Force required to keep objects in orbit.

[the7joker7]

Homework Statement

Neutron stars are extremely dense objects that are formed from the remnants of supernova explosions. Many rotate very rapidly. Suppose that the mass of a certain spherical neutron star is twice the mass of the Sun and its radius is 10.0 km. Determine the greatest possible angular speed it can have for the matter at the surface of the star on its equator to be just held in orbit by the gravitational force.

Homework Equations

F = G*$$\frac{v^{2}}{r}$$

The Attempt at a Solution

The way I see it, r = 10,000 meters and mass = 3.977 *10$$^{30}$$ kilograms. So I used the equation above. to get...

F = (6.67*10^-11)*$$\frac{v^{2}}{10000}$$

Now, I think I'm supposed to find out what F should be, and solve for V, but I'm not sure what I'm supposed to make F be.

 

[blochwave]

I don't know why you tossed the gravitational constant into the equation for centripetal force

You have two forces going on here, centripetal and gravity, but what IS actually causing the centripetal force? What is holding the mass in place?

Anyways this proceeds extremely similarly to the question "how fast is a satellite at such and such orbit around the Earth traveling?" Note that I didn't give the mass of the satellite. You find that speed, and that is THE speed for that orbit. If you slow it down, it falls into a closer orbit, if you speed it up...

 

[tiny-tim]

Homework Equations

F = G*$$\frac{v^{2}}{r}$$

Hi!

You need two relevant equations: one for the acceleration (which you've got), and one for the gravity.

I think you confused yourself by writing G instead of m in the acceleration equation, which made you think you'd already written the gravity equation!

Oh, and you've been asked for angular speed, not the ordinary speed, v.

 

[the7joker7]

So you're saying the other equation I need is...

F = G$$\frac{m_{1}m_{2}}{r^{2}}$$

So at this point I have...

F = 3.977 * 10$$^{30}$$$$\frac{v^{2}}{10000}$$

and...

F = 6.67*10$$^{11}$$$$\frac{3.988*10^{30}*m_{2}}{r^{2}}$$

Now what do I do. :/

 

[blochwave]

Well why use the mass of the planet for the centripetal force equation?

For the gravitational force you're looking at some mass, whatever it may be, that you denoted m2, located at the surface of the neutron star

This same mass is experiencing the centripetal force, which is being caused by gravity.

 

[tiny-tim]

So you're saying the other equation I need is...

F = G$$\frac{m_{1}m_{2}}{r^{2}}$$

Yes!

Your first F, $$m_2v^2/r$$, is the force needed to keep a mass $$m_2$$ in that circle of radius r.

Your second F is the gravitational force on a mass $$m_2$$ at a distance r.

You need the actual force to be equal to the force that's needed!

So you just make them equal.

(I suspect what you're finding confusing is that there seem to be two forces for no reason. It would be clearer if you went back to Newton's second law, force = mass x acceleration, or F = m x A, and wrote A = $$v^2/r$$: same equation, but only one F, so less confusing.)

 

[matthewpowers]

Yeah, if you write the two equation symbolically, and keep your masses straight, it should become clear.