Suppose I built a track with up and down humps like a roller coaster track. Slightly concave so a ball bearing would stay in the center while rolling. The ball would start rolling on a downhill slope higher than the highest hump on the track.
Would the density of the metal used to make the ball bearing affect how high I would have to start the roll? Ignore air resistance as a factor.
Nope! Not unless there was some kind of track friction that mattered.
What if it was a spherical cow?
A spherical cow would roll more slowly because of squishing and sloshing.
Ignoring the friction of the track, as long as the starting height is above the hump of the next hill, the ball will have enough energy to clear the hump, regardless of the mass of the ball.
As above.
This is basically Galileo’s (likely apocryphal) experiment. Take balls of the same material but different weights and drop them from the Tower of Pisa. They fall at the same rate (or close enough since wind resistance is near negligible for them).
Ditto the NASA experiment on the moon. Drop a feather and a hammer at the same time on the moon and they fall at the same rate.
Frictional losses (whether wind resistance or against the track) would be the deciding factors, rather than the density of the objects.
Edited to add: that said, frictional losses are often correlated with density, so there is that.
It’s not clear to me that the Feather drop experiment applies to the op. It’s not the rate of fall that gets the ball over the hump it’s whether or not the ball has enough energy. It seems to me that since the rate is constant; the mass of the object being the variable would determine whether or not there’s enough energy.
True, strictly speaking, but if we are to ignore frictional losses, the energy in the system remains the same.
It’s the friction against the track that will matter, and that will depend more on the mass and surface properties of the material rather than the density directly.
Ah yes, my beverage of choice from New Gla…
The mass factor cancels when converting between kinetic and potential energy if there is no other energy gain or loss.
Absent friction, the mass of the object also determines how much energy it has. The Energy it gains going down is directly/strictly proportional to mass, and the energy required to go up is directly/strictly proportional to mass.
Absent friction, the size of the object does not matter, and the mass of the object cancels going down and up, and since neither the size nor the mass matters, the density does not matter.
I reiterate what has been said: ignoring friction, as long as the ball starts at the highest point on the track, the ball will make it to the end. Of course, in the real world there is always friction so you have to be sufficiently higher to account for that.
The real reason I’m posting is because I just can’t let a spherical cow reference go by without linking this video:
The density (and hence total mass) of the ball doesn’t matter in the least if you ignore all the resistive forces (rolling friction and air resistance). But in the real world, it matters a lot. Consider in one case a ping-pong ball, and in the other case an equally smooth solid steel ball of the same size. The steel ball will have a lot of gravitational potential energy at the top of the incline, translating quickly into kinetic energy, which would be so much greater than the energy lost to resistive forces that the starting point would only need to be slightly higher than the highest point of the track. But a ping-pong ball would have to start much higher in order to make it to the end because it would proportionately lose much more of its energy.
Right; in textbook-land where we get to ignore dissipative forces, the size and mass don’t matter. But in the real world, size and mass do matter, because we can’t ignore those forces.
E=1/2 MV^2 - the energy produced by going down a ramp is lost going up; but with no friction, no air resistance, the energy converted from potential to kinetic on the way down is also enough to return to the same height.
Of course, with a rolling mass, there is also energy converted to to rotational energy - but wait! There’s no friction, so there’s no rotation. A mass will rotate because friction causes the bottom of the rolling mass to “stick” to the track. Instead it’s basically a ball bearing on a Luge track, it slides down without rotating and slides back up the other side with no loss or gain of rotation. Spherical cows with no friction have consequences…
If it rolls up the next bump, the rotational energy is recovered just like the translational energy is recovered.
However, as discussed in the 1960’s, and repeated in high school physics experiments, the effect of rotational energy can be used to demonstrate that some early renaissance physics experiments were ‘thought experiments’, not actual physical measurements – because the published results for rolling a ball down a chute don’t match what you get in high school physics experiments, which do match what you get when you include rotational mass in your theory.
For a rolling object, the distribution of mass is going to matter, too. Consider a hollow ball, versus a solid one (of the same size and total mass) - it’s going to take longer for the hollow one to accelerate, because its moment of inertia is higher.
Which reminds me of another oddity that will work without external friction. Try rolling two eggs down the track - the hard-boiled one will behave differently than the uncooked one.
We are so used to the effects of friction that a frictionless environment is counter-intuitive. I recall some quote from Aristotle or some Greek explaining why an arrow did not slow down like things slid across a floor. he used the excuse arrow had feathers, and because feathers came from birds, they wanted to fly, caused the air to rush past the feather and keep the arrow flying fast. Even when Newton said “a body in motion will tend to remain in motion” that was counter to real life experience and took some deep thinking.
in the experiment we are discussing - if there is friction, resulting in the ball bearing rolling on the way down, as long as there is enough friction for the rotational momentum to drive the bearing going upward also, then all the energy - rotational and velocity - will be recovered. (Steel bearing on steel track will transform very little of the friction into non-velocity energy like heat or deformation of the materials).
Also consider a bowling ball rotating the wrong way as it skids down the alley - until it slows enough to grip, and then funny things can happen…
In general, a rolling (without slipping) object will have a kinetic energy of c*m*v^2, where the value of c will depend on the mass distribution. This will result in an object with the mass distributed far out, like a hollow ring, rolling slower than one with the mass tight in, like a solid sphere (and all rollers will be slower than a frictionless slider). But the energy’s all still there and available, and so if any roller or slider will make it over a given frictionless hill, they all will (just not all at the same time).
My egg example was intended as an exception to that - because some of the kinetic energy will be wasted in the lossy process of spinning up the internal goo (and thus heating it) and not be available to get the egg rolling over the hill. But I could be wrong about how big an effect that is.