A thin, massless, horizontal pipe rotates about a vertical axis with angular speed ω as shown in the figure.
A small bug, located at a distance r from the axis, starts crawling inside the pipe at constant speed v (relative to the pipe).
Because of the bug, the motor must develop the power
P_m=Cmω^2vr in order to keep the system rotating at constant angular speed ω.
Determine the numerical coefficient C.
To determine the numerical coefficient C, we can use the fact that power is defined as the rate at which work is done or energy is transferred. In this case, the power P_m is the rate at which the motor does work to keep the system rotating at a constant angular speed ω.
First, let's consider the work done by the motor. The work done is equal to the force applied by the motor multiplied by the distance over which the force is applied. In this case, the force applied by the motor is the torque τ, which is equal to the product of the moment of inertia I and the angular acceleration α. The distance over which the force is applied is the circumference of the pipe, which is equal to 2πr.
So, the work done by the motor is given by:
Work = Force × Distance
Work = τ × 2πr
Now, the power P_m is the rate at which this work is done. Therefore, we can write:
Power = Work / Time
Since the system is rotating at a constant angular speed, the angular acceleration α is zero. Therefore, the torque τ is also zero. This means that the motor does not have to do any work to keep the system rotating at a constant angular speed.
So, the power developed by the motor is zero:
P_m = 0
Now, let's look at the expression for the power P_m given in the problem:
P_m = Cmω^2vr
Since P_m is zero, we have:
0 = Cmω^2vr
To find the numerical coefficient C, we need to isolate it in the equation. From the equation above, we can solve for C:
C = 0 / (mω^2vr)
Since we know that anything divided by zero is undefined, the numerical coefficient C is undefined.
Therefore, the numerical coefficient C cannot be determined using the given information.