Find the net work required to accelerate a cylinder

[blixel]

Homework Statement

A merry-go-round has a mass of 1440kg and a radius of 7.50m . How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 7.00s? Assume it is a solid cylinder.

Homework Equations

I believe my relevant equations are:
I_CYL=½mr²
ω=(Δθ/Δt)
α=(Δω/Δt)
tau=Iα (moment of inertia * alpha)
W=tau*Δθ

The Attempt at a Solution

First I calculated the moment of inertia as:
I_CYL=½(1440kg)(7.50m)²=40500 kg⋅m²

Then I calculated ω as 2π/7.00s

Knowing ω allowed me to calculate α as: (2π/7.00s)/7.00s=2π/49.00s²

Then I used the tau equation to calculate tau as Iα=(40500 kg⋅m²)*(2π/49.00s²)=5193.244999J

Then I used the work equation to calculate work as tau*Δθ=5193.244999J*2π=32630.12067J

With 3 sig figs, this would be 3.26*10⁴J

The problem is, the book says the answer is 1.63*10⁴J and I can't figure out why.

 

[TJGilb]

Your answer is twice as much. Look for a rogue 2 somewhere in your calculations.

 

[blixel]

Thanks for the reply. I realize my answer is exactly 2 times as high. (I wouldn't have gone through the trouble of creating an account here and posting a question for something as obvious as that). But after carefully going over the steps multiple times, I cannot account for the fact that the answer should be (my answer / 2). So I am hoping someone can explain what I'm doing wrong which logically explains the problem. I can't just arbitrarily divide my answer by 2 and call it a day.

 

[TomHart]

Then I calculated ω as 2π/7.00s

This is correct.

Knowing ω allowed me to calculate α as: (2π/7.00s)/7.00s=2π/49.00s2

This is an assumption you made - that it took 7 seconds to reach the angular velocity above. No such information was given in the problem statement. For all we know, it might have taken 3 weeks to get up to that angular velocity. But I think we can still work with your assumption.

Then I used the tau equation to calculate tau as Iα=(40500 kg⋅m2)*(2π/49.00s2)=5193.244999J

That looks good based on your assumption of 7 seconds to accelerate up to the final angular velocity.

Then I used the work equation to calculate work as tau*Δθ=5193.244999J*2π=32630.12067J

Now you are multiplying the torque times the angle over which it was applied, which is 2π. But is 2π really the correct number? We know, based on your assumption, that it took 7 seconds to accelerate up to the final velocity. At your calculated (or assumed) angular acceleration, what angle of rotation would it take to reach the final angular velocity of 2π/7??

Edit: Oh yeah, welcome to Physics Forums.

 

[blixel]

This is an assumption you made - that it took 7 seconds to reach the angular velocity above. No such information was given in the problem statement. For all we know, it might have taken 3 weeks to get up to that angular velocity.

Ah, yes. I see what you are saying. That was sloppy reading on my part. Since I don't know how long it took to reach that angular velocity, I can't calculate angular acceleration, so my entire approach is wrong. The fact that my answer comes out twice as high as it should be was just coincidental. Back to the drawing board.

 

[TomHart]

Actually, I think you should find your error in the solution that you have started with. Even though you made that assumption, you can still find the right answer based on that assumption.

 

[PeroK]

Ah, yes. I see what you are saying. That was sloppy reading on my part. Since I don't know how long it took to reach that angular velocity, I can't calculate angular acceleration, so my entire approach is wrong. The fact that my answer comes out twice as high as it should be was just coincidental. Back to the drawing board.

Work is closely related to which dynamic quantity?

 

[blixel]

Thanks for the replies. I went back over the problem using the work energy theorem. W≡K-K₀=ΔK where K=½Iω². This allowed me to get the same answer as the book without any assumptions.

Using the same I and ω that I already found from the original post: K=½(40500 kg⋅m²)([2π/7.00s])²=16315.06034J

So that takes care of that problem. Thanks again to everyone who replied.