Subtitled: is this the correct way of looking at this problem?
In high school physics, we debated which of two balls rolling down the same incline would get to the bottom first: the one in a deep slot or in a shallow slot. Intuition told me that the shallow slot would finish first, but my three-dimensional vectors were not up to the task, so I justified my belief by just looking at the forces and energy transfer, namely:
– In order to go the same distance down the plane, the ball in the deeper slot must rotate faster.
– If the ball in the deeper slot finished at the same time as the ball in the shallow slot, both balls have the same forward velocity.
– But this means that the ball in the deeper slot would have a greater energy than the other one, because it also has a greater rotational energy.
– Since both balls transferred equal amounts of gravitational potential to kinetic energy, this cannot be the case.
– Therefore, the ball in the deeper slot will finish slower.
Experimentation showed that I was correct, but are there any holes with this line of reasoning, as opposed to adding the force of gravity to the force of the angled ground on the ball? One that I can think of is that it assumes a uniform acceleration, but I’m not sure this objection is valid.
I think I understand the OP’s scenario, but it was stated poorly.
Imagine two spheres, each 10 inches across.
Imagine an inclined plane with two slots parallel to its slope. Both slots are ten inches deep, but one slot is one inch wide, and the other slot is nine inches wide.
Set both spheres at the top of the incline, such that one sphere is straddling one slot, and the other sphere is straddling the other slot.
The sphere on the 1-inch-wide slot will be almost tangent to the inclined plane; its center of rotation will be 4.98 inches above the plane, so it can travel at a high forward speed with modest RPM.
The sphere on the 9-inch-wide slot will ride significantly deeper in that slot; its center of rotation will be only 2.18 inches above the plane, so it can’t achieve high forward speed without also spinning at high RPM.
The sphere on the narrow slot will reach bottom first.
It took me awhile to visualize this as well as like I said I am not good at 3-D physics. But imagine the two grooves that the ball is touching while it is travelling down the slot. Then imagine a circle drawn on the ball as it is travelling down this groove. For a [wider] slot, these circles will be smaller, and closer to the edge, than a [narrower] slot. (For a nonexistent slot, the circle will be the ball’s circumference).
One full rotation is needed for the ball to go through each of these circles, no matter how big the circle is. Yet the smaller circles permit the ball to travel over less displacement. So, in order to for the ball [that is sitting deeper in the] groove to travel the same distance down the incline, it needs to rotate more times, therefore it needs to rotate faster for the same velocity down the incline.
ETA: Machine Elf describes the scenario correctly, as well.
Deep vs shallow seems to be a poor way to describe it. Wide vs narrow is better, given the assumption that the ball rests upon the edge of the slot.
If the slot is rounded like the balls are rounded, there would not be a rotational difference between the slots. At that point, you’re talking friction differences.
I think it might be simpler to imagine a ball rolling down two shafts, rather than a groove. The farther apart the shafts are, the faster the ball must rotate.
The ball in the deeper wider slot has less potential energy because it sits lower on top.
When the deep wide slotted ball reaches the tape at the finish line it will be contacting the tape above its mid section resulting in a delay in contact.