The net total torque of 51.5 N · m on a wheel rotating around an axis through its center is due to an applied force and a frictional torque at the axle. Starting from rest, the wheel reaches an angular speed of 13.2 rad/s in 5.00 s. At t = 5.00 s, the applied force is removed, and the frictional torque brings the wheel to a stop in 40.0 s.
(c) What is the total number of revolutions the wheel undergoes during this 45.0-s interval?
To find the total number of revolutions the wheel undergoes during the 45.0s interval, we first need to calculate the angular displacement of the wheel during this time period.
We are given that the wheel starts from rest and reaches an angular speed of 13.2 rad/s in 5.00s. We can use the equation:
θ = ω_i * t + (1/2) * α * t^2
where:
θ is the angular displacement,
ω_i is the initial angular velocity,
t is the time, and
α is the angular acceleration.
Since the wheel starts from rest, the initial angular velocity ω_i is 0 rad/s. The angular acceleration α can be calculated using the formula:
α = torque / moment of inertia
where torque is the net total torque of 51.5 N·m and the moment of inertia depends on the geometry of the wheel.
After finding the angular acceleration α, we can substitute the values into the equation for angular displacement θ to find the initial angular displacement.
Next, we need to find the final angular displacement at the end of the 45.0s interval when the wheel comes to a stop. We know that the wheel stops after 40.0s, so we can use the equation:
θ_f = ω_i + ω_f * t + (1/2) * α * t^2
again, where:
θ_f is the final angular displacement,
ω_i is the initial angular velocity (13.2 rad/s),
ω_f is the final angular velocity (0 rad/s),
t is the time (45.0s), and
α is the angular acceleration.
Now that we have the initial and final angular displacements, we can calculate the total angular displacement by subtracting the initial angular displacement from the final angular displacement:
total angular displacement = θ_f - θ_i
Finally, to find the number of revolutions, we need to convert the total angular displacement to revolutions. We know that 2π radians is equivalent to one revolution, so we can use the conversion factor:
number of revolutions = total angular displacement / (2π)