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Angular Momentum often comes up in forum discussions.
A popular use of angular momentum is the idea of giving a payload an impulse in a desirable direction by launching the payload from a geostationary point some distance from the surface of the object.
Because this is a useful concept, I decided to ask Google's Gemini for suggestions about how to think about the problem, and others that involve angular momentum in space.
Conversation with Gemini
Hi Gemini! Thank you (again) for your help with previous questions. Today I'm hoping you can find educational material that would help a student to understand angular momentum. The context of the question is use of the rotation of an asteroid as a way to launch mass from the asteroid. It may not be obvious to everyone that energy must be expended to give a payload that starts at the surface of an asteroid the angular momentum at the end of a rope that is extended from the surface of the asteroid. What I'm hoping you can find would be instructional material that shows how force must be applied to the payload in two directions: First, force must be applied to overcome the gravitational pull of the asteroid. Second, force must be applied in the direction of rotation of the asteroid.; so that the payload remains directly above the starting point as it rises.Absolutely! This is a fascinating problem that blends orbital mechanics, rotational dynamics, and a bit of practical engineering. Let's break down the educational materials and concepts a student would need to grasp, focusing on your specific context of launching mass from a rotating asteroid.
Key Concepts to Understand:
Angular Momentum:
Definition: Angular momentum (L) is a measure of an object's tendency to keep rotating. It's calculated as L = Iω, where I is the moment of inertia and ω is the angular velocity.
Conservation: In a closed system, angular momentum is conserved. This is crucial for understanding how the asteroid's rotation affects the launched mass.
Changing Angular Momentum: To change an object's angular momentum, a torque (a rotational force) must be applied.
Rotational Kinematics:Angular Velocity (ω): The rate of change of angular position.
Tangential Velocity (v): The linear velocity of a point on a rotating object, calculated as v = rω, where r is the distance from the axis of rotation.Centripetal Force: The force required to keep an object moving in a circular path.
Gravitational Force:Newton's Law of Universal Gravitation: Understanding the gravitational force exerted by the asteroid is essential to calculate the required energy to lift the payload.
Frames of Reference:Rotating Frame: Understanding how motion appears in a rotating frame of reference is crucial. Concepts like Coriolis force (though less important in this specific problem where the payload is kept directly above the launch point) are relevant here.
Inertial Frame: Understanding the motion from a non-rotating or "fixed" point of view.
Educational Materials and Approaches:Visualizations and Simulations:
Interactive Simulations: Search for online simulations that demonstrate angular momentum conservation and rotational motion. PhET
Interactive Simulations (phet.colorado.edu) often has relevant content.
3D Animations: Animations that show the asteroid rotating and the payload being lifted, with vectors representing the forces and velocities involved, would be extremely helpful.
Orbital Mechanics Simulators: Programs that simulate orbital mechanics can be used to show how the payloads trajectory changes based on the applied forces.
Worked Examples and Problem Sets:
Simple Rotational Motion Problems: Start with basic problems involving rotating objects, such as a spinning disk or a rotating platform.
Problems Involving Angular Momentum Transfer: Create or find problems that involve transferring angular momentum between objects, such as a person jumping onto a rotating platform.
Asteroid Launch Scenario Problems: Develop problems that specifically address the asteroid launch scenario, breaking it down into steps:
Calculating the asteroid's angular momentum.Calculating the gravitational force on the payload.
Determining the tangential velocity needed for the payload to maintain its position above the launch point.
Calculating the required change in angular momentum of the payload.
Calculating the forces that must be applied to the payload.
Explanations and Diagrams:Free-Body Diagrams: Draw free-body diagrams of the payload at different stages of the launch, showing the gravitational force, the applied force in the radial direction (to counteract gravity), and the applied force in the tangential direction (to match the asteroid's rotation).
Vector Diagrams: Use vector diagrams to illustrate the velocity and acceleration of the payload in both the radial and tangential directions.
Step-by-Step Explanations: Provide clear and concise explanations of each step of the launch process, using diagrams and equations to support the explanations.
Khan Academy: Khan Academy has very good videos and problems related to rotational motion and angular momentum.
Real-World Examples:Figure Skating: Explain how figure skaters use angular momentum conservation to control their spin.
The Earth-Moon System: Discuss how the Earth and Moon's rotation and orbital motion are related to angular momentum.Centrifugal Force Demonstrations: Demonstrations of centrifugal force can help students visualize the forces involved in rotational motion.
Addressing the Specific Challenge:
Tangential Velocity Matching: Emphasize that to keep the payload directly above the launch point, its tangential velocity must match the tangential velocity of the asteroid's surface at all times. This means that as the payload moves further from the axis of rotation (as the rope extends), its tangential velocity must increase.
Force Application: Make it clear that this increase in tangential velocity requires an applied force in the direction of rotation. This force is what provides the necessary change in angular momentum.
Energy Expenditure: Connect the force application to energy expenditure. Explain that work must be done to apply these forces, and this work translates to an increase in the payload's kinetic and potential energy.
By combining these resources and approaches, you can help students develop a strong understanding of the physics involved in launching mass from a rotating asteroid.
Thank you for these helpful suggestions!
You're very welcome! I'm glad they were helpful. If you have any further questions or want to delve deeper into any of these concepts, feel free to ask. Good luck with your studies!
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This post is reserved for an index to what I hope will be helpful posts contributed by members.
Links to animation that already exists on the Internet would be helpful.
In addition, there may be web sites that have a page that discusses this exact situation.
Post #5: Calliban: http://newmars.com/forums/viewtopic.php … 22#p230022
In this post, Calliban describes how a particular object (Haumea) could provide energy if it's rotation momentum is tapped.
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Gemini suggested a visit to this site:
There is a lot on offer there. If anyone in the NewMars membership has time to explore, a post or two about the offerings would be appreciated.
The focus of interest for this topic is Angular Momentum, and in particular, the practical application of Angular Momentum to launch payloads from rotating objects in space.
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Update regarding Angular Momentum as applied to the rotating asteroid launch problem...
Since I don't understand the complex dynamics of the physical system our NewMars members are describing in multiple topics, I asked ChatGPT4o if it had advanced enough to be able to tackle animation of:
1) A payload to be launched
2) A tether/line to hold the payload to an anchor on the surface of the asteroid
3) The asteroid itself, which is rotating.
To my astonishment, ChatGPT4o confidently announced that it could indeed create such a program, and it did so.
I should explain that I have been testing ChatGPt4o's powers as they advance, and the Python program contains a new feature that I have not seen before.
It will be ** very ** interesting to see if the program runs at all, let alone achieves the objective.
I'll report after giving the program a test run.
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The dwarf planet Haumea is the fastest rotating object of its size in the solar system. It is an ellipsoid, with its longest axis being roughly twice as long as the shortest.
https://en.m.wikipedia.org/wiki/Haumea
It is therefore an interesting case study for harnessing rotational energy. This would be accomplished using a space elevator mounted onto the tip one of the dwarf planet's lobes, i.e along the longest axis. The space elevator will continue beyond the geostationary point, such that payloads are released with enough velocity to escape the gravity of the dwarf planet into interplanetary space.
As objects pass the geostationary point, they begin to move at a tangential velocity that exceeds the local orbital speed. The further they move along the tether above geostationary, the greater their tangential velocity becomes. This is because the tether must have a constant angular velocity that matches the angular velocity of the planet. Tangential velocity increases linearly with distance along the tether, until it exceeds local escape velocity. Any object attached to the tether beyond geostationary, will exert tension on the tether. This can be used to generate power if the tether is replaced by a cable loop, with pulleys at both ends. Masses of material are pulled up the tether from the surface, to geostationary point. Up to this point, a net force is needed to lift mass out of the gravity well. Beyond this point, any mass will begin to pull on the cable, as its tangential velocity exceeds local orbital velocity.
This can be used to generate power, as net energy can be extracted from the mass lifted by the tether, prior to it being released into interplanetary space at the end of the tether. How much energy is available to extract?
The rotational energy contained in any rotating body if given by:
Er = 0.5×I×w^2
Where I is moment of inertia, and w is angular velocity (rad/s).
The moment of inertia of an ellipsoid is given by:
I = 0.2×M×(a^2 + c^2)
M is mass (4E21 kg), a is the short axis length and c is the long axis length, 2100km. The dimensions of Haumea are listed as being 2,100 × 1,680 × 1,074km. The short axes of haumea are therefore not the same length. The value of a is taken as the mean of the two minor axis lengths, which equates to 1377km. The gives an overall moment of intertia of 5.045E+33.
Haumea rotates once every 3.915 hours. This equates to an angular velocity of 4.458E-4 rad/s. The rotational energy can now be calculated:
Er = 0.5×I×w^2 = 0.5×(5.045+E33)×(4.458E-4)^2 = 5.014E+26 joules
This amounts to 15.885 billion GW-years. 1GW-yr is the amount of electrical energy produced by a large nuclear powerplant in 1 year. Only a fraction of this energy will be available to generate electric power, as a significant portion would be used lifting material out of the gravity well and accelerating it to escape velocity. But the resource available is clearly huge. Enough to power a civilisation of many millions of people, for millions of years. Colonists heading to Haumea would take a small nuclear reactor with them. But once a tether is established, it will meet all the energy needs of a developing ciivilisation.
Last edited by Calliban (2025-02-25 16:38:28)
"Plan and prepare for every possibility, and you will never act. It is nobler to have courage as we stumble into half the things we fear than to analyse every possible obstacle and begin nothing. Great things are achieved by embracing great dangers."
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I returned to work with ChatGPT4o on a Python program to deliver animation showing movement of:
1) An object (such as an asteroid or small planet)
2) A tether
3) A payload
At the moment, in version 03, we have an animation that shows the payload attached to a tether moving from the object to the limit of the tether.
We added a Start/Restart button so that as we add input options, the operator can rerun the program.
We added an input window for rotation of the object, and I chose to use RPM as the input value, since that is such a widely recognized value. That can be changed if something else proves more suitable. The input will be a small decimal number in the case of the Earth.
Per Google, the rotation rate of the Earth is .000694 RPM
Rotation rate of RobertDyck's Large Ship is given as 3 RPM.
The "rotation rate" currently manifests in the program as a change of velocity of the payload as it moves out along the tether.
In the next iteration, we will add rotation to the central object, and set up the velocity of the payload as it's own variable.
Calliban recently reminded us that the ** real ** Space Elevator (as originally intended for Earth) had a very large counterweight outside the center of balance, to insure that the tether remained taut while climbers ascended and descended.
It should be noted that Angular Momentum lost by the Counterweight when climbers ascend the elevator would be returned by climbers that descend, assuming that the mass on descent is the same as the mass on ascent.
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