The Conservation of Angular Momentum in Polar Coordinates

[WaiYan]

Homework Statement

I'm confused about angular momentum conservation and the polar component velocities.

Relevant Equations

I know that angular momentum is conserved in planetary motion.When I think the motion in terms of polar coordinate system, theta component of velocity seems to be constant.But I think if theta component of velocity is constant, angular momentum is not conserved.So, theta component must change somehow.But I can't think why that component of velocity changes.

 

[Cutter Ketch]

Yes, the angular velocity changes. I am guessing your question is something like, if the force is always “toward the center”, why does the angular velocity change? Well, “toward the center” is the wrong way to think about it. That only makes sense for circular orbits. Consider an elliptical orbit. There are two places where the force is perpendicular to the velocity vector. Consider the apogee. Yes the force is perpendicular to the path and does not cause angular acceleration. However, the speed is too slow. The force is greater than is required to keep moving in a circle. The path deviates from a circle, and so the force is no longer perpendicular to the velocity vector. Now the force is partially pulling in the direction of motion and the speed increases as does the angular velocity.

in short, if the speed is wrong for a circle you don’t get a circle, and if it isn’t a circle then the force isn’t always perpendicular to the motion.

 

[WaiYan]

I don't think the force is acting towards centre.But I think the force is always towards the big mass M.Right?When I think this situation in terms of polar coordinates, I put the big mass at the origin of coordinate system(But for ellipse, at one focus).In polar coordinates, I can resolve the velocity into the r component and theta component.R direction and theta direction are always perpendicular.Force is always along r direction and so there is no acceleration in theta direction.So component of v in theta direction must not change in magnitude.I mean that.Is that correct?

 

[WaiYan]

Yes, the angular velocity changes. I am guessing your question is something like, if the force is always “toward the center”, why does the angular velocity change? Well, “toward the center” is the wrong way to think about it. That only makes sense for circular orbits. Consider an elliptical orbit. There are two places where the force is perpendicular to the velocity vector. Consider the apogee. Yes the force is perpendicular to the path and does not cause angular acceleration. However, the speed is too slow. The force is greater than is required to keep moving in a circle. The path deviates from a circle, and so the force is no longer perpendicular to the velocity vector. Now the force is partially pulling in the direction of motion and the speed increases as does the angular velocity.

in short, if the speed is wrong for a circle you don’t get a circle, and if it isn’t a circle then the force isn’t always perpendicular to the motion.

I don't think the force is acting towards centre.But I think the force is always towards the big mass M.Right?When I think this situation in terms of polar coordinates, I put the big mass at the origin of coordinate system(But for ellipse, at one focus).In polar coordinates, I can resolve the velocity into the r component and theta component.R direction and theta direction are always perpendicular.Force is always along r direction and so there is no acceleration in theta direction.So component of v in theta direction must not change in magnitude.I mean that.Is that correct?

 

[jbriggs444]

Force is always along r direction and so there is no acceleration in theta direction.So component of v in theta direction must not change in magnitude.I mean that.Is that correct?

No, that is not correct.

The problem is that as the particle's angle changes, the component of velocity in the radial direction is being traded for a component of velocity in the tangential direction. An incremental clockwise change in theta turns a bit of inward radial velocity into a bit of clockwise tangential velocity. And it turns a bit of tangential velocity into a bit of outward radial velocity.

A similar way of thinking about it is that by considering component-wise acceleration relative to the current rotation rate, you have implicitly adopted a rotating frame and are neglecting the Coriolis force.

 

[WaiYan]

No, that is not correct.

The problem is that as the particle's angle changes, the component of velocity in the radial direction is being traded for a component of velocity in the tangential direction. An incremental clockwise change in theta turns a bit of inward radial velocity into a bit of clockwise tangential velocity. And it turns a bit of tangential velocity into a bit of outward radial velocity.

A similar way of thinking about it is that by considering component-wise acceleration relative to the current rotation rate, you have implicitly adopted a rotating frame and are neglecting the Coriolis force.

Thank you for your explanation.

 

[Cutter Ketch]

I don't think the force is acting towards centre.But I think the force is always towards the big mass M.Right?When I think this situation in terms of polar coordinates, I put the big mass at the origin of coordinate system(But for ellipse, at one focus).In polar coordinates, I can resolve the velocity into the r component and theta component.R direction and theta direction are always perpendicular.Force is always along r direction and so there is no acceleration in theta direction.So component of v in theta direction must not change in magnitude.I mean that.Is that correct?

I think you have to be careful with thinking in polar coordinates. Forget about orbits and gravity for a second. Imagine a particle moving at constant velocity in a straight line toward, but not directly toward the origin of a polar coordinate system. When the particle is far away it’s angle is hardly changing. It’s angular velocity is very small. As the particle gets close to the origin the rate of change of its angle grows rapidly and maximizes at the point the path is tangential. The closer to the origin the larger the angular velocity. Here there are no forces. The particle travels in a straight line without changing speed, and yet the angular velocity changes an arbitrarily large amount. This is what is confusing you. Your intuition from linear motion is misleading you. You have the idea that if there are no tangential components of force then the angular velocity will not change. That just isn’t true.

but what about Newton’s law in circular motion. If there is no torque there is no angular acceleration, right? Well, think about those equations. They always have the rotational inertia. It is the product of the angular velocity and rotational inertia that are conserved. If the mass is moving toward the origin not only can the angular velocity change, it MUST because if there are no torques the product of velocity times inertia must be conserved. Try that particle moving in a straight line and you will find the angular momentum is conserved.