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Recently I am stuck with a question about an ice cube that is orbiting the earth from a certain radius and it starts to melt down by the sun.

Which of the followings are wrong?

  1. The cube will start to move away from Earth.

  2. The cube's velocity will start to increase.

  3. The cube's period will decrease.

For the solution I came up with an equation which is I'll write down in a second: .($M$ = mass of the Earth, $m$ = mass of the cube, $r$ = radius between cube and Earth.)

$$G.\frac{M.m}{r^2} = m.\frac{v^2}{r}$$

$$G.\frac{M}{r} = v^2$$

$$G.M = v^2.r = v.(v.r)$$

$$L= m.(v.r)$$

$$\frac{L}{m}=(v.r)$$

$$G.M =v.(v.r)$$

$$G.M=\frac{L}{m}.v$$

$$\frac{G.M}{L}=\frac{v}{m}$$

In the final form all the things at the left side are constant(Since there is no torque involved, cube's angular momentum must be conserved, mass of the Earth and G are also constant.)

From this I can only see if mass decreases than the velocity must also decreases since they are directly proportional.

If I assume what I came up with is true than for first following (The cube will start to move away from Earth.) if I rewrite the angular momentum formula ''$L=m.v.r$'' I can see if ''$m$'' and ''$v$'' are decreasing then the radius must increase to keep ''$L$'' constant but the answer is saying that is wrong.(other two seems to be okay with my conclusions only first one is not.)

So I am here to hear some opinions to see if I am thinking wrong or if there are somethings other I am missing.

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  • $\begingroup$ We are neglecting the momentum imparted by sunlight? If not, we will have to consider the orientation of the orbit $\endgroup$ – Mauricio Jun 12 at 14:07
  • $\begingroup$ Could maybe all answers be wrong? $\endgroup$ – denklo Jun 12 at 14:20
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In my opinion, there's something wrong with the question. What does it mean by melting? Even if the sun melted it, the remaining water and ice should move together, as they did before, in the same orbit.

Now i am assuming that the question setter meant by 'melting' that the melted water is being removed and the total mass of the ice is decreasing. Then, there's another problem. It would need external force to remove the water. Then the total momentum/angular momentum/energy of the whole system would not be constant. But, for the remaining ice, the quantities should be the same as before. Still, i think, the period and all the orbital parameters should remain the same. As the circular motion only depends on the central body, as long as earth is same. I think, all of them are wrong

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