Assumption on central forces between two particles

In summary, the conversation discusses Newton's force law for two particles interacting through a central force and the Galilean transformation in relation to Newtonian mechanics. Part A requires making assumptions about the force and the Galilean transformation, while Part B focuses on the assumptions about the force. The main assumption for Galilean relativity is that the laws of physics are the same in all inertial reference frames. In terms of the force, it is assumed that the forces between the particles are equal and opposite, and that the problem can be simplified to a single axis since there are only two particles involved.
  • #1

Homework Statement



Consider Newton’s force law for two particles interact through a central force F12(r1',r2',u1,u2), where by Newton’s third law F12 = -F21.

m1(d^2r1/dt^2) = F12(r1,r2,u1,u2)

m2(d^2r2/dt^2) = F21(r1,r2,u1,u2)

A. Show that Newtonian mechanics is form invariant with respect to a Galilean transformation?
B. What assumptions about the force must be made?

Homework Equations


Newton's third law:
F12 = -F21

Galilean Transform:
t'=t
m'=m
r'=r-vt

The Attempt at a Solution



I did the A. without a problem. It's the B. that troubles me a bit. I'm sure I'm missing what my prof wants me to find...

I wrote that the forces are equal and have opposite directions. The problem must lie on a single axis since there is only two particles. I'm not sure what other assumption we can make..

Thanks for the help!
 
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  • #2
When you did part a, you had to make some assumptions.
What were they?

If you did not notice any assumptions, you did not complete part a.
Try going back over your working in more detail... check each step for its reasoning.
Though you may just be expected to list the assumptions for Galilean relativity.
 
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What is the meaning of a central force?

A central force is a type of force that acts on a particle in such a way that the direction of the force is always directed toward or away from a fixed point. This fixed point is referred to as the center of force, and the magnitude of the force is dependent only on the distance between the particle and the center of force.

What are the assumptions made when studying central forces between two particles?

The main assumptions made when studying central forces between two particles are that the particles are point masses, the force acts along the line connecting the particles, and the force is inversely proportional to the square of the distance between the particles.

How are central forces different from non-central forces?

Central forces differ from non-central forces in that they only act along the line connecting the two particles, whereas non-central forces can act in any direction. Additionally, central forces are dependent on the distance between the particles, while non-central forces may also depend on other factors such as velocity or acceleration.

What is an example of a central force?

An example of a central force is the gravitational force between two objects. The force acts along the line connecting the two objects and is inversely proportional to the square of the distance between them.

How are central forces used in scientific research and applications?

Central forces are used in many scientific fields, including physics, astronomy, and engineering. They are used to study the motion of celestial bodies, to understand the behavior of particles in a system, and to design systems with stable orbits. They are also applied in fields such as aerospace engineering, where understanding central forces is crucial in designing spacecraft trajectories and orbital mechanics.

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