A long hollow non-conducting cylinder of radius 0.060 m and length 0.70 m carries a uniform charge per unit area of 4.0 C/m^2 on its surface. Beginning from rest, an externally applied torque causes the cylinder to rotate at constant acceleration of 40 rad/s^2 about the cylinder axis. Find the net power entering the interior volume of the cylinder from the surrounding electromagnetic fields at the instant the angular velocity reaches 200 rad/s.
To find the net power entering the interior volume of the cylinder, we need to calculate the rate of work done on the cylinder by the electromagnetic fields.
Power is defined as the rate at which work is done. Mathematically, power can be calculated using the formula:
Power = Work / Time
In this case, we want to find the net power entering the interior volume of the cylinder. To do this, we need to calculate the work done on the cylinder and determine the time it takes for this work to be done.
The work done on the cylinder can be calculated using the formula:
Work = Torque * Angle
Where Torque is the applied torque causing the rotation, and Angle is the angular displacement.
Given:
Applied torque (τ) = 40 rad/s^2
Angular velocity (ω) = 200 rad/s
We need to find the angular displacement (θ) at the instant the angular velocity reaches 200 rad/s.
The angular displacement can be calculated using the formula:
θ = 1/2 * α * t^2
Where α is the angular acceleration and t is the time taken.
To find the time taken (t), we can use the formula:
ω = α * t
Now, let's calculate the angular displacement and time:
Given:
α = 40 rad/s^2
ω = 200 rad/s
Using the formula ω = α * t, we can rearrange it to find t:
t = ω / α
Substituting the given values, we get:
t = 200 rad/s / 40 rad/s^2
t = 5 seconds
Now, let's calculate the angular displacement (θ) using the formula:
θ = 1/2 * α * t^2
Substituting the given values of α and t, we get:
θ = 1/2 * 40 rad/s^2 * (5 s)^2
θ = 500 rad
Now that we have the torque (τ) and the angular displacement (θ), we can calculate the work done (W) on the cylinder:
Work = Torque * Angle
Work = 40 rad/s^2 * 500 rad
Work = 20,000 Joules
Now we need to determine the time it takes to do this work, which is the same as the time for the angular displacement.
Time taken (t) = 5 seconds
Finally, we can calculate the power:
Power = Work / Time
Power = 20,000 J / 5 s
Power = 4000 Watts
Therefore, the net power entering the interior volume of the cylinder from the surrounding electromagnetic fields at the instant the angular velocity reaches 200 rad/s is 4000 Watts.