PC1141: :Our Moon Has been Earth’s Best Friend in a Lonely Universe for Billions of Years Let us Investigate its Motion: Introduction to Classical Mechanics homework, NUS, Singapore

Problem 2: The Original Waterbender

Our Moon has been Earth’s best friend in a lonely universe for billions of years. Let us investigate its motion. For this problem, you are allowed to search up numerical values for the Moon and the Earth when specified. Of course, cite your sources for such values.

(a)Let us first consider the orbit along which the Moon revolves around the Earth. A hallmark discovery of classical mechanics is that the orbit of the Moon around the Earth is an ellipse, with the Earth centred at one of its focal points. In polar coordinates (r, 0), an ellipse is represented as:

where a is the semi-major axis length, and e is the eccentricity of the ellipse, a dimensionless number 0 < e < 1 represents how “squished” the ellipse is. If e = 0, the ellipse is a circle, while if e = 1, it is a line (an infinitely squished ellipse).

Look up the values for a and e of the Moon’s orbit. You’ll see that it’s a non-circular ellipse. Based on how gravity is the source of this orbit and the expression for the orbit alone, i.e. via explicit calculation and no invocation of physical laws other than the known form of gravity, does the Moon’s orbit conserve its angular momentum, i.e. is the angular momentum constant throughout the orbit?

(b) The answer to (a) means that the gravity between the Earth and the Moon is strictly radial. This allows us to disregard any non-radial forces to our Earth-Moon system, and instead consider the results of a single instant the Moon’s revolution, at some separation R between the centres of the Earth and the Moon.

Let us label the radius of the Earth rE and the radius of the Moon, What is the net force acting on a water molecule of mass on the surface of the Earth?

(c) What is the acceleration of the water molecule in the perspective of an observer on Earth?

(d) Drawing an appropriate diagram with accelerations of water molecules across the Earth’s surface indicated on it, what is the shape of a stable layer of water on the surface of the Earth?

(e) What is the difference (as a numerical quantity in SI units) between the highest height of the water layer from the surface of the Earth to its lowest height? You may want to look up the values of Rm, RE and the time-average R, and compare them with each other. You should also lookup values for C, the mass of the Moon ML, and the mass of the Earth ME.