# Moment of inertia problem involving a cylinder rolling down an incline

• as2528

#### as2528

Homework Statement
A metal can containing condensed mushroom soup has a mass of 215 g, a height of 10.8 cm, and a diameter of 6.38 cm. It is placed at rest on its side at the top of a 3.00-m-long incline that is at an angle of 25.0° to the horizontal and is then released to roll straight down. Assuming energy conservation, calculate the moment of inertia of the can if it takes 1.50 s to reach the bottom
of the incline. Which pieces of data, if any, are unnecessary for calculating the solution?
Relevant Equations
vf=vi+at
K=1/2*I*w^2+1/2*m*v^2
PE=mgh
v=rw
a=2/3*g*sin(B)
a=2/3*g*sin(25*(pi/180))=>a=2.8507 m/s^2
vf=vi+at=>vf=0+2.8507*1.50=>vf=4.2760 m/s

So the translational motion of the cylinder is 4.2760 m/s.

4.2760=R*w
PE=mgh=>PE=215*9.8*.108=>PE=227.56 J
PE = KE at the end of the roll because of energy conservation.

227.56 = 1/2*I*w^2+1/2*m*v^2
227.56=0.5*I*(134.04)^2+0.5*215*(4.276)^2
I=-0.1931 kg*m^2

The answer is 1.21 *10^4 kg*m2

How is this solved and why is my approach wrong? I think I did all the calculations correctly, so I must have gone wrong in applying the physics.

Relevant Equations::

PE=mgh
When calculating the PE, careful with the units. The mass is given in grams.

a=2/3*g*sin(B)
You shouldn't assume that the can of soup is a uniform solid cylinder. So, this formula might not apply. You are given enough information to find the acceleration without this assumption.

The answer is 1.21 *10^4 kg*m2
Looks like you dropped the negative sign for the power of 10.

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• as2528, jbriggs444 and Lnewqban
As an aside, I should add that the conservation of energy in this situation is temperature-dependent. I used to do a lecture demonstration in which two cans of the same soup (chunky beef) raced down an incline. One was at room temperature, the other had spent the night in the freezer and was still frozen. Guess which can won the race.

• as2528
As an aside, I should add that the conservation of energy in this situation is temperature-dependent. I used to do a lecture demonstration in which two cans of the same soup (chunky beef) raced down an incline. One was at room temperature, the other had spent the night in the freezer and was still frozen. Guess which can won the race.
Try a similar demonstration with a raw egg versus a hard-boiled egg.

• as2528
PE=215*9.8*.108
In addition to the mass units error, where does the .108 come from? It is not 3sin(25°).

• as2528
As an aside, I should add that the conservation of energy in this situation is temperature-dependent. I used to do a lecture demonstration in which two cans of the same soup (chunky beef) raced down an incline. One was at room temperature, the other had spent the night in the freezer and was still frozen. Guess which can won the race.
Is that to do with conservation of energy or just the fact that the contents are not rotating as fast?

• as2528
In addition to the mass units error, where does the .108 come from? It is not 3sin(25°).
The .108 was supposed to be the height of the incline.

The .108 was supposed to be the height of the incline.
3m at 25°, only 11cm? Looks like you used 2°.

• as2528
3m at 25°, only 11cm? Looks like you used 2°.
No I used it from the question. It said that the height was 10.8 cm. I converted that to meters and reasoned that since gravity is conservative I could use .108 as the height of the soup can.

It said that the height was 10.8 cm.
That is the height of the can as a cylinder, or length if that's clearer, not the height at which the can is placed.

• as2528
That is the height of the can as a cylinder, or length if that's clearer, not the height at which the can is placed.
Oh! I did not realize that. Thanks!

Is that to do with conservation of energy or just the fact that the contents are not rotating as fast?
The latter. I believe that the energy goes into viscous losses between the solid tidbits in the can and the broth plus the coupling between the fluid inside and the walls.

On edit:
Here is a homemade video of a can of tomato sauce given an impulse on my kitchen counter. What happens to the initial energy?

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• as2528