2 ramps each 100ft tall. 1st ramp 1,000ft long, second ramp 10,000 ft long. Would a ball bearing be traveling at approx. the same speed at the bottom of these ramps?
Ignoring friction, the two ramp speeds would be identical, as you have the same potential energy to start, which is converted to the same kinetic energy at the bottom of the ramp. However, once you factor in rolling friction, the longer ramp represents more time spent rolling, and a greater normal force pressing against the ramp and compressing the ball bearing as it rolls, so more of the energy is lost due to friction.
Nvm misread the question
Would the longer ramp benefit in lower air resistance even though it is exposed for a longer time?
Since the ball on the long ramp is subject to the same aero drag/speed relationship as the other, but covers more distance at each intermediate speed, the total mechanical work done by drag force (and therefore the total reduction in final velocity) would end up being greater.
I wasn’t considering rotational energy, Any affect there?
If we’re talking the real world, and an ordinary sized ball bearing (say, half an inch), then in both cases the bearing will easily be in equilibrium with drag forces (aero, friction) by the end of the ramp. But the short ramp is steeper, so the bearing on that one will be moving faster.
If the ball bearing or the ramp isn’t perfectly uniform, a ball bearing on a 1:100 slope might not even roll at all.
No distinction between two different ramps, though it will make a considerable difference between different sorts of rollers. Given a solid sphere, a solid cylinder, a hollow sphere, and a hollow cylinder, all rolling down the same ramp, the speeds will be in that order (solid sphere fastest, hollow cylinder slowest).
The force paralell to the ramp is the sin of the angle of the ramp, times the acceleration due to gravity, times mass.
Is the ramp actually 10:1 and 100:1, or more accurately - since the ramp is 100’ or 1,000’ not the horizonatal distance is that… the ramps horizontal runs are sqrt(990,000) and sqrt(99,990,000) respectively.
By congruent triangles rule, the angle we need to calculate the force along the ramp is the same as the angle between the ramp and the flat surface. Θ=arcsin(100/1,000) or arcsin(100/10,000) respectively.
Which implies our accelertation force down the ramp is *g·(100/1000) etc. F=ma where m is the mass of the ball bearing.
But F will be consumed both by pulling the ball bearing down and creating rotational energy.
The moment of inertia of a ball bearing (the internet tells me) is I=(7/5)Mr^2
Need to do more math, but it seems to me the potential-to-kinetic argument seems correct.
2/5. It’d have to have mass outside of its radius to have a moment of 7/5.
Doh! The website I was reading was actually 5/7.
No, that website said 7/5, but it was also taking that with respect to the edge of the ball, rather than the more usual with respect to the ball’s center of mass. Though I don’t know why that person is doing a force analysis at all: The problem is much easier in terms of energy.
On further reflection - for a sliding mass (i.e. a sperical chicken with no friciton, in a vacuum)
F=ma
E=Fd=Fma
But we said F along the slope was g(height h/slope distance d)
So those d cancel out, showing that E=mg(height h) which is what we expect for convertin potential energy to kinetic.
I’m too lazy to look up exactly what the moment of inertia and force needed to make a ballbearing roll, the angular momentum and energy, etc. so let’s call it mX.
The force pulling the ball bearing down the slope is partly contributing to its velocity, and partly to increasing its angular velocity
F=ma + mX
So ma will be reduced by the amount of force required to induce angular momentum.
However, all I’m seeing is that the resultant velocity and angular velocity is irrelevant of the length of the slope since it cancels out. Friction depends on the quality of the ball bearing and the ramp but my guess is a longer ramp means more friction, it depnds on distance all else being equal.
However, something I read about aircraft once said air resistance is the 4th power of speed, so the faster ball bearing will have much more air resistance.