Two cans rolling down a ramp with the same mass

[henry3369]

Homework Statement

Two cans of the same size, mass, and shape are released from a ramp at the same height. One of the cans has milk and the other has refried beans. Which will reach the bottom first?

Homework Equations

K_total = K_translation + K_rotational

The Attempt at a Solution

Since they start at the same height, their total kinetic energy at the bottom is the same. The can of milk has a higher moment of inertia, so in order total kinetic energy to be the same, it has to have lower translational kinetic energy; therefore, a smaller velocity. So why does the can of milk still reach the bottom first?

 

[gneill]

What's the difference between milk and refried beans?

 

[henry3369]

What's the difference between milk and refried beans?

Well the demonstration I watched said that the can with the milk acts more like a hollow can while the can with the refried beans is similar to a solid cylinder.

 

[gneill]

Well the demonstration I watched said that the can with the milk acts more like a hollow can while the can with the refried beans is similar to a solid cylinder.

Can you think why that might be? Again, what's the difference between milk and refried beans?

 

[henry3369]

Can you think why that might be? Again, what's the difference between milk and refried beans?

The milk moves around slides around in the can while the refried beans don't move around much. I already know that the moment of inertia is higher for the milk because of this. I don't understand how you can use this to determine which will reach the bottom first.

 

[AlephNumbers]

Exactly- the milk slides against the can, while the refried beans rotate. The milk does not rotate. What does this mean in terms of translational and rotational kinetic energy for the milk by the time it reaches the bottom?

 

[AlephNumbers]

*What does this mean in terms of translational and rotational kinetic energy for the milk as gravity does work on it?

 

[henry3369]

Exactly- the milk slides against the can, while the refried beans rotate. The milk does not rotate. What does this mean in terms of translational and rotational kinetic energy for the milk by the time it reaches the bottom?

Rotational kinetic energy is higher for the milk because the moment of inertia is higher. Doesn't that mean that the linear speed is lower for the can of milk because the total kinetic energy has to be the same?

 

[AlephNumbers]

No, the moment of inertia of the milk is not higher because, as you said, the milk slides against the can and does not rotate. No torque is applied to the milk, and the milk does not rotate. If the milk does not rotate, does that not mean that all of the work done on the milk is converted into translational kinetic energy?

 

[henry3369]

No, the moment of inertia of the milk is not higher because, as you said, the milk slides against the can and does not rotate. No torque is applied to the milk, and the milk does not rotate. If the milk does not rotate, does that not mean that all of the work done on the milk is converted into translational kinetic energy?

Ok that makes sense. The demonstration I watched explained how the can with the milk with was as if it was a hollow. If it is hollow, and the mass is still the same, wouldn't the mass be concentrated on the sides of the can; therefore increasing radial distance of the particles from the axis of rotation which in turn, increases the moment of inertia?

 

[AlephNumbers]

The fact that the milk is a fluid (and a relatively low friction fluid at that) effectively prevents the milk from having anything to do with the rotation of the can. The can exerts a force on the milk, but because fluids move around so easily, it really only affects the few particles directly in contact with the can. The milk really has very, very little effect on the moment of inertia of the can. Yet, the milk still has mass, and affects the force of gravity on the can.