Kinematics - Two bodies revolving around each other question?

[Vckay]

Homework Statement

Hi. I am a newbie to the Physics forum. I am actually attempting to figure out a problem where there are two bodies forced to revolve around each other. The only constraints that would exist are the distance of separation between the two bodies is assumed to be constant. And there might be an initial angular velocity.
Intuitively I believe that if they are given initial equal angular velocities, the entire system will move in a straight line. And if they are having different angular velocities, the system will tend to spiral. Can anyone tell me if this is right?

Homework Equations

Here is where I get stuck...My initial attemp was to design parametric equations in the form :
x1(t)=x2(t)+Rcostheta1
y1(t)=y2(t)+Rcostheta1

x2(t)=x1(t)+Rcostheta2
y2(t)=y1(t)+Rcostheta2

Now both are simultaneously varying..I am confused as to how model the same..So any tips on as to whether I am on the right track or whether I have to consider acceleration also for them to maintain that distance of separation would be appreciated.

 

[dirk_mec1]

Can you give a drawing? It's unclear to me now what you're talking about.

 

[dynamicsolo]

The only constraints that would exist are the distance of separation between the two bodies is assumed to be constant. And there might be an initial angular velocity.
Intuitively I believe that if they are given initial equal angular velocities, the entire system will move in a straight line.

Here is where I get stuck...My initial attemp was to design parametric equations in the form :
x1(t)=x2(t)+Rcostheta1
y1(t)=y2(t)+Rcostheta1

x2(t)=x1(t)+Rcostheta2
y2(t)=y1(t)+Rcostheta2

Your physical intution seems to be right. The two bodies must be on a single line rotating about the center of mass of the system, so they will have the same angular velocity. You don't say whether the two have the same mass, but that does not alter this statement; it just means the center of mass isn't halfway between them. Also, since the forces the bodies exert on each other are equal and opposite, the net internal force on the system is zero. So the center of mass will move through space at a constant velocity.

What that means for your equations is the following:

1) the center of mass moves at constant velocity, so both bodies can have their positions referenced to that center;

2) the bodies will move at the same angular velocity, so a single angle $$\theta = \omega \cdot t$$ can be used for both; thus

3) both bodies move on circles about the center of mass;

4) since the center of mass is not necessarily halfway between them, the two "orbital" circles may have different radii; and

5) since the two bodies are on "opposite sides" of the center of mass, the trig functions involved in their coordinates should have opposite signs.

So your equations will look more like, say:

$$x_1(t) = X(t) + R_1 cos(\omega t)$$
$$y_1(t) = Y(t) + R_1 sin(\omega t)$$

$$x_2(t) = X(t) - R_2 cos(\omega t)$$
$$y_2(t) = Y(t) - R_2 sin(\omega t)$$

I'll leave you to work out what the radii are, in terms of the masses of the bodies and how omega relates to the period of the system. You didn't say whether this is a gravitationally bound system (like, for instance, a simplest possible binary star), so I've ignored any connection to the accelerations here. These equations will be true for any attractive force in this idealized system.