A centrifuge takes 1 minute to turn through 20 complete revolutions before reaching it's final speed. What is the angular acceleration (assumed constant) and what was it's final angular speed in rpm? How do I solve this without knowing the radius of the centrifuge?
To solve this problem without knowing the radius of the centrifuge, we can use the concept of angular velocity.
Angular acceleration (α) is defined as the rate of change of angular velocity (ω) with respect to time. It can be calculated using the formula:
α = (ωf - ωi) / t
where ωf is the final angular velocity, ωi is the initial angular velocity, and t is the duration of time.
In this case, since the centrifuge starts from rest (initial angular velocity ωi = 0) and reaches a final angular velocity ωf after 1 minute (t = 1 minute = 60 seconds), we can use the formula to find the angular acceleration:
α = ωf / t
To find the final angular speed, we need the conversion factor to convert from revolutions per minute (rpm) to radians per second (rad/s).
1 revolution = 2π radians
1 minute = 60 seconds
Therefore, the conversion factor is:
1 rpm = (2π radians) / (60 seconds)
To find the final angular speed in rpm, we can use the formula:
ωf (in rpm) = (ωf (in rad/s) * (60 seconds)) / (2π radians)
Now we can calculate the angular acceleration and the final angular speed:
1. Calculate angular acceleration (α):
α = ωf / t
= (20 revolutions / 1 minute) / (60 seconds)
= (20 revolutions / 60 seconds)
= 1/3 revolution per second^2
2. Convert final angular speed to rad/s:
ωf (in rad/s) = (20 revolutions / 1 minute) * 2π radians / (60 seconds)
= (20/60) * 2π radians/second
= (1/3) * 2π radians/second
= 2π/3 radians/second
3. Convert final angular speed to rpm:
ωf (in rpm) = (2π/3 radians/second) * (60 seconds) / (2π radians)
= (1/3) * 60
= 20 rpm
Therefore, the angular acceleration is 1/3 revolution per second^2 and the final angular speed is 20 rpm.
Let angular acceleration = alpha
(1/2)*(alpha)*t^2 = 20*2 pi
= 40*pi radians
alpha = 80*pi/60^2 = 0.06981 rad/s^2
Final angular speed = alpha*t = 4.19 rad/s
OR
= 2*(average angular speed)
= 2*(40 pi/60)= (4/3)*pi
(same answer)