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
Pm=Cmω2vr
in order to keep the system rotating at constant angular speed ω. Determine the numerical coefficient C. m is the mass of the bug.
To determine the numerical coefficient C in the equation Pm = Cmω^2vr, we need to analyze the forces acting on the bug and the work done by these forces.
The bug is crawling inside the rotating pipe, so it experiences two forces: the centrifugal force and the frictional force. The centrifugal force acts radially outward due to the rotation, while the frictional force opposes the motion of the bug.
The centrifugal force is given by Fc = mv^2/r, where m is the mass of the bug, v is its speed relative to the pipe, and r is the distance of the bug from the axis of rotation.
Since the bug is moving at a constant speed relative to the rotating pipe, the frictional force must provide the necessary centripetal force to keep the bug on its circular path. The centripetal force is given by Fp = mω^2r, where ω is the angular speed of the rotation.
The work done by the frictional force is given by Wp = Fp * d, where d is the distance traveled by the bug inside the pipe.
Now, the power developed by the motor is equal to the work done by the frictional force per unit time. Therefore, we have:
Pm = (Wp / t)
Since the bug crawls a complete circle in time t = (2πr) / v, the distance traveled inside the pipe is equal to d = 2πr.
Substituting the values, we have:
Pm = (Fp * d) / t
= (mω^2r * 2πr) / [(2πr) / v]
= mω^2v
Comparing this to the given equation Pm = Cmω^2vr, we can determine that C = v.
Therefore, the numerical coefficient C in the equation Pm = Cmω^2vr is equal to the speed of the bug relative to the pipe, v.