Rotational vs Translational Velocity

[Dreamer29]

Two objects of equal mass and center of gravity are placed at the top of an incline plane and given an initial velocity v. The µs and the µk of both objects are identical. One is a ball and one is a block. If both objects have the same contact surface area, which will hit the ground first?

Answer: Ball

My reasoning: Both the ball and the block were at the same initial height upon release. The only difference between the two would be that the ball has rotational kinetic energy and translational KE, whereas the block only has translational. My reasoning was that since the two started at the same PE, that PE for the ball would have to be "divided" between the rotational and linear KE terms, whereas the PE for the block only becomes linear KE.

PE_ball= 1/2mv^2+ 1/2Iw^2
PE_block= 1/2mv^2

Thus, because the PE of the ball is divided between the two terms, the final linear velocity would be less than that of the block.

Can anybody help me out here? What am I doing wrong?

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[V5RED]

Two objects of equal mass and center of gravity are placed at the top of an incline plane and given an initial velocity v. The µs and the µk of both objects are identical. One is a ball and one is a block. If both objects have the same contact surface area, which will hit the ground first?

Answer: Ball

My reasoning: Both the ball and the block were at the same initial height upon release. The only difference between the two would be that the ball has rotational kinetic energy and translational KE, whereas the block only has translational. My reasoning was that since the two started at the same PE, that PE for the ball would have to be "divided" between the rotational and linear KE terms, whereas the PE for the block only becomes linear KE.

PE_ball= 1/2mv^2+ 1/2Iw^2
PE_block= 1/2mv^2

Thus, because the PE of the ball is divided between the two terms, the final linear velocity would be less than that of the block.

Can anybody help me out here? What am I doing wrong?

Replace the block and ball with a car with its parking brake on and a car in neutral.

It is pretty obvious that the car in neutral will reach the bottom first.

For a rolling object(the ball or the car in neutral), static friction causes it to roll. This is why we have rubber tires. The friction is of benefit to accelerating a thing that is rolling.

For a sliding object(the block or the car with its parking brake on), static friction can prevent it from moving at all. If the object overcomes the static friction, then its kinetic friction will inhibit its sliding.

 

[Dreamer29]

Ok, gotcha on the friction. Thanks!

So my interpretation of the question w/ regards to PE and KE is wrong?

 

[V5RED]

Ok, gotcha on the friction. Thanks!

So my interpretation of the question w/ regards to PE and KE is wrong?

It led you to the wrong answer didn't it?

 

[Dreamer29]

Yeah, it did. But I'm more concerned that I'm just not understanding rotational energy or that I haven't been applying it in an appropriate manner.

 

[V5RED]

I have not looked at rotational energy for a while since it isn't an MCAT topic, but one thing stands out to me that you missed.

You forgot to include the heat transfer from the kinetic friction when the block slides along the ramp. That will cost it quite a bit of its energy.

The ball loses no energy to friction because it is rolling(static friction), not sliding. Friction only results in heat transfer if there is a rubbing of surfaces, but the ball is just pushing off the surface to continue rolling.

 

[Dreamer29]

I have not looked at rotational energy for a while since it isn't an MCAT topic, but one thing stands out to me that you missed.

You forgot to include the heat transfer from the kinetic friction when the block slides along the ramp. That will cost it quite a bit of its energy.

The ball loses no energy to friction because it is rolling(static friction), not sliding. Friction only results in heat transfer if there is a rubbing of surfaces, but the ball is just pushing off the surface to continue rolling.

Thanks! I guess I was missing the point of the question

 

[johnnybravo19]

Don't use math if you don't need it.

The ball has less friction as opposed to the block. Think it through.

 

[Dreamer29]

Right, intuition tells us that the ball will go faster than the block.
Work done by friction in translation and rotation of the ball are equal and opposite. Thus, no work is done by friction in rolling; whereas friction is clearly at work on the block.

But I'm still wondering where and how I've misapplied the work-energy theorem? If the total kinetic energy of a ball is divided between linear and rotational energies, wouldn't the linear energy thus be less?

 

[V5RED]

Right, intuition tells us that the ball will go faster than the block.
Work done by friction in translation and rotation of the ball are equal and opposite. Thus, no work is done by friction in rolling; whereas friction is clearly at work on the block.

But I'm still wondering where and how I've misapplied the work-energy theorem? If the total kinetic energy of a ball is divided between linear and rotational energies, wouldn't the linear energy thus be less?

There is less total energy in the block because it is lost as heat.

Your most obvious error in the application of the theory is the fact that you neglected the heat loss from the block. The block will have less total energy at the bottom of the ramp than the ball does because it lost energy as heat. Even if some of the energy of the ball is in rotational energy, it still will have far more energy at the bottom on the ramp than the block, and it will have far more translational kinetic energy.

 

[Dreamer29]

Thanks for your help.
I've figured out my mistake!