k4ff3 said:
I have corrected the rotational PE typo, I meant of course rotational KE.
Yes, I caught that.
What does it mean that the point of application of the force do not move? Why does it move when you apply the same force on the COM?
The instantaneous speed of the bottom of the cylinder as it rolls down the incline is zero. (Otherwise it would be sliding.) The speed of the COM of the cylinder is not zero, of course.
By leaking I meant that the body was constantly losing PE, i.e PE was transformed. Anyway, I understood you explanation - see if I got this right:
(Arrow here means "is being transformed to")
a) Rolling case: PE -> translational KE & rotational KE
b) Sliding case: PE -> translational KE & internal E to the "thing" that exerts the "braking force" f
c) Sliding case without any other force than gravity acting: PE -> KE
In a) and b) the translational KE is the same. However, in a) the PE transforms to energy that stays within the system, namely rotational energy. Thus, the total Mechanical Energy in a) will be conserved - as opposed to what happens in b): here energy is being transferred out to the environment (like the book). Therefore: Mechanical Energy is conserved in a), not in b) - but the translational KE are in both cases the same! The race will be a tie.
In c) the Mechanical Energy is clearly conserved, the sliding body will gradually lose PE in favor of gaining more (trans.) KE. No potential energy is lost to the environment, or to making rot. KE. Since the trans. KE and trans. velocity are connected, the cylinder in c) obviously wins the race!
Sounds good to me!
Can you please elaborate on this? My English isn't too good.. "In order for an external force to perform work, the point of application of that force must be displaced". Can you say this sentence with other words?
Work requires force X displacement, not just force. Since, as pointed out above, there's no instantaneous motion of the point of contact of the friction force, there's no instantaneous displacement--and no work done. This is a subtle point.
Here's another, totally different, example where a force acts to change the motion of something yet no work is done. Crouch down, then leap into the air. The ground exerts a force on your feet, propelling you upward, yet the ground doesn't move and no work is done. So where does your KE come from? (Answer: You've transformed internal energy--chemical energy in your muscles--into translational KE.)
Another example, involving static friction. Driving along, you step on the gas to accelerate your car. Assuming no slipping of the tires, the friction is static friction. And, again, the instantaneous speed of the tire patch in contact with the ground is zero so no work is done, even though, obviously, the ground is what propels the car forward. Where does the energy come from? The car converts chemical energy into mechanical energy--it burns gas.
Thank you very much! I feel like I'm on the verge of an epiphany :)
I think you are very close. As I said, some of these concepts are subtle, so don't be so hard on yourself.
Just for fun, why not use Newton's law to fully analyze the motion of the rolling cylinder? You can solve for the static friction required and the acceleration of the cylinder. And you can use the 'rolling without slipping' condition to relate rotation to translation.
