Proving F is a Central Force: Alonso & Finn's Physics

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Homework Help Overview

The discussion revolves around proving that a force defined as F = k⋅u × v results in circular or spiral motion. The context is based on a problem from Alonso & Finn's physics textbook, focusing on the nature of central forces and angular momentum.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the implications of proving that F is a central force and question whether this leads to meaningful conclusions about the motion. There are attempts to relate constant angular velocity to circular motion, and some participants suggest deriving equations for acceleration to further explore the motion's characteristics.

Discussion Status

The discussion is active, with participants providing feedback and prompting further exploration of the problem. Some guidance is offered regarding the need to derive equations related to the motion, and there is an acknowledgment of the complexity involved in demonstrating the nature of the motion as circular or helical.

Contextual Notes

Participants are navigating the definitions and implications of central forces, angular momentum, and the specific nature of the motion described in the problem. There is an emphasis on deriving equations and understanding the motion's characteristics rather than reaching a definitive conclusion.

Anastasis Pk
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Good afternoon!
Exercise: Prove that if a body moves under the action of a force F = k⋅u × v, where u is an arbitrary unit vector and v the velocity, the motion is circular with angular velocity ω = k ⋅ u or, in a more general case, a spiral parallel to u.

Source: (Alonso & Finn: Fundamental university physics)

I attempted to prove that F is a central force, so that the angular momentum is constant. Is that wrong?
Thank you in advance!
 
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Would showing F is a central force buy you anything? The force of the Sun on a planet is a central force. but the orbits aren't circular.
 
Anastasis Pk said:
I attempted to prove that F is a central force
That won't do it. F = k.v would satisfy that. Get an equation for the acceleration.
 
Thank you for your responses! In order to prove that it's motion is circular wouldn't it be sufficient to prove that ω is constant?
Could you "boost" me a bit?
 
Anastasis Pk said:
Thank you for your responses! In order to prove that it's motion is circular wouldn't it be sufficient to prove that ω is constant?
Not really. The question is asking you to demonstrate that the equation of motion corresponds to a particle moving at constant speed in a helix. (It says spiral, but it means helix.)
It might help if you could write down the general equation for such a motion, so that you could see what the answer has to look like. Hint: pick a suitable set of orthogonal unit vectors.
Anyway, the first (easy) step, as I posted originally, is to obtain an equation for the acceleration of the particle.
 

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