Help me with physics pls

Question

Encke’s comet revolves around the sun in a period of just over three years. The closest it approaches the sun is 4.95 × 10^7 km, at which time its orbital speed is 2.54 × 10^5 km/h. At what distance from the sun would Encke’s comet have a speed equal to 1.81 × 10^5 km/h?

In 1981, Sammy Miller reached a speed of 399 km/h on a rocket-powered ice sled. Suppose the sled, moving at its maximum speed, was hooked to a post with a radius of 0.20 m by a light cord with an unknown initial length. The rocket engine was then turned off, and the sled began to circle the post with negligible resistance as the cord wrapped around the post. If the speed of the sled after 20 turns was 456 km/h, what was the length of the unwound cord?

Earth is not a perfect sphere, in part because of its rotation about its axis. A point on the equator is in fact over 21 km farther from Earth’s center than is the North pole. Suppose you model Earth as a solid clay sphere with a mass of 25.0 kg and a radius of 15.0 cm. If you begin rotating the sphere with a constant angular speed of 4.70 × 10^–3 rad/s (about the same as Earth’s), and the sphere continues to rotate without the application of any external torques, what will the change in the sphere’s moment of inertia be when the final angular speed equals 4.74 × 10^–3 rad/s?

Answer 1

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_sunG/r +mv²/2

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 = mr² and its rotational velocity is ω = v/r, thus the angular momentum is L = Iω = mr²(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)mr². I hope this helps. Feel free to ask follow up questions.