A centrifuge rotor rotating at 13,000 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?
To find the number of revolutions the rotor will make before coming to rest, we need to calculate the initial angular velocity of the rotor and then use it to find the time it takes to come to rest.
Step 1: Calculate the initial angular velocity (ωi)
The angular velocity of the rotor can be determined using the formula:
ωi = 2πf
where ωi is the initial angular velocity and f is the initial frequency in Hz (revolutions per second). Since the rotor is rotating at 13,000 rpm, we need to convert this to Hz by dividing by 60.
f = 13,000 rpm / 60 = 216.67 Hz
ωi = 2π × 216.67 = 1361.37 rad/s
Step 2: Calculate the deceleration (α)
The deceleration of the rotor can be determined using the formula:
α = τ / I
where α is the deceleration, τ is the frictional torque (1.30 m·N), and I is the moment of inertia of the rotor.
The moment of inertia of a solid cylinder is given by the formula:
I = (1/2)mr²
where m is the mass of the rotor and r is the radius of the rotor.
m = 4.80 kg
r = 0.0710 m
Let's calculate I:
I = (1/2) × 4.80 kg × (0.0710 m)² = 0.01216928 kg·m²
Now, let's calculate α:
α = 1.30 m·N / 0.01216928 kg·m² = 106.60 N·m/kg
Step 3: Calculate the time it takes to come to rest (t)
The time it takes for the rotor to come to rest can be determined using the equation:
ωf = ωi - αt
where ωf is the final angular velocity (0 rad/s), ωi is the initial angular velocity (1361.37 rad/s), α is the deceleration (106.60 N·m/kg), and t is the time in seconds.
Let's plug in the values:
0 = 1361.37 rad/s - 106.60 N·m/kg × t
Solving for t:
t = 1361.37 rad/s / 106.60 N·m/kg = 12.78 s
Step 4: Calculate the number of revolutions (N)
The number of revolutions can be determined using the formula:
N = (ωi - ωf) / (2π)
where N is the number of revolutions, ωi is the initial angular velocity (1361.37 rad/s), ωf is the final angular velocity (0 rad/s), and 2π is the number of radians in one revolution (approximately 6.283).
Let's calculate N:
N = (1361.37 rad/s - 0 rad/s) / (2π) = 216.67 revolutions
Therefore, the rotor will complete approximately 216.67 revolutions before coming to rest.
To determine the number of revolutions the rotor will turn before coming to rest, we need to use the concept of rotational kinematics.
First, we need to find the initial angular velocity (ω₀) of the rotor when it is rotating at 13,000 rpm. The angular velocity is given by:
ω₀ = 2π * n₀ / 60,
where ω₀ is the initial angular velocity in rad/s, and n₀ is the initial rotation rate in rpm.
Plugging in the values, we have:
ω₀ = 2π * 13,000 / 60 = 1372.79 rad/s.
Next, we can use the torque (τ) and moment of inertia (I) to find the angular acceleration (α) of the rotor. The torque is given by:
τ = I * α,
where τ is the torque in N·m, I is the moment of inertia in kg·m², and α is the angular acceleration in rad/s².
The moment of inertia of a solid cylinder is given by:
I = 0.5 * m * r²,
where m is the mass of the rotor and r is the radius.
Plugging in the values, we have:
I = 0.5 * 4.80 kg * (0.0710 m)² = 0.0124 kg·m².
Substituting τ and I into the torque equation, we can solve for α:
α = τ / I = 1.30 m·N / 0.0124 kg·m² = 104.84 rad/s².
Now, we need to find the time (t) it takes for the rotor to come to rest. The angular velocity decreases uniformly until it reaches zero, so we can use the equation:
ω = ω₀ - α * t,
where ω is the final angular velocity (which is zero in this case), ω₀ is the initial angular velocity, α is the angular acceleration, and t is the time.
Substituting the known values, we have:
0 = 1372.79 rad/s - 104.84 rad/s² * t.
Solving for t, we get:
t = 1372.79 rad/s / 104.84 rad/s² = 13.10 s.
Finally, we can find the number of revolutions (N) using the formula:
N = (ω₀ * t) / (2π),
where N is the number of revolutions.
Plugging in the values, we have:
N = (1372.79 rad/s * 13.10 s) / (2π) = 702.06 revolutions.
Therefore, the rotor will turn approximately 702 revolutions before coming to rest.
calculate the moment of inertia I for a solid cylinder.
Torque= I * alpha
then
displcement= wi*t-1/2 alpha*t^2
you are given wi, t, and have calculated alpha