Angular and linear velocity of a circle

Let's assume I have a circle of radius $$r$$ and I apply a constant force at a point on the circumference of the circle that is perpendicular to the distance vector which goes from the centre of the circle to the point. If I wanted to find how fast it is rotating (angular velocity) at a specific time, I can integrate the acceleration vector ($$F/m$$) to get the velocity vector and then do the cross product between the distance vector from the centre to the point and the velocity vector (and divide by $$r^2$$) to find the angular velocity. Now if the circle is tied down (or fixed) at it's centre, it will simply rotate in place. If it's not fixed, would it be correct to assume that on top of spinning around it's centre of mass, the centre of mass will also be moving? If so, how can I find the velocity at which the centre of the circle is moving (not rotating). We're not talking about a circle that is rolling like wheels, we're talking about a circle that is flat with the ground and we can assume that the circle has no friction.

• Hi and welcome to physics SE. Please, consider rephrasing your question, because it is quite unclear right now. If you can include some pictures of what you're asking, that would help quite much as well. Thank you Jun 8, 2021 at 20:33
• ok I posted a picture, I don't know if it's clearer now. I want to know if the circle is not held at it's centre of mass, would it rotate in place or would it also move forward/backward. In other words, in this example, would you end up with the circle only rotating or also having a linear motion? Also how would you find the value of the linear motion if it does move. Jun 8, 2021 at 20:55
• Now I'm getting you, you're saying that you have a solid circle, right? I thought you had a particle in circular motion. Okay. Then the other answer is fine. One last thing: is the force constantly upwards or constantly perpendicular? Jun 8, 2021 at 21:44
• In reality the circle is actually a cylinder, but I just want to know how it works in 2d first (on a disk, or a ring, doesn't matter at this point, I'm just trying to get a feel of how it moves when the center of mass is not tied down). You can see the problem as a cylindrical boat floating on water, with the force being an engine (assume mass of engine is 0) that is attached at that specific point and pushes the boat with a force perpendicular to the hull. I don't want to go into the details of the entire system since i still don't understand the basic as explained in the picture. Jun 8, 2021 at 21:45
• The force is going to be constantly perpendicular, so you can assume that as the circle rotates, the force shifts. So if it rotates by 45 degrees, now the force applied is at 45 degrees from the original one. Jun 8, 2021 at 21:46

The center of mass moves with an acceleration given by $$\vec{F} = m \vec{a}$$, just as if it were a point mass. You can then integrate $$\vec{a}$$ to find the position of the center as a function of time.

I should also note that the method you describe will only work for a hoop where all the mass is concentrated at a fixed radius $$r$$. It will not work for a solid disk, for example. The better method is to find the torque exerted by the force about the center of mass $$\tau$$, use Newton's Third Law for rotations ($$\tau = I \alpha$$) to find the angular acceleration, and then integrate that to find the angular velocity as a function of time.

• Just want to clear something up. Once I find the angular acceleration (and angular velocity), in which direction does the centre of mass move? In the +x/-x, +y/-y or some combination of those or does it stay fixed? From the looks of it, it seems like the hoop should both rotate and move, kind of like how it works when you throw a baseball where it both spins and moves in a direction. Jun 8, 2021 at 21:17
• @SamuelSnerden It moves just as a point particle would move under the same force. In particular, if the center of mass is initially at rest, then the object's trajectory will be along a straight line parallel to the force. Jun 8, 2021 at 21:19
• So we simply shift the force to overlap directly on the center of mass, find the velocity from that and that would be the direction it should be moving correct? Jun 8, 2021 at 21:40

This is a problem of rigid body mechanics. The movement can be separated into trasnlational motion and rotational motion.

Translation of the center of mass.

The center of mass moves LIKE if all forces were acting on it. No matter where the forces are, the center of mass moves like if all forces were acting on the center of mass.

Then, in addition, you can have motion around the center of mass. In fact, you can describe rotation around any point, but it is usual to refer rotation to the center of mass, and your particular geometry is quite suitable for it.

To know how it rotates around the center of mass, you need to calculate the torque of the force w.r.t. the center of mass. Then, use the formula $$\tau=I\cdot \alpha$$, where $$I$$ is the moment of inertia of the disk, $$\alpha$$ is the angular acceleration, and $$\tau$$ is the torque.

• okay, that's exactly what I was asking for. So just checking to make sure I understand it correctly, if we have a force, to find the translational motion, we simply take all the forces and apply it at the center of mass, then find the net force and integrate to find the velocity correct? Jun 8, 2021 at 22:04
• Yes, that's pretty much it. If you that, you'll find that the CoM also moves in circles, which makes sense Jun 8, 2021 at 23:11
• perfect thank you so much. Jun 9, 2021 at 0:18