Pilots can be tested for the stresses of flying high-speed jets in a whirling "human centrifuge," which takes 1.8 min to turn through 21 complete revolutions before reaching its final speed.
1. What is the angular acceleration?(rev/min^2)
2. What is the final angular speed in rpm?
Thanks in advance
Angular acceleration is usually expressed in radians/s^2 and angular speed in radians/s.
In your case, 21/1.8 = 11.67 is the average speed in rev/min. The final speed is twice that, or 23.33 rev/min
The angular acceleration is 23.33 rev/min divided by 1.8 min, which is 12.96 rev/min^2.
To solve these problems, we need to use the equations of rotational motion. We will also convert the time from minutes to seconds to ensure consistent units.
Given:
Time taken to turn = 1.8 min = 1.8 × 60 s = 108 s
Number of complete revolutions = 21 revolutions
1. To find the angular acceleration (α):
We know that angular acceleration (α) is defined as the change in angular velocity divided by the time taken.
Since the initial angular speed is 0 (starting from rest), the final angular speed (ω) can be calculated using the equation:
ω = (Number of revolutions × 2π) / Time taken
Substituting the known values:
ω = (21 × 2π) / 108
= 1.222 π rad/s
Now, the angular acceleration α can be calculated:
α = (Change in angular velocity) / Time taken
= (ω - 0) / Time taken
= 1.222 π / 108
≈ 0.0358 π rad/s^2
≈ 0.113 rev/min^2
Therefore, the angular acceleration is approximately 0.113 rev/min^2.
2. To find the final angular speed (ω) in rpm:
We can use the same formula using the known values:
ω = (21 × 2π) / 108
= 1.222 π rad/s
To convert ω to rpm, we know that 1 revolution = 2π radians, and 1 minute = 60 seconds. Therefore:
1 revolution/min = 2π rad/s × 60 s/1 min = 120π rpm
Converting ω to rpm:
Final angular speed (in rpm) = ω × (1 revolution/min) / (2π rad/s)
= 1.222 π × 120π / (2π)
= 73.3 rpm
Therefore, the final angular speed is approximately 73.3 rpm.
To find the angular acceleration, we need to use the formula:
Angular acceleration (α) = (final angular speed - initial angular speed) / time
Here, the time is given as 1.8 minutes, and we need to find the final angular speed. However, the initial angular speed is not given, so we need to find it first.
We know that one revolution is equivalent to 2π radians. So, the total angle covered by the centrifuge is 21 revolutions multiplied by 2π radians per revolution.
Total angle covered = 21 revolutions * 2π radians/revolution
Now, we can calculate the initial angular speed using the formula:
Initial angular speed (ω_initial) = Total angle covered / time
Substitute the values into the formula:
ω_initial = (21 * 2π) radians / 1.8 minutes
Once we have the initial angular speed, we can use it along with the final angular speed and the given time to calculate the angular acceleration.
To find the final angular speed, we can use the formula:
Final angular speed (ω_final) = (Total angle covered) / (time)
Substitute the values into the formula:
ω_final = (21 * 2π) radians / 1.8 minutes
Now you can calculate the angular acceleration (α) and the final angular speed (ω_final) by performing the necessary calculations.