Suppose that an object is in a potential U(r) that only depends on r, the distan

ID: 2114035 • Letter: S

Question

Suppose that an object is in a potential U(r) that only depends on r, the distance from

the origin, but not theta , the angle in the plane. We know that the kinetic energy can be written

as: KE =1/2m(Vr^2+r^2w^2)

where w is the time derivative of theta. We also know that the angular momentum L = mr^2w

is conserved.

Do the following:

1)Use the facts above to eliminate w from the kinetic energy and write it in terms of

the conserved quantity L, as well as r and the radial component of velocity vr.

2)Add U(r) to the kinetic energy, to get the total energy.

3) Compute the derivative of the total energy with respect to r. This will be the radial

component of force. Your force should contain the expected dU/dr term, and also another

term that depends on L.

Please all the work you did. THANKS!

(1)

L = m*vt*r

vt = L/m*r

w = vt/r

Put the values of w and Vt in KE and we get

KE = 1/2m(vr^2) + 1/2(L^2)/(mr^2)

(2)Total energy = U(r) + KE

= U(r) + 1/2m(vr^2) + 1/2(L^2)/(mr^2)

(3)dT/dr = dU/dr - L^2/m*r^3

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