How Do You Calculate the Angular Acceleration and Torque of a Merry-Go-Round?

[cristina]

The radius of a park merry-go-round is 2.2m. To start it rotating, you wrap a rope around it and pull with a force of 260N for 12s. During this time, the merry-go-round makes one complete rotation. a) Find the angular acceleration of the merry-go-round. b) What torque is exerted by the rope on the merry-go-round? c) What is the moment of inertia of the merry-go-round?

A)
The angular acceleration is alpha = Torque/I
I did conceder the merry-go-round as a solid cylinder so I =1/2(MR^2)
Torque = Fr
Acceleration = (260N * 2.2m)/(1/2(MR^2))
I don’t know how to get rid of the M?

B)
Torque = Fr = (260N * 2.2m) = 572 Nm

C) The moment of inertia of the merry-go-round is I =1/2(MR^2)
Can I get the M in the torque from F = ma (after I know the acceleration from a) )?

Thank you

 

[Chen]

The merry-go-round completes one rotation, i.e 2π radians, in 12 seconds. I think it's valid to apply the normal motion equations here, so:
$$d\alpha = \alpha _0 + \omega _0t + \frac{1}{2}a_rt^2$$
$$2\pi = \frac{1}{2}a_r(12s)^2$$
$$a_r = \frac{4\pi }{144s^2}$$

Now that you know the angular acceleration you can find the merry-go-round's moment of inertia since a = Torque/I as you said. With this method you don't even need to know the shape of the device (and I don't think you are allowed to assume it is a cylinder).

 

[cristina]

472Nm*144s^2 = 4(pi)I
I=67968/4(pi) = 5408.72 (I feel there is something with the answer)

 

[Chen]

It's 572N, not 472N. According to this the mass of the merry-go-round would be 2.7 tons... do you know what the answer should be?

 

[cristina]

I = 6554.64 ? I think this is the corect answer but I don't know the rounding if its correct.

 

[Chen]

I also get 6,544. I'm not sure what you're asking, is this not the correct answer?

 

[cristina]

Yes, is it the correct answer? Is the unite of I is the same as the unite of mass?

 

[Chen]

No, the unit of I, the moment of inertia, is m^2kg. I don't know if it's the right answer or not, but as far as I'm concerned my calculations are correct.

 

[cristina]

I really appreciate your help. I think the answer is correct But I am not sure if I should round it to 6555 m^2.kg?

 

[cristina]

like in this case what would be the significant figures rule?

 

[Integral]

The general rule would be that you cannot have more significant digits then the numbers used in the calcultion. I would say the answer should be

6.6x 10³ m²kg