# Different diameter rotational inertia

1. Dec 16, 2015

### David Earnsure

1. The problem statement, all variables and given/known data
I am doing the old frozen can vs liquid can experiment and one of the questions asks that if cans with different diameters were used with the same contents (so same mass relative to diameter size) how would this effect the time taken for the cans to go down a ramp (how would if differ from the cans I used).

2. Relevant equations
Rot Inertia of a cylinder = 0.5mr^2

3. The attempt at a solution
Straight away I know the larger the diameter means the larger the rotational inertia is going to be, therefore the more likely it is going to remain rest and the slower it will reach the bottom of the ramp and the opposite would be true for a smaller diameter - that works for the frozen contents, for the liquid contents the same would happen as even though the liquid is not rotating with the can the can still has an increase/decrease in its mass and radius. I feel like I'm missing something here, or maybe I'm not and I'm over complicating it. A nudge in the right direction or confirmation on what I'm saying would be appreciated.

Thanks.

2. Dec 16, 2015

### haruspex

Quite sure about that, are you? Tried any equations?
A complication is that the can and contents will have different densities, and the way the moments of inertia and masses of the contents and the end caps vary with radius is different from the way those of the sides of the can vary.
Again, I would not trust intuition. Write some equations.

3. Dec 16, 2015

### David Earnsure

Hmm some good points..
So lets say for example my original frozen can has a mass of 1kg and a radius of 2cm.
I = ½ 1kg x 2cm2 (saying that frozen can is a solid cylinder as for the purpose of this experiment I am limited to rotational motion)
= 2 kg cm2
Now lets double the radius assuming this doubles the mass also
I = ½ 2kg x 4cm2 = 16 kg cm2
Then lets half the radius assuming this halves the mass also
I = ½ 0.5kg x 1cm2 = 0.25 kg cm2
To me this confirms what I was thinking - the bigger the more rotational inertia it has therefore the longer it takes to get moving.

With the can with liquid in it I'm pretty lost as nothing in our textbooks talks about a liquid mass inside a hollow cylinder that doesn't move in a rotational motion with the hollow cylinder its in. Since this experiment is almost entirely meant to be relating to rotational motion I have made the assumption that the liquid contents can has the rotational inertia of a hoop so I = MR2 which would then also mean the bigger the radius/mass the larger the inertia and the opposite if it got smaller.

I know my assumptions on mass and radius are rather broad but it has been stated in the question that the contents are the same (as the radius increases so will the mass, make these "new" cans scaled up or down versions of the original ones).

Where is the gap in my thinking?

4. Dec 16, 2015

### haruspex

That would be an unusual consequence.
Yes, more rotational inertia, but more weight, and the greater radius gives that weight more torque, and it will have more linear velocity in relation to angular velocity. Write an equation for the acceleration. You can't solve this with intuition.

5. Dec 18, 2015

### David Earnsure

I am still completely stuck, I have looked up the equations for constant rotational acceleration and found:

acceleration = Torque/rotational inertia

Using the above formula I plugged in all sorts of different scenarios with radius increasing and mass increasing (or even staying the same) and for all of them when radius increased acceleration decreased, which brings me back to what I originally thought about a larger radius taking longer to reach the bottom of the ramp.

But when weight increases so does the gravitational potential energy of the cylinder which I feel will make the cylinder move faster but am not sure where it all fits in to this.

You say it is an unusual consequence with the radius doubling for the mass to double but all I am trying to simulate is the can itself doubling in size to try get some numbers to use in the equations for it having a larger/smaller diameter, I realise that doubling the radius wouldn't necessarily double the mass.

Can you please go into more detail or provide some examples as I am feeling even more stuck than when I started!

Thanks!

6. Dec 18, 2015

### haruspex

If the radius doubles but the density and cylinder length stay the same then the mass would quadruple.

So, cylinder radius r, length L, density p on a slope at angle theta. Draw it.
We will ignore the mass of the can.
1. What is the mass of the cylinder? Call this m.
2. Draw a vertical line down from the centre of the cylinder (O); draw a horizontal line across from the point of contact with the slope (P) to meet the vertical line at Q. How far is PQ?
3. What is the moment of the weight (mg) about P?
4. For the frozen case, what is the moment of inertia of the cylinder about P? (Use the parallel axis theorem.)
5. What is the angular acceleration?
6. What is the linear acceleration of the cylinder's centre?

7. Dec 19, 2015

### David Earnsure

Oh this is so obvious how could I of missed that!

So I worked out the accelerations for 3 different radius lengths and they're all exactly the same!

8. Dec 19, 2015

### haruspex

Quite so.
For a given density of disc, the mass rises as r2 and the MoI rises as r4. The gravitational force rises as the square; multiplying that by moment arm gives a moment rising as r3. So the angular acceleration goes as 1/r, but the linear acceleration is constant.