A centrifuge rotor rotating at 12000 rpm is shut off and is eventually brought to rest by a frictional torque of 1.50 m·N. If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest? How long will it take?

To find out how many revolutions the rotor will turn before coming to rest, we can use the concept of angular momentum.

1. First, let's calculate the initial angular velocity of the rotor:
- Given that the rotor is rotating at 12000 rpm (revolutions per minute).
- We need to convert it to radians per second. Since 1 revolution = 2π radians, we have:
Angular velocity = (12000 rpm) * (2π radians/1 revolution) * (1 minute/60 seconds)
= 400π radians/second

2. Next, we can calculate the initial angular momentum of the rotor:
- The formula for angular momentum is: L = Iω, where L is angular momentum, I is the moment of inertia, and ω is angular velocity.
- The moment of inertia for a solid cylinder rotating around its central axis is given by the formula: I = ½ * m * r^2, where m is mass and r is radius.
- Plugging in the values, we get: I = ½ * (4.80 kg) * (0.0710 m)^2 = 0.01226 kg·m^2
- Therefore, initial angular momentum, L = (0.01226 kg·m^2) * (400π radians/second) = 4.885 kg·m^2/s

3. Now, let's calculate the time it takes for the rotor to stop:
- The formula for angular acceleration is: α = τ/I, where α is angular acceleration, τ is torque, and I is moment of inertia.
- Rearranging the formula, we have: α = (τ/I) = α
- Substituting the given values, we get: α = (1.50 N·m) / (0.01226 kg·m^2) = 122.3594 radians/second^2
- Since the rotor is slowing down, angular acceleration is negative, so: α = -122.3594 radians/second^2
- We can now use the formula: ω = ω0 + αt, where ω0 is the initial angular velocity, ω is the final angular velocity, α is the angular acceleration, and t is the time taken.
- Rearranging the formula and plugging in the values, we get: 0 = (400π radians/second) + (-122.3594 radians/second^2) * t
- Solving for t: t = (400π radians/second) / (122.3594 radians/second^2) ≈ 10.299 seconds

4. Finally, we can calculate the number of revolutions (N) by multiplying the time (t) by the final angular velocity (ω):
- N = (ω * t) / (2π)
- Substituting the values: N = ((400π radians/second) * 10.299 seconds) / (2π) ≈ 2061 revolutions

Therefore, the rotor will turn approximately 2061 revolutions before coming to rest, and it will take approximately 10.299 seconds for it to stop.

To find out how many revolutions the rotor will turn before coming to rest, we need to calculate the initial angular velocity, final angular velocity, and the angular acceleration.

1. Calculate the initial angular velocity (ω0) in radians per second:
The formula to convert revolutions per minute (rpm) to radians per second (rad/s) is:
ω0 = 2π * (rpm / 60)

Given that the rotor is rotating at 12000 rpm, we can substitute the values into the formula:
ω0 = (2π * 12000) / 60
ω0 ≈ 1256.637 rad/s

2. Calculate the final angular velocity (ωf):
At rest, the final angular velocity will be zero (ωf = 0 rad/s).

3. Calculate the angular acceleration (α):
Using the formula: α = (ωf - ω0) / t, where t is the time taken to come to rest.

Rearranging the formula, we get:
t = (ωf - ω0) / α

The frictional torque (τ) acting on the rotor causes the angular acceleration. Torque (τ) is given by τ = I * α, where I is the moment of inertia.

4. Calculate the moment of inertia (I) of the rotor:
The moment of inertia of a solid cylinder is given by the formula: I = 0.5 * m * r^2, where m is the mass of the rotor and r is the radius.

Substituting the values, we get:
I = 0.5 * 4.80 kg * (0.0710 m)^2
I ≈ 0.0122 kg·m^2

5. Calculate the angular acceleration (α):
Using the formula τ = I * α and rearranging, we get:
α = τ / I

Substituting the values, we get:
α = 1.50 m·N / 0.0122 kg·m^2
α ≈ 122.95 rad/s^2

6. Calculate the time it takes to come to rest (t):
Using the formula t = (ωf - ω0) / α and substituting ωf = 0 rad/s:
t = (0 - 1256.637) / 122.95
t ≈ 10.22 s

7. Calculate the number of revolutions (n) the rotor will turn before coming to rest:
The formula to convert angular velocity (in rad/s) to revolutions (n) is:
n = ω0 / (2π)

Substituting the values, we get:
n = 1256.637 rad/s / (2π)
n ≈ 200 revolutions

Therefore, the rotor will turn approximately 200 revolutions before coming to rest, and it will take approximately 10.22 seconds to do so.

a. calculate the moment of inertia for the rotor (solid cylinder). Personnally, I recmmmend changing rpm to radians/sec

Torque=momentI* angular acceleration
solve for angular acceleration.

Then,
wfinal=winitial+ angAcc*time
solve for time

displacement=winitial*time+1/2 angacc;*time^2
solve for displacement in radians, divide by 2PI to get revolutions.