Central force motion and particles

In summary, the problem involves two particles of masses m and M, where m is initially at a distance of infinity with a velocity of v_o. The mass M is then found at a distance d, where both particles are acting on each other due to gravitational pull. The equations used include the initial energy of m, the final energy including the effective potential, and the magnitude of the angular momentum. The question is to find the mass M in terms of the given quantities.
  • #1
cardamom
3
0

Homework Statement



The problem involves two particles of masses m and M; initially, m is at [tex]r=∞[/tex] and has a velocity [tex]v=v_o[/tex]. The path of m is deflected, ie pulled towards M due to its gravitational pull.

Question: Find the mass M (in terms of the quantities given) at a distance d where the particles are now acting on each other.

Homework Equations



Initial energy of m

[tex]

E_i = \frac{1}{2}mv_o^2
[/tex]

[tex]E_f = \frac{1}{2}μv_o^2 + U(r) = \frac{1}{2}μv_o^2 + (\frac{-GMm}{d})
[/tex]

[tex]\frac{1}{2}μv_o^2 = \frac{l^2}{2μd^2}


[/tex]

The Attempt at a Solution



I've tried using combinations of the above, but in the end, I am not confident that I am correct in my assumptions of E_f, otherwise this would be an easy algebraic game. I also considered that E_f should include the effective potential, but at distance d, the two particles haven't yet crossed, though they are at a distance such that the vector between them is orthogonal to the path of m at that point. Any guidance is appreciated!
 
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  • #2
Am I missing something obvious?
 
  • #3
Is this a question that you made up?

What is mu? What is l (lower-case L)?
 
  • #4
No, it's on an assignment... mu=Mm/(M+m) and l is the magnitude of the angular momentum. I can use l since the direction of L is constant as it's only in one plane - yes?
 
  • #5




In this problem, we are dealing with central force motion, where the only force acting on the particles is directed towards the center of mass. This is evident from the fact that the path of m is deflected towards M due to its gravitational pull.

To find the mass M at a distance d where the particles are now acting on each other, we can use the conservation of energy principle. Initially, the total energy of m is given by E_i = 1/2 mv_o^2, where m is the mass of the particle and v_o is its initial velocity.

At a distance d, the total energy of m is given by E_f = 1/2 μv_o^2 + U(r), where μ is the reduced mass of the system and U(r) is the effective potential due to the gravitational interaction between the particles.

Since the particles are now acting on each other, we can equate E_i and E_f to get:

1/2 mv_o^2 = 1/2 μv_o^2 + U(r)

Solving for U(r), we get:

U(r) = (m - μ)v_o^2/2

At a distance d, the effective potential is given by U(r) = -GMm/d, where G is the gravitational constant.

Equating the two expressions for U(r), we get:

(m - μ)v_o^2/2 = -GMm/d

Solving for M, we get:

M = (m - μ)vd_o^2/2GM

Therefore, the mass M at a distance d where the particles are now acting on each other is given by (m - μ)vd_o^2/2GM.

I hope this helps. If you have any further questions or concerns, please let me know.
 

1. What is central force motion and particles?

Central force motion and particles refer to the motion of particles under the influence of a central force, which acts towards or away from a fixed point. Examples of central forces include gravity, electric and magnetic forces, and nuclear forces.

2. What is the difference between central and non-central forces?

Central forces act towards or away from a fixed point, while non-central forces act in other directions. Central forces result in orbital motion, while non-central forces can lead to more complex trajectories.

3. How is the motion of central force particles described?

The motion of central force particles can be described using Newton's laws of motion and the law of universal gravitation. These laws can be used to derive equations of motion, such as the radial equation and the angular equation, which describe the position, velocity, and acceleration of the particle.

4. What is the significance of angular momentum in central force motion?

Angular momentum is conserved in central force motion, meaning that the total angular momentum of the particle remains constant throughout its motion. This can be explained by the fact that the central force acts along the line joining the particle and the central point, resulting in no torque and thus no change in angular momentum.

5. How is central force motion related to celestial mechanics?

Central force motion plays a crucial role in celestial mechanics, which is the study of the motion of celestial bodies such as planets, stars, and galaxies. The gravitational force between celestial bodies is a central force, and the laws of central force motion can be used to accurately predict the orbits of these bodies.

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