Avg Power in a Rotational Energy/Work Problem

[dcmf]

Homework Statement

Consider a motor that exerts a constant torque of 25.0 N⋅m to a horizontal platform whose moment of inertia is 50.0 kg⋅m^2. Assume that the platform is initially at rest and the torque is applied for 12.0 rotations. Neglect friction.

What is the average power Pavg delivered by the motor in the situation above? Enter your answer in watts to three significant figures.

Relevant Equations

K = (1/2)Iw^2
W = τΔθ
P = W/Δt

This question has multiple parts and according to all the work done up to this point...

How much work W does the motor do on the platform during this process?​

1885 J​

What is the rotational kinetic energy of the platform K_rot,f at the end of the process described above?​

1885 J​

What is the angular velocity ω_f of the platform at the end of this process?​

8.68 rad/s​

How long Δt does it take for the motor to do the work done on the platform calculated in Part A?​

17.4 s​

Now I'm being asked...

What is the average power Pavg delivered by the motor in the situation above?​

???​

I assumed to find average power I would need average work, especially because the question's hint prompted me to find the average angular velocity, which it confirmed to be 4.34 rad/s. So I did the following calculations:

27.1 W was not accepted as the answer. Any advice on adjustments to make to my equation? Is there a rounding error?

 

[haruspex]

I would need average work

How would you define that?
What is the definition of average power? You wrote ##P_{avg}=W/\Delta t##, which is correct if you define that W appropriately.

 

[dcmf]

How would you define that?
What is the definition of average power?

In class, we actually did not address work or power in the context of problems involving rotation. This homework question seems to to be presented like a way to self-learn the material, but we were not provided with an equation to work with, so I assumed I should use the P=W/Δt equation.

I'm not sure how P differs from P_avg. According to one website I just found, "average power as the total energy consumed divided by the total time taken". Would this be accurate?

If so, would doing the calculation as...

P_avg = W _tot/Δt_tot = 1884.95559 J / 17.36646 s = 108.54 W​

(using unrounded versions of earlier values) and then rounding to 3 sig figs (109 W) be the right way to set up and solve the equation according to the above definition?

 

[dcmf]

How would you define that?
What is the definition of average power? You wrote ##P_{avg}=W/\Delta t##, which is correct if you define that W appropriately.

I just noticed that you may have updated your reply. Would it be appropriate to use the work as defined in the first part of the question that asks "How much work W does the motor do on the platform during this process?"

 

[kuruman]

I would set the work aside. You are asked to find the average power over time. The time average of a function ##f(t)## over an interval ##T## is $$f_{\text{avg.}}=\frac{\int_0^T f(t)~dt}{\int_0^T dt}.$$ Think of the equivalent linear situation when you have a constant force acting on an object and the velocity is not constant. In that case, the power delivered to the object is given by $$P(t)=Fv(t)$$where ##v## is the instantaneous linear velocity. In this case you have a constant torque. What do you think the equivalent equation would be for rotations?

Put it together and you will discover why the problem asked you to find the average angular velocity first.

 

[haruspex]

I would set the work aside.

Why?

The time average of a function ##f(t)## over an interval ##T## is $$f_{\text{avg.}}=\frac{\int_0^T f(t)~dt}{\int_0^T dt}.$$

And in the present case, the numerator is the work done, no?

 

[kuruman]

Why?

Please see PM on this.

And in the present case, the numerator is the work done, no?

Yes.