Angular Momentum: Constant Velocity, Elliptical Orbit

• vivinisaac
In summary, the conversation discussed the relationship between angular momentum, velocity, and radius in a planet's orbit around the sun. It was determined that if the angular momentum is constant, then the product of velocity and radius must also be constant. However, this only holds true for circular orbits, not elliptical ones. This was shown through the equation GM = v^2r, which only applies to circular orbits. It was also mentioned that a planet's velocity changes as it moves along an elliptical orbit, resulting in a change in angular momentum. This is expected, as a planet's velocity at perihelion is greater than that needed to counter centripetal force, while at aphelion it is less.
vivinisaac
if the angular momentum of planet is constant when its orbiting around the sun then
velocity X radius is a constant
vr=constant

but if we consider the velocity and distance from the sun of the planet in an elipticall orbit to be v and r
the centrepital force is provided by gravitational force then
GMm/r^2 =mv^2/r
GM=v^2r
ie. v^2r=constant
ie. angular momentum is not same in two different position in the elipticall orbit

is what i wrote correct or is there something wrong in the equation

L=m v r^2, not m r v^2.
Put the two equations together and you get Kepler III.

clem said:
L=m v r^2, not m r v^2.
Put the two equations together and you get Kepler III.

L=mvr

vivinisaac said:
if the angular momentum of planet is constant when its orbiting around the sun then
velocity X radius is a constant
vr=constant

but if we consider the velocity and distance from the sun of the planet in an elipticall orbit to be v and r
the centrepital force is provided by gravitational force then
GMm/r^2 =mv^2/r
GM=v^2r
ie. v^2r=constant
ie. angular momentum is not same in two different position in the elipticall orbit

is what i wrote correct or is there something wrong in the equation

Your mistake is in assuming that $GM = v^2r$ holds for all parts of an elliptical orbit. It holds for all points of a circular orbit, but not elliptical ones.

What you have shown that two bodies of equal mass with different orbital radii don't have the same angular momentum. We don't expect them to.

For an eliptical orbit v at perhelion is larger than that needed to counter centripetal force and at aphelion it is less. If this were not true than a planet at perhelion would not climb out to aphelion and a planet at aphelion would not fall in towards perhelion.

1. What is angular momentum?

Angular momentum is a physical quantity that describes the rotational motion of an object. It is defined as the product of an object's moment of inertia and its angular velocity.

2. How is angular momentum related to constant velocity?

When an object is moving in a straight line at a constant velocity, its angular momentum is zero. This is because its moment of inertia is zero, since it is not rotating, and its angular velocity is also zero.

3. What is an elliptical orbit?

An elliptical orbit is a type of orbit where the distance between an object and its central point varies throughout the orbit. It is shaped like an ellipse, with the central point located at one of the foci of the ellipse.

4. How does angular momentum change in an elliptical orbit?

In an elliptical orbit, the distance between the object and its central point changes, resulting in a change in its moment of inertia. This causes the object's angular momentum to also change, as it is directly proportional to the moment of inertia.

5. What is the significance of angular momentum in an elliptical orbit?

Angular momentum is conserved in an elliptical orbit, meaning it remains constant throughout the orbit. This is because the net external torque acting on the object is zero, and thus there is no change in its angular momentum.

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