Angular Momentum vs Latus Rectum

In summary, the problem involves a shuttle in circular orbit around a planet that fires thrusters at a point P, causing an increase in speed. The task is to find the angular momentum, gravitational force, centripetal acceleration, and radius of curvature at point P, and to compare the latus rectum of the new orbit to that of the circular orbit. While the velocity increase results in a larger angular momentum, the latus rectum of the new orbit is actually smaller, causing a decrease in angular momentum. It is also important to consider the orientation of the thruster's acceleration compared to the centripetal force.
  • #1
Lucretius
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0

Homework Statement



There is a shuttle following a circular orbit around a planet. At some point P in the orbit, the shuttle fires thrusters causing the speed at that point to increase. I am supposed to a) find the angular momentum, gravitational force at the point P, centripetal acceleration at the point P, and the radius of curvature at point P. Part B is to say whether the latus rectum of my new orbit is smaller, larger, or equal to that of the circular orbit.

Homework Equations



I used L=R x P, a=v^2/r, and F=Gmm/r^2.

The Attempt at a Solution



I didn't have much problem with part A. The velocity perpendicular to the radius changed, so the cross product for L increased, making L larger. The gravitational force stayed the same at the point, since that only depended on the radius. The centripetal acceleration increased, since v increased, while r remained the same. And lastly, the radius of curvature at the point didn't change, because the radius didn't change instantaneously.

My problem is that, it is known that L, the angular momentum, is directly proportional to the latus rectum, c. But for the ellipse just made, the latus rectum is smaller than that for the circle. Thus my new L should be SMALLER than my first L for the circular orbit. But the velocity increase tells me something different! In other words, when analyzing two different aspects of this orbit, I get both an increase and a decrease in angular momentum, depending on whether I look at the velocity or the latus rectum. What am I missing here?

EDIT: Also, my professor gives a hint that this point P will lie along the semi-major axis of the newly elliptical orbit. But he has drawn the point at a 3' o clock location on the circular orbit. If the thrusters went off here, I would expect the point to lie along the semi-minor axis because the shuttle should move further now in this direction than it has previously! Perhaps this is just a typo?
 
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  • #2
Lucretius said:
The centripetal acceleration increased, since v increased, while r remained the same.
No, centripental acceleration does not change, because the force of gravity does not change.
 
  • #3
Major axis/minor axis are determined based on the orientation of the
elliptical orbit, so they are not necessarily alligned to your coordinate sistem.
 
  • #4
You have the "particle" moving in a circle so that its instantaneous velocity is tangential to the circle and it experiences a centripetal acceleration towards the center of the circle (due to gravity).

I might be mistaken, but it doesn't seem like you have considered the directionality of the thruster's acceleration compared to that of the centripetal force correctly in your logic - but I apologize if I'm wrong in this interpretation!
 

1. What is angular momentum?

Angular momentum is a measure of the rotational motion of an object. It is defined as the product of an object's moment of inertia and its angular velocity. In simpler terms, it is the tendency of an object to keep spinning unless acted upon by an external force.

2. What is latus rectum?

Latus rectum is a term used in mathematics and physics to describe the distance between a focus and a point on a conic section, such as an ellipse or a hyperbola. In the case of an ellipse, it is the chord that passes through the focus and is perpendicular to the major axis.

3. How are angular momentum and latus rectum related?

Angular momentum and latus rectum are not directly related. They are two different concepts used in different contexts. Angular momentum is used to describe rotational motion, while latus rectum is used to describe the shape of a conic section.

4. How is angular momentum calculated?

Angular momentum is calculated by multiplying an object's moment of inertia (a measure of the object's resistance to rotational motion) by its angular velocity (rate of change of angular displacement). The formula for angular momentum is L = Iω, where L is angular momentum, I is moment of inertia, and ω is angular velocity.

5. Can latus rectum be used to calculate angular momentum?

No, latus rectum cannot be used to calculate angular momentum. As mentioned before, they are two different concepts and are not directly related. Angular momentum is calculated using an object's moment of inertia and angular velocity, while latus rectum is used to describe the shape of a conic section.

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