Rotational Inertia

jaumzaum

I was taught there was two types of inertia. The translational inertia and the rotational inertia. If in the Earth for example, that has a translational motion around the Sun, and a rotational motion around itself, all forces disappeared, it would follow an uniform rectilinear motion tangent to its previous translational motion around the Sun, and would continue in uniform circular motion around itself.

But would would happen if in the ball, that was initially moving around the red axis below, suddenly disappeared all the external forces? Consider the distance of the axis to the center of the ball is R/2 for example.

cwilkins

I don't know if this is a physically meaningful question to ask. The star and planet example is just a thought experiment meant to illustrate Newton's first law. Both conservation of energy and momentum would be violated if the ball suddenly disappeared.

In your analogous thought experiment, I don't think anything would happen. In reality, either the fields holding the ball in orbit would have to absorb the ball's momentum, or the ball would have to radiate away its energy and momentum in some other way.

jaumzaum

Actually it's not the ball which disappear, the forces are. I would like to know what type of movement the ball would follow in this case, due to its inertia.

cepheid

What forces?

Doc Al

At any point in time, the center of mass of the ball has some translational velocity and the ball has some rotational velocity. If the axis somehow disappears, and no other external forces act on the ball, then the ball's center of mass will continue to move at whatever translational velocity it has at that moment and the ball will continue to rotate about its center of mass at whatever rotational velocity it has at that moment.

xAxis

Why wouldn't it be? Sun for instance moves in a similar way, except that the centre of rotation is closer to its centre of mass.

cwilkins

I misunderstood your question. The ball will move in a straight line from wherever it was in a direction tangent to its original orbit. If the ball was initially spinning it will continue to do so.

jaumzaum

Thanks

e.chaniotakis

Well, as cwilkins very nicely phrased, once the external forces are shut down ,earth will move tangentially towards the 'way it was going'.
Now, about the rotational motion, all the forces acting on earth are central . therefore angular momentum will be conserved and it will keep rotating with 24hrs period.

xAxis

It will actualy rotate a little faster than that. It will make 366 rotations a year as it has to conserve spin.

e.chaniotakis

Can you explain a bit more? Afraid I lost you

AlephZero

Wmat xAxis said might be misleading. The earth's rotation speed would not change when the external forces stopped.

The point is that a "day" is defined by the earth's rotation relative to the sun, not relative to distant stars. There are 366 rotations relative to the stars in a year of 365 days, because the earth also rotates once around the sun.

That is why the same stars appear rise and set about 4 minutes earlier on each successive night. 4 minutes = 1/365 of a day, approximately.

DrZoidberg

The earth takes 23 hours, 56 minutes and 4.0916 seconds to rotate once around it's own axis. So it makes 366 rotations per year. The reason a year has 365 days is because we are additionally also rotating around the sun so after one full rotation the sun doesn't have the same position relative to the earth's surface. So that means if the gravity between sun and earth were to disappear, we would keep rotating at 366 rotations per year just as we are now.

xAxis

I ment faster than once in 24 hours exactly what AlephZero and DrZoidberg explained.
Take the Moon for example. We always see the same side of the moon which means that relative to the Earths direction it doesn't spin. But it still makes one revolution in 28 days, so if we magically removed the earth, it would spin once in 28 days.

e.chaniotakis

I see your point and thank you for that.