- What happens when you put a ball on a turntable
and then start spinning the turntable?
Intuitively, you might expect the ball
to be flung from the turntable.
That's certainly what I thought would happen.
But look, instead the ball goes into a kind
of circular motion on the plate.
The ball does eventually slide off the turntable,
but it takes ages.
(turntable spins)
Compare that to the behavior of a penny,
as you would expect, the penny slides from the turntable.
(penny drops)
Or a toy car if, if a toy is pointing outwards,
it flies off the turntable quite quickly.
But even if it's pointing tangentially,
it still leaves the turntable.
What about if we start the ball off center?
What if we simply drop the ball onto the turntable,
then it flies off?
But what if I hold the ball
in position so it gets up to speed and then let go?
By the way, this is a child-sized pool ball
as opposed to a giant hand.
Look, when I let go, the ball just sits there,
and it maybe that's the first clue
as to what's going on, actually.
But before we dive in,
look what happens when I give the ball a little nudge.
We're now back to that circular motion,
except this time it's not circling around the center
of the turntable, which is quite surprising.
And here's something really crazy.
Let's compare how long it takes for the turntable to turn
versus the ball to go round one of those circular orbits.
I'll count the turntable turns out loud,
you keep an eye on the ball orbits.
1, 2, 3, 4, 5, 6, 7.
And look, in the time it took
for the turntable to rotate seven times,
the ball completed two orbits.
And that's always the way.
It doesn't matter how heavy the ball is,
it doesn't matter how big the ball is.
It doesn't even matter how fast the turntable is turning,
there will always be this seven to two ratio,
unless the ball is hollow, like this ping pong ball,
in which case it's 1, 2, 3, 4, 5.
It's a five to two ratio.
Why is that?
Well, let's first explain the stationary ball.
Why does the ball just sit there spinning
at exactly the right speed?
Well actually, why not?
There's a single point of contact.
The ball only knows about that one point
of contact, and that one point
of contact is moving to the right at a certain speed.
And if we assume that the force
of friction is strong enough to overcome any sliding motion,
and assuming that we have held onto
the ball until it's brought up to speed,
then we should expect it to keep spinning at that speed.
There's very little air resistance
because it's not moving through the air.
Some areas dragged along the surface of the turntable.
But generally speaking, because of Newton's first law
of motion, the ball should just keep spinning on the spot.
And crucially, there's no force pushing the ball outwards,
or inwards for that matter.
But what about that circular motion?
Well, when the ball is in steady state like this,
the spinning speed
of the ball matches the turning speed of the turntable.
To say that more precisely, the fastest moving part
of the ball's surface, the equator if you like,
is moving at the same speed
as the surface of the turntable directly beneath it.
But if I nudge the ball inwards,
well, the ball is now spinning faster
than the turntable directly underneath it, right?
Because this point on the turntable is moving more slowly
than this point on the turntable.
So, if the ball is rotating faster
than the ground beneath it, it will progress forwards
in the direction that it is rotating.
In other words, the ball will roll in this direction.
Let's analyze that a bit to see what happens next.
Remember, the ball is moving
with a relatively small velocity,
shown here with a yellow arrow as a result
of two much larger velocities being added together.
You've got the large velocity of the turntable
in this direction, and you've got
the slightly larger velocity of the ball relative
to the turntable in the opposite direction.
The velocity of the turntable doesn't quite cancel out
the velocity of the ball relative to the turntable,
so you have a small absolute velocity
of the ball shown in yellow.
But what happens when the ball finds itself
in this new position that it's rolled into?
Well, the velocity vector
of the turntable here shown in blue, got a little bit longer
because it's further from the center of the disc,
but also the direction has changed.
Again, the absolute velocity
of the ball is the yellow vector.
It's the sum of the blue and green vectors,
and you can see how it's changed
as a result of the changing turntable vector.
In reality, this change of direction would happen smoothly.
So we expect this curved path for the ball.
You can do a similar analysis
at different points around the orbit,
and you find the same thing,
that the ball follows this curved path.
In fact, if you do the proper mathematical analysis
linked to a paper in the comments, you find
that that curved path is in fact a perfect circle.
So with no slippage, no air resistance,
you get a perfectly circular path.
In reality, that circular path drifts a little bit.
I believe that's mostly because
of slipping as opposed to air resistance.
But the result is a spiral.
The ball drifts towards the center
of the disc until the center of the disc
is inside the orbit of the ball, and then slippage
of the ball causes it to eventually leave the turntable.
I've tried various things to reduce slippage.
One that worked quite well was to use a blank vinyl record
whose surface is shiny but also quite sticky
and it takes much longer for the ball to be removed.
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Why does the ball orbit twice
for every seven turns of the turntable?
Well, I've given quite a hand wave the explanation
of what's going on here,
but if you do the proper mathematical analysis,
eventually an equation drops out
for the orbital period of the ball, and it looks like this.
The orbital period of the ball is equal
to the orbital period
of the turntable, multiplied by this thing.
This thing is the mass of the ball multiplied
by the radius of the ball squared divided
by the moment of inertia of the ball plus one.
The moment of inertia of something is just like,
how hard is it to get the thing spinning?
And that certainly depends on the geometry of the thing,
and you can calculate the moment
of inertia for spherical geometry,
in other words, a ball,
by integrating all these little slices.
The details don't matter, but it comes out as the moment
of inertia is equal to 2/5 of the mass
of the ball multiplied by the radius of the ball squared.
Rearranging that, we get that the mass
of the ball multiplied by the radius squared divided
by the moment of inertia is equal to five divided by two.
But look, we have that exact same expression
in our original equation.
So we can substitute 5/2 into that equation.
5/2 plus one is 7/2, so we have that the orbital period
of the ball is equal to 7/2 multiplied
by the orbital period of the turntable.
In other words, for every seven turns at the turntable,
you get two orbits of the ball.
The reason it's different for a hollow ball
is because the moment of inertia calculation
for a hollow ball is different,
because all the mass is on the outside,
the integration is different and you end up
with a ratio of five to two instead.
You can also think about this in terms
of the Coriolis force.
The Coriolis force you might know is a fictitious force
that you only experience when you're in
a rotating reference frame,
that's a non-inertial reference frame.
And because we're analyzing this from the lab frame,
if you like, which is inertial, then it's maybe
not that useful to think in terms of the Coriolis force.
But in the same way that winds traveling north
in the Northern Hemisphere are deflected to the east,
when I nudge this ball
towards the center of the disc, it's deflected to the right.
Something else quite cool
is if you tilt the turntable.
Look, if the turntable isn't spinning and I tilt it,
obviously the ball rolls off.
But look, when the turntable is spinning,
instead of the ball rolling in the direction of the tilt,
it rolls perpendicular to that,
which is very reminiscent of gyroscopes, isn't it?
When you apply a force to a gyroscope in one direction,
it moves perpendicular to that.
That's elegant, isn't it?
Or maybe it's a coincidence.
I don't know.
I mean, sure, there's a link there.
It's like spinning things, you know?
I dunno.
Maybe discs also behave interestingly on turntables.
That's quite fun, isn't it?
So there you go.
Balls and other round things behave weirdly on turntables.
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