Show that the areal velocity is constant for a particle moving under the influence of an attractive force given by F(r) = -kr

Physics Sep 18, 2020

Show that the areal velocity is constant for a particle moving under the influence of an attractive force given by F(r) = -kr. Calculate the time averages of the kinetic and potential energies and compare the results of the viral theorem.

Expert Solution

Areal velocity w = (r * V)/2

Where * denotes the vector product, r is the position of the particle and V its velocity.

dw/dt = (dr/dt) * V/2 + (r/2) * (dV/dt)

Now dr/dt = V

Further dV/dt = acceleration = Force/Mass

Force F = -k r (i.e., it is along the radius vector)

Therefore dV/dt = - (k/m) r

Plugging it back in we have

dw/dt = V * V/2 - (k/2m) r * r

But V*V = r*r = 0

Therefore dw/dt = 0

or w is a constant in time.

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