A turntable must spin at 33.7 rpm (3.53 rad/s) to play an old-fashioned vinyl record. How much torque must the motor deliver if the turntable is to reach its final angular speed in 1.5 revolutions, starting from rest? The turntable is a uniform disk of diameter 30.5 cm and mass 0.27 kg.
Kinematic of rotational motion gives
2•π•N=ε•t^2/2
ω= ε•t.
Therefore,
ε = ω^2/4•N.
The moment of inertia (disc)
I = m•R^2/2.
M =I• ε
To find the torque required to reach the final angular speed, we need to use the principle of torque and the rotational dynamics equation.
The torque (τ) is given by:
τ = I α
where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
The moment of inertia for a solid disk rotating about its central axis is given by:
I = (1/2) * m * r^2
where I is the moment of inertia, m is the mass, and r is the radius.
We are given the diameter of the disk, which can be used to calculate the radius:
r = d/2 = 30.5 cm / 2 = 15.25 cm = 0.1525 m
Thus, the moment of inertia is:
I = (1/2) * 0.27 kg * (0.1525 m)^2 = 0.004481 kg·m^2
The angular acceleration (α) can be calculated using the following equation:
α = (ωf - ωi) / t
where α is the angular acceleration, ωf is the final angular velocity, ωi is the initial angular velocity, and t is the time taken.
We are given that the turntable starts from rest, so the initial angular velocity (ωi) is 0 rad/s. The final angular velocity (ωf) can be calculated using the formula:
ωf = 2πf
where ωf is the final angular velocity and f is the frequency.
We are given that the record needs to spin at 33.7 rpm. We need to convert it to rad/s:
ωf = 2π * (33.7 rpm / 60) = 3.53 rad/s
The number of revolutions undergone by the turntable can be calculated by dividing the final angular velocity by 2π:
n = ωf / (2π) = 3.53 rad/s / (2π) = 0.561 revolutions
We are given that the turntable reaches its final angular speed in 1.5 revolutions, so the time (t) can be calculated using:
t = n / (f / 60)
where t is the time, n is the number of revolutions, and f is the frequency.
t = (0.561 revolutions) / (33.7 rpm / 60) ≈ 0.998 seconds
Now we can calculate the angular acceleration:
α = (3.53 rad/s - 0 rad/s) / 0.998 s ≈ 3.54 rad/s^2
Finally, we can calculate the torque required:
τ = I α = 0.004481 kg·m^2 * 3.54 rad/s^2 ≈ 0.0158 N·m
Therefore, the motor must deliver a torque of approximately 0.0158 N·m to reach the final angular speed in 1.5 revolutions, starting from rest.