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.
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is the answer 0.25
by what method did you find 0.25?
To determine the numerical coefficient C, we can use the fact that power is the rate at which work is done, and work is equal to the force multiplied by the distance covered.
In this case, the force is generated by the motor to maintain the constant angular speed ω. The distance covered is the circumference of the circle traced by the bug, which is given by 2πr.
The work done on the bug can be calculated as the product of force and distance covered:
Work = Force x Distance = (F x 2πr)
The force can be obtained using Newton's second law, which states that force equals mass times acceleration. In this case, the acceleration can be calculated in terms of angular velocity ω:
a = rω^2
Thus, the force F is equal to mass times acceleration:
F = m x rω^2
Substituting this into the work formula, we have:
Work = (m x rω^2 x 2πr)
The power developed by the motor P_m is the rate at which work is done, so we divide the work by the time taken t:
P_m = Work / t
The time taken to complete one full circle is given by the formula:
t = (2πr) / v
Substituting this into the power formula:
P_m = (m x rω^2 x 2πr) / ((2πr) / v)
Simplifying, we get:
P_m = mvrω^2
Comparing this to the given expression for power P_m = Cmω^2vr, we can determine the value of the numerical coefficient C:
C = mvr/mvrω^2 = 1
Therefore, the numerical coefficient C is equal to 1.