A ceiling fan is rotating at 0.50 rev/s. When turned off, it slows uniformly to a stop in 11 s.
(a) How many revolutions does it make in this time?
(b) Using the result from part (a), find the number of revolutions the fan must make for its speed to decrease from 0.50 rev/s to 0.25 rev/s.
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(a) Well, if the ceiling fan is rotating at 0.50 rev/s for 11 seconds, then it will complete 0.50 revolutions every second. So, in 11 seconds, it would make a total of... 0.50 x 11 = 5.50 revolutions. That's quite a spin!
(b) Okay, let's calculate this. We know that the fan's speed decreases uniformly, so we can assume that the rate of decrease is constant. If it takes 11 seconds for the fan to go from 0.50 rev/s to 0 rev/s, then it takes the same amount of time to go from 0.25 rev/s to 0 rev/s.
So, if the fan is at 0.25 rev/s, it would take 11 seconds for it to come to a stop. Therefore, it would make 0.25 x 11 = 2.75 revolutions during that time. The fan might be slowing down, but it's still going around a few times before it stops!
To solve this problem, we need to use the basic formulas of rotational motion, specifically the formula for angular velocity and the formula for angular deceleration.
(a) To find the number of revolutions the ceiling fan makes in 11 seconds, we first need to find the initial angular velocity (ω₀) and the final angular velocity (ω).
Given:
ω₀ = 0.50 rev/s (initial angular velocity)
t = 11 s (time taken to slow down)
We can find the angular deceleration using the formula:
α = Δω / Δt
Since the fan is slowing down uniformly, the change in angular velocity (Δω) is simply the difference between the initial and final angular velocities:
Δω = ω - ω₀
Δω = 0 - 0.50 rev/s (final angular velocity is 0 since the fan comes to a stop)
Substituting the values:
α = (0 - 0.50 rev/s) / 11 s
Now, we can rewrite the formula for angular velocity as:
ω = ω₀ + αt
Substituting the values we have:
0 = 0.50 rev/s + (0 - 0.50 rev/s) / 11 s * t
Simplifying the equation:
0 = 0.50 rev/s - 0.50 rev/s * t / 11 s
Now, rearranging the equation to solve for t:
0.50 rev/s * t / 11 s = 0.50 rev/s
Simplifying further:
t / 11 = 1
t = 11 s
Therefore, the fan takes 11 seconds to slow down to a stop. To find the number of revolutions it makes, we can substitute this value back into the formula for angular velocity:
ω = ω₀ + αt
ω = 0.50 rev/s + (0 - 0.50 rev/s) / 11 s * 11 s
ω = 0.50 rev/s + 0
ω = 0.50 rev/s
So, the fan makes 0.50 revolutions in 11 seconds.
(b) Now we need to find the number of revolutions the fan must make for its speed to decrease from 0.50 rev/s to 0.25 rev/s.
Using the same formula:
α = Δω / Δt
We can find the angular deceleration:
Δω = ω - ω₀
Δω = 0.25 rev/s - 0.50 rev/s
Substituting the values into the formula for angular deceleration:
α = (0.25 rev/s - 0.50 rev/s) / t
Now, rearranging the formula:
t = (0.25 rev/s - 0.50 rev/s) / α
Substituting the value of α we found earlier:
t = (0.25 rev/s - 0.50 rev/s) / [(0 - 0.50 rev/s) / 11 s]
Simplifying the equation:
t = (0.25 rev/s - 0.50 rev/s) / (-0.50 rev/s) * (11 s / 1)
t = (0.25/0.50) * (11 s)
t = 0.50 * 11 s
t = 5.50 s
Therefore, the fan must make 0.50 revolutions in 5.50 seconds to decrease its speed from 0.50 rev/s to 0.25 rev/s.
3. A pulley of radius 0.5m, on a motor is turning at 30 Rev/s and slows down uniformly in 2.0s. Calculate;
a. The angular acceleration of the motor.
b. The number of revolutions it makes in this time.
c. The length of the belt it winds in this time
Wo = 0.5rev/s
Wf = 0.25rev/s
t = 11s
Revolution = ?
a) Acceleration = 0.5rev/s / 11s
Acceleration = -0.045rad/s²
Θ = Wot + 1/2(-a)t
Θ = 0.5rev/s(11s) - 1/2(0.045rad/s²)(11²)
Θ = 2.75rev <-- Answer
b) Average speed = (0.5rev/s + 0.25rev/s) / 2
Average speed = 0.375rev/s
Θ = Vav * t / 2
Θ = 0.375rev/s * 11s / 2
Θ = 2.06rev <-- Answer