1. When we related Newton\'s Laws to conservation of momentum and energy, we sho

Question

1. When we related Newton's Laws to conservation of momentum and energy, we showed that for a conservative force like gravity: where U is the potential energy. For Newtonian gravity: GMm U- where M and m are the two masses that are interacting via the force of gravity. This is closely related to the so-called gravitational potential. This is a convenient way to describe how a mass (or mass distribution) acts on a test particle. It is essentially the potential energy per unit mass: Thus for point mass, M: GM And if instead of 1 point mass, we had many point masses: where subscript i indicates different particles at different s -, and i is the location of mass i, and is the location at which the potential is evaluated. Thus we can see that if instead of a bunch of point masses, we have a continuous mass distribution (recall the spherical shell from class), we would rewrite the above equation as an integral: d M (a) Now consider a simple version of our solar system. Let's just think about the Sun, the Earth, and Jupiter. We know the potential that the Earth feels from the Sun, but what about Jupiter? Let's imagine that we can treat Jupiter

Explanation / Answer

1. a. consider soalr system

mass of jupiter = Mj

if we consider having mass of jupiter extended ionto a uniform at its current orbit

radius oif orbit of jupiter = Rj

hence

Potential V = -GMj/Rj

for r < R ( R = Rj)

considering Sun ( Ms) earth ( Me) and jupiter only

V = -G(Ms/r + Me/r + Mj/Rj)

b. dV/dr = G(Ms/r^2 + Me/r^2)

the acceleration felt by the test particel depends on the relative position of the particel eartha nd the sun, and the acceleration can be increased or deceresed due to acceleration of the sun

c. Ve/Vj = (R(Ms + Me)/r + Mj)/(Ms + Me + Mj)

Ms = 2*10^30 kg

Me = 5.972*10^24 kg

Mj = 1.898*10^27 kg

R = 5.2 AU

Ve/Vj = 5.19601799

d. if ring of jupiter is removed, the acceleration observed by the earth still remains the same hence there is no change of net force on the earth

this means the accel;ration of the earth shall remain constant

hence ther eshall be no change in timne period of earth isscilation around rthe sun

hence the calenders shall need no change

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