω1) / t
= (38.83 rad/s - 24.13 rad/s) / 5.80 s
≈ 2.54 rad/s²

Therefore, the magnitude of the angular acceleration of the blades is approximately 2.54 rad/s².

Part B:

To determine the number of revolutions the blades go through while the fan is accelerating, we can use the equation:

number of revolutions = (final angular velocity - initial angular velocity) / (2π)

Using the same angular velocities as in Part A:

number of revolutions = (ω2 - ω1) / (2π)
= (38.83 rad/s - 24.13 rad/s) / (2π)
≈ 2.35 revolutions

Therefore, the blades go through approximately 2.35 revolutions while the fan is accelerating.