A centrifuge rotor rotating at 10700 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 2.48 m * N.
1. If the mass of the rotor is 5.61 kg and it can be approximated as a solid cylinder of radius 0.0820 m, through how many revolutions will the rotor turn before coming to rest?
2. How long will it take?
To find the number of revolutions the rotor will turn before coming to rest, we can use the concept of angular deceleration.
1. First, let's calculate the initial angular velocity of the rotor. We know that angular velocity is given in radians per second (rad/s), so we need to convert rpm to rad/s. One revolution is equal to 2π radians, so the initial angular velocity is:
Initial angular velocity (ω_i) = 10700 rpm * 2π rad/1 min * 1 min/60 s
2. Next, we can find the final angular velocity (ω_f) when the rotor comes to rest. The final angular velocity is 0, as the rotor is brought uniformly to rest.
3. The angular deceleration (α) can be obtained using the formula:
Torque (τ) = Inertia (I) * Angular acceleration (α)
In this case, the torque is given as 2.48 m*N and the inertia of a solid cylinder is given by:
I = (1/2) * m * r^2
where m is the mass of the rotor and r is the radius of the rotor.
4. Now, using the formula for angular acceleration:
Angular acceleration (α) = (ω_f - ω_i) / t
where t is the time taken to come to rest.
5. With the angular acceleration, we can find the number of revolutions (N_revolutions) using the formula:
N_revolutions = (ω_i^2 - ω_f^2) / (2 * α * 2π)
Let's calculate the values step by step:
1. Initial angular velocity (ω_i):
ω_i = 10700 rpm * 2π rad/1 min * 1 min/60 s
2. Final angular velocity (ω_f):
ω_f = 0 rad/s (as the rotor comes to rest)
3. Torque (τ):
τ = 2.48 m*N
4. Inertia (I):
I = (1/2) * m * r^2
= (1/2) * 5.61 kg * (0.0820 m)^2
5. Angular acceleration (α):
τ = I * α
2.48 m*N = (1/2) * 5.61 kg * (0.0820 m)^2 * α
α = (2.48 m*N) / [(1/2) * 5.61 kg * (0.0820 m)^2]
6. Time taken to come to rest (t):
t = (ω_f - ω_i) / α
= (0 - ω_i) / α
7. Number of revolutions (N_revolutions):
N_revolutions = (ω_i^2 - ω_f^2) / (2 * α * 2π)
By calculating the values of these formulas, we can determine the answer to your question.