Why Do Planets and Stars Revolve Around Their Center of Mass?

[erisedk]

Homework Statement

From my book--
"In motion of a planet round the sun we have assumed the mass of the sun to be too large in comparison to the mass of the planet. Under such a situation, the sun remains stationary and the planet revolves round the sun. If, however, masses of sun and planet are comparable and motion of sun is also considered, then both of them revolve around their center of mass with same angular velocity but different linear speeds in circles of different radii. The center of mass remains stationary."

Homework Equations

The Attempt at a Solution

Please explain the bold lines. Why do they revolve with the same angular velocity? Why about the centre of mass? Why does the centre of mass remain stationary? I get that the acceleration of centre of mass should be zero because there is no external force. But why does the COM itself remain stationary?

 

[BvU]

What would be needed to change one of the two angular velocities ?

And your feeling about the com remaining stationary is justified: the two can move through space at any constant speed without changing the physical situation.

 

[ecastro]

This is due to the change of reference frame.

 

[erisedk]

A torque would be needed to change angular velocity. But my question is why do they have the same angular velocities in the first place.

 

[BvU]

Because there never was any torque: gravity works from com to com...

I realize never is a big word, but the context makes it plausible one can say that in this exercise

 

[erisedk]

I get that there is no torque, but I still don't get why angular velocities of two different bodies about the COM is the same. And I still don't get why COM is stationary.

 

[ecastro]

The center of mass is stationary according to its own reference frame.

 

[BvU]

That's also true for a body in free fall. Or when rotating violently around an axis that doesn't go through its center of mass.

Here they mean stationary or moving at a constant velocity (in the absence of other forces) in an inertial frame of reference. Like in Newton's first law.

(The exercise started with a "stationary sun" -- same difference -- and the subject is a correction to that in m_planet/m_sun)

 

[gneill]

At all times you can draw a line connecting the centers of the co-orbiting objects. The center of mass must lie along that line (obvious from the geometry and definition of center of mass). Without external forces or torques acting on the system the center of mass of the isolated system must obey Newton's first law. So if you choose a frame of reference where the center of mass happens to be stationary, then it must remain stationary.

So it's as though the two objects were connected by a rigid rod that pivots about the center of mass (we're assuming circular orbits here so the radii have fixed lengths). You should be able to see that the two lengths of the rod on either side of the pivot cannot have different angular velocities --- the "rod" could not be rigid otherwise.

 

[mfb]

A torque would be needed to change angular velocity. But my question is why do they have the same angular velocities in the first place.

Imagine they would move around the center of mass at different angular velocity - then at some point, both objects would be at the same side of the center of mass, in obvious violation to the definition of the center of mass which has to be between them.

 

[ehild]

How is the centre of mass and its velocity is defined: If there are two spheres (with individual centres of mass in the centre of the spheres)

$\vec r _{CM}=\frac {\vec r_1 m_1+ \vec r_2 m_2}{m_1+m_2}$ .

$\vec r _{CM}-\vec r_1= \frac {m_2}{m_1+m_2}( \vec r_2-\vec r_1)$

The CM is on the line connecting the centres of the spheres. If one mass turns around the CM by a certain angle θ, the other mass has to turn with the same angle, so as the line connecting them still go through the CM.

 

[erisedk]

Thank you everyone! I finally get it :D
And Merry Christmas :)