What takes more energy rolling a ball up a hill or carrying it up

[curiouschris]

My Son asked me a question, I am not sure how to answer it.

It feels easier to push or roll a heavy ball up a hill than it is to pick it up and carry it up the same hill.

The ball has the same potential energy at the top of the hill no matter how it got there so it would appear to be deceptive?

Is the amount of energy required for both cases the same?

How do I calculate the energy used in either case?

What about for a cube?
The cube rolls end over end, friction prevents the cube from rolling back down (moderate slope) does it just feel easier because of the small rest periods in between? or is it actually easier than carrying the cube up because much of the weight is resting on the ground at all times?

Thanks in advance.

CC

 

[CWatters]

That's quite a difficult question to answer. Once the ball is stationary at the top of the hill the potential energy that the ball has gained is indeed the same. So the short answer is that it takes the same energy to raise the ball up the hill no matter what method is used.

A longer answer is that there is a significant difference that depends on the efficiency of the "machine" or method used. For example if you were to drag it up there there would be losses due to friction with the ground. If it was rolled up there the frictional losses might be lower. If carried up in a lorry there are losses in the engine etc. If transported up by human effort there is the problem of quantifying the efficiency of the human body when moving a load in different ways. Humans turn food, water and air into mechanical energy with an efficiency of somewhere between 20 and 40% depending if it's respiration or food energy (according to wikipedia). I think the only way to tell if rolling of carrying is more efficient is to try and measure it, perhaps by measuring the volume of breath consumed?

 

[curiouschris]

Thank you

I continued to think of the problem afterwards and I realized that the reason it feels a harder when carrying the ball or block is because not only are we attempting to climb a hill but the weight of the object is also consuming energy. Even standing still it would require an expenditure of energy just to hold the object.

So I guess that covers the "apparently harder" portion of being more difficult but the end over end block is still fascinating.

Thinking about it each time its rolled over, the center of gravity reaches a point which is halfway between opposite edges (assuming uniform weight distribution) but then rolls onto the next side, therefore its a case of a step forward and half a step back.

I suppose it would not be too hard to work out how much energy is required to raise the block onto its edge and then multiply that by the length of the slope divided by the length of each side. The energy used in raising the block onto its side would need to account for the angle of the slope as well.

Sometimes the hardest part is picturing what you want to find out in your head, after that things start to fall into place.

 

[CWatters]

When the cube "falls over" onto the next face some of the "excess" PE needed to raise the cube onto an edge would be recovered (indirectly) because the cube ends up higher up the hill. However as far as I can see the rolled cylinder would allways win because not all of that energy could be recovered.