Question
I know I can calculate average power by using the work done perunit time.
I thought I could also use (for a rotational problem) torque timesangular velocity.
I tried to use torque times angular velocity for the followingproblem and got it wrong. Why?
Calculate the average power required to accelerate the Earth in 4days from rest to its present qngular speed about its axis.
I used I = 2/5 MR2 = 9.71e37 kg m 2, = 7.27e-5 rad/s , = 2.104e-10 rad/s2,
= 2.04e28 N m ; KE = 2.57e29 J
If I use P = KE/t I get P = 7.42e23 W, if I use P = (dot) I get P = 1.48e24 W
Why aren't they equal? What am I missing?
Explanation / Answer
You have the right idea, you just made one little mistake (andthe little mistakes are always hardest to find!) . You're using the final speed of the Earth as if it was at thatspeed for the entire 4 days. . The relation power = force *velocity or power =torque * angular velocity is onlyvalid if velocity is constant . Otherwise you can modify thisto: average power = force * average velocity or . average power = torque * averageangular velocity . Now, if acceleration is constant the average velocity is justthe average of the initial and final velocities. If the initial iszero... . then the averagevelocity is simply one half the final velocity! . using this, we can now modify your lastcalculation: . P = * average = 2.04 x 1028 * (1/2) * 7.27 x10-5 = 7.42 x1023 Watts . which of course matches your other method, final KEdivided by time. Notice that your two original calculations differby a factor of 2. . Hope this helps!