# The Conservation of Angular Momentum in Polar Coordinates

• WaiYan
In summary, the angular velocity changes when the speed is wrong for a circle. If the speed is wrong for a circle, the force is not always perpendicular to the motion. However, when the speed is right, the force is always towards the big mass.
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.

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?

Cutter Ketch said:
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?

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WaiYan said:
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
jbriggs444 said:
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 said:
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.

## 1. What is an elliptical orbit?

An elliptical orbit is a type of orbit in which an object follows an elliptical path around another object, with the second object located at one of the two foci of the ellipse.

## 2. What causes planets to have elliptical orbits?

The elliptical orbit of a planet is caused by the gravitational pull of the sun. As the planet moves around the sun, the gravitational force varies, causing the planet's orbit to be elliptical rather than circular.

## 3. How does the eccentricity of an elliptical orbit affect a planet's motion?

The eccentricity of an elliptical orbit determines how elongated the orbit is. A higher eccentricity means a more elongated orbit, which results in the planet moving at different speeds at different points in its orbit. This can lead to changes in the planet's distance from the sun and the amount of time it takes to complete one orbit.

## 4. Can planets have perfectly circular orbits?

No, according to Kepler's laws of planetary motion, all planets have elliptical orbits with the sun located at one of the two foci. However, some planets have orbits with very low eccentricity, making them appear almost circular.

## 5. How do scientists measure the eccentricity of a planet's orbit?

Scientists use mathematical calculations and observations to determine the eccentricity of a planet's orbit. One method is to measure the planet's distance from the sun at different points in its orbit and then use a formula to calculate the eccentricity. Another method is to use radar or telescopes to observe the planet's position and velocity over time and plot its orbit to determine its eccentricity.

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