# Speed of two objects orbiting around a central axis

This question has been bugging me for quite some time and i'm not too sure if i'm missing out some important factor in the understanding of this scenario:

Given two objects of equal mass and size both orbiting around a central axis x where ObjectA is distance n away from x and ObjectB is distance 2n away from x.

Now lets assume some force f is being applied to ObjectA and force 2f being applied to ObjectB such that it takes the exact same time for ObjectA and ObjectB to do one full orbit around x.

Logically, there will have to be a greater force being applied to ObjectB to make it travel faster, as it has a greater circumference around x, in order for it to complete the orbit at the same time ObjectA completes its orbit.

However, if we were to join ObjectA to ObjectB via a long pipe for example and only apply a single force f on the pipe such that both objects will orbit around x at the same time.

How is it now that both objects will complete the orbit at the same time using one single force but yet ObjectB still has a larger circumference which means it has a greater distance to travel in order to complete the orbit?

P.S. Clearly i'm no physics expert and lack the knowledge for the proper terminology for the concepts I described above.

How is it now that both objects will complete the orbit at the same time using one single force but yet ObjectB still has a larger circumference which means it has a greater distance to travel in order to complete the orbit?

As you have connected the two identical objects by a pipe (of course a rigid body) and applying a force on the pipe such that a point on the pipe- length is fixed and the pipe is rotating about an axis passing through the point . As you have suggested the point fixed is off centre - one body is at distance n and another at 2n from the fixed axis.

Though you are not fixing the point of application of the force f being applied, say its acting on the body A. The moment of the force f x n will act and rotate it around the fixed axis this torque will act and produce angular acceleration such that

Torque = moment of inertia of the rigid body. Angular acceleration

very much identical to force acting on a body producing linear acceleration. now the speed generated will depend on the torque i.e. in turn on the moment of inertia which is kept constant. now the time period for one complete rotation will depend on where you applied f-i.e. how far from the axis of rotation ; as it changes value of the torque.

your question as to why earlier you applied two forces without the pipe-connectivity f and 2f on the two bodies and their period of rotation was same and now you only apply f and both are moving with exactly same period but one point is missed that the pipe has converted the two bodies as part of a rigid body and internal forces are there to see that they are connected and move identically. As to the time period of rotation one has to check whether they change or not in two situations.

In the first case when your masses were circling at distances n and 2n radii They must need two forces F(A) =(mass.n. angular velocity^2 ) and F(B) =(mass.2n.angular velocity^2 )

as masses are same and angular velocity is same (as per your specification) you were taking the centripetal force towards the point x as F(B) = 2.F(A) or 2f and f, but they were essentially forces directed to the centre of motion. If you apply f only in the new situatiion towards the centre , along the pipe no roational motion will take place as the torque will be zero.

As Torque is cross product of force and radius vector from the fixed axis and both are collinear.

However, if we were to join ObjectA to ObjectB via a long pipe for example and only apply a single force f on the pipe such that both objects will orbit around x at the same time.

Perhaps by this you mean that you can make this happen by applying only a single additional force. Because you are accelerating the masses, they are necessarily applying forces to the rod.

Imagine that you simply set the rod spinning. As it spins around it will reach a point where mass A is pulling the rod (and the other mass) to the left, while mass B is pulling the rod (and the other mass) to the right.

At this point you can apply an additional force $f$ to the rod. If your force is applied to the left, you are decreasing the net force on mass A and increasing the net force on mass B. You are applying one force, but that force is not equal to the net force on either object.