Different diameter rotational inertia

[David Earnsure]

Homework Statement

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).

Homework Equations

Rot Inertia of a cylinder = 0.5mr^2

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.

 

[haruspex]

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

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.

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.

Again, I would not trust intuition. Write some equations.

 

[David Earnsure]

Hmm some good points..
So let's 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 let's double the radius assuming this doubles the mass also
I = ½ 2kg x 4cm² = 16 kg cm²
Then let's 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?

 

[haruspex]

lets double the radius assuming this doubles the mass also

That would be an unusual consequence.

the bigger the more rotational inertia it has therefore the longer it takes to get moving.

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.

 

[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 into 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 realize 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!

 

[haruspex]

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 realize that doubling the radius wouldn't necessarily double the mass.

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?

 

[David Earnsure]

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

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!

 

[haruspex]

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!

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.