Eric C. answered • 02/05/20

Former high school teacher with decades of tutoring experience.

The first problem can be solved using conservation of energy. The sum of the comet's gravitational potential energy and kinetic energy will remain constant. So the sum:

mM_{sun}G/r +mv^{2}/2

will have the same value at any point in the orbit. You can use this to solve the problem.

The second problem can be solved using conservation of angular momentum. Once the sled's motor is turned off, there are no more torques on the sled, so it will maintain a constant angular momentum. The moment of inertia of the sled about the pole is I = mr^{2} and its rotational velocity is ω = v/r, thus the angular momentum is L = Iω = mr^{2}(v/r) = mvr. This quantity remains constant as the rope wraps around the pole.

The final problem also has to do with conservation of angular momentum. For a solid sphere, the moment of inertia is I = (2/5)mr^{2}. I hope this helps. Feel free to ask follow up questions.