A particle of mass m moves in a central potential V(r) = - k/r * e ^ (- gamma) ', with k, gamma 0 What should be the initial velocity of the particle to execute circular motion? Describe a rough idea to estimate the value of from the earth-sun system?

Question

A particle of mass m moves in a central potential V(r) = - k/r * e ^ (- gamma) ', with k, gamma > 0 What should be the initial velocity of the particle to execute circular motion? Describe a rough idea to estimate the value of from the earth-sun system?

Answer

The initial velocity for circular motion:
In order for a particle to execute circular motion in a central potential, the net force acting on the particle must always be directed towards the center of the motion. This means that the centrifugal force should balance the centripetal force.

The centripetal force is given by the equation F_c = m*v^2/r, where m is the mass of the particle, v is its velocity, and r is the radius of the circular path.

The centrifugal force is given by the equation F_cf = m*r*d^2V/dt^2, where dV/dt is the derivative of the potential function V with respect to time.

For circular motion, the two forces must be equal, so we have:
m*v^2/r = m*r*d^2V/dt^2

We can rewrite this equation as:
v^2 = r^2*d^2V/dt^2

Estimating the value of γ in the Earth-Sun system:
To estimate the value of γ in the Earth-Sun system, we can start by considering the gravitational potential between the two bodies. The gravitational potential is given by the equation V(r) = -GMm/r, where G is the gravitational constant, M is the mass of the Sun, and m is the mass of the Earth.

We can rewrite this equation as:
V(r) = -k/r * e^(-γ)
where k = GMm and γ = 1

By comparing the two equations, we can see that γ = 1. This means that the value of γ in the Earth-Sun system is approximately 1.

Summary:
- In order to execute circular motion, the initial velocity of the particle should be such that the centrifugal force balances the centripetal force.
- The value of γ in the Earth-Sun system can be estimated by comparing the gravitational potential equation with the given potential function. In this case, γ is approximately equal to 1.

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