A centrifuge rotor rotating at 7,000 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 1.30 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?
_________ rev
how long will it take?
____________ s
To find the number of revolutions the rotor will turn before coming to rest, we can use the principle of conservation of angular momentum. Initially, the angular momentum of the rotor is equal to its final angular momentum when it comes to rest.
The angular momentum of a rotating object is given by the formula: L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
The moment of inertia of a solid cylinder rotating around its central axis is given by the formula: I = 0.5 * m * r^2, where m is the mass of the cylinder and r is its radius.
Given:
Angular velocity (ω) = 7,000 rpm
Frictional torque (τ) = 1.30 m·N
Mass (m) = 4.80 kg
Radius (r) = 0.0710 m
Using the given values, let's calculate the initial angular momentum (Li) and final angular momentum (Lf) of the rotor.
Li = I * ω
Li = (0.5 * m * r^2) * ω
Lf = 0
(Since the rotor comes to rest, the final angular velocity is zero)
Setting the initial angular momentum equal to the final angular momentum and solving for the number of revolutions (N), we get:
Li = Lf
(0.5 * m * r^2) * ω = 0
Now, let's solve for N:
N = (Li / Lp)
N = [Li / (0.5 * m * r^2 * ω)]
Plug in the given values to calculate N:
N = [(0.5 * 4.80 kg * (0.0710 m)^2 * 7,000 rpm) / (0.5 * 4.80 kg * (0.0710 m)^2 * 0)]
N = 7,000 rpm / 0
Since the denominator is zero (due to the final angular velocity being zero), the rotor will not complete any revolutions before coming to rest. Therefore, the number of revolutions will be 0 rev.
To calculate the time it takes for the rotor to come to rest, we can use the equation of torque:
τ = I * α,
where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
Since the rotor comes to rest, the final angular velocity is 0. We can express this as:
α = (ωf - ωi) / t,
where ωf is the final angular velocity, ωi is the initial angular velocity, and t is the time taken for the rotor to come to rest.
Substituting α = (ωf - ωi) / t into the torque equation, we get:
τ = I * [(ωf - ωi) / t]
1.30 m·N = (0.5 * m * r^2) * [(0 - 7,000 rpm) / t]
Solving for t, we have:
t = [(0.5 * m * r^2) * (0 - 7,000 rpm)] / 1.30 m·N
Substituting the given values, we can calculate t:
t = [(0.5 * 4.80 kg * (0.0710 m)^2) * (0 - 7,000 rpm)] / 1.30 m·N
Calculating the expression above will give us the time it takes for the rotor to come to rest in seconds.