Newton's law of gravitation, Gauss' law

The gravitational force and potential energy

Problem:

Consider a rotating spherical planet. The speed of a point on its equator is V.  Particles near the surface of the planet accelerate downward (towards the center of the planet) with acceleration g' = ½g, where g is the downward acceleration of particles at the poles of the planet.  What is the escape velocity for a polar particle on the planet expressed as multiple of V?

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Problem:

You are an astronaut and you observe a small planet to be spherical.  After landing on the planet, you set off, walk straight ahead, and find yourself returning to your spaceship from the opposite side after completing a lap of 25 km.  You hold a hammer and a falcon feather at a height of 1.4 m, release them, and observe them to fall together to the surface in 29.2 s.  What is the mass of the planet?

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Problem:

An expedition of space treasure hunters discovered a remote spherical planet made of pure gold with density ρ = 19.3*103 kg/m3.  The expedition circled the planet and decided to come back later.  Upon their return, they were astonished to find that someone had sliced off exactly half of the planet, leaving a perfect hemisphere.  The treasure hunters landed at the center of the flat surface of the remaining hemisphere and discovered that the acceleration due to gravity there was the same as that on the surface of the Earth.  Help the treasure hunters determine the radius of the planet and the acceleration due to gravity on the surface before it was sliced in half.

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Problem:

Use the principle of conservation of mechanical energy to find the velocity with which a body must be projected vertically upward, in the absence of air resistance, to rise to a height above the earth's surface equal to the earth's radius, R.

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Problem:

Suppose the Moon were to have the same mass as the Earth, and you are trying to throw one of your physics books from the Earth to the Moon.  With what minimum velocity must the book leave the surface of the Earth?  Neglect the relative motion of the Earth and them Moon, and the Earth's rotation.  The mass of the Earth is ME = 6.0*1024 kg, the radius of the Earth is RE = 6.4*106 m, and the distance from the center of the Earth to the center of the Moon is REM = 3.8*108 m.  Compare your answer to the escape velocity from Earth alone.  The gravitational constant is G = 6.67*10−11 N m2/kg2.

Solution:

Problem:

The tidal force ΔFφ/m of the Moon on the Earth is defined as the difference between the gravitational field of the Moon at a point on the Earth's surface defined by the angle φ and the gravitational field of the Moon at the Earth's center.  (See figure.)  Derive an expression for ΔFφ/m.  Simplify by assuming that R, the radius of the Earth, is much smaller than r, the Earth-Moon distance.
Find the x- and y components and the tangential and radial components of ΔFφ/m.

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Problem:

Consider a neutron star, a macroscopic body composed of neutrons, at a density ρ = 1017 kg/m3. The temperature of the star's interior is approximately 107 K.  For this problem, consider the star to be a non-interacting Fermi gas of neutrons.
For an ideal non-relativistic 3D Fermi gas comprising N non-interacting fermions, the Fermi energy (the energy difference between the highest and lowest occupied single-particle states at T = 0) is given by EF = [ħ2/(2m)](3Nπ2/V)2/3, and the average energy per fermion at absolute zero <E> = (3/5) EF.
(a)  Determine the Fermi energy of the neutrons in the neutron star.  Are the neutrons relativistic or nonrelativistic?
(b)  Determine whether or not the neutrons are reasonably well considered to be a zero temperature (EF >> kT) Fermi gas.
(c)  Estimate the pressure in the neutron star P = -∂U/∂V.  (quasi-static, adiabatic, fixed # of particles)  In equilibrium this pressure balances the pressure due to gravity.
(d)  Use (c) to estimate the mass of this neutron star.

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Gauss' law

Problem:

Assuming that the earth is a uniformly dense sphere and that the acceleration of gravity is g at the surface, what is the acceleration of gravity below the surface as a function of distance from the center?

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Problem:

Imagine a tunnel bored through a homogeneous spherical Earth with radius R and mass M.  Suppose a particle is dropped from the surface into the tunnel.  Find its motion.

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Problem:

Use Newtonian gravity to derive an expression for the time to collapse a spherical mass distribution if the only force acting is gravity (the dynamical or free-fall timescale).  Use this result to estimate the expected response time of the Sun to a significant gravitational disturbance, given that the radius of the Sun is about 7*105 km and its mass is about 2*1030 kg.

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Problem:

The shortest way from the US to Australia is via a tunnel that goes thought the center of the Earth.  If one could build such a tunnel and make it friction free, then an object dropped at the US side with zero initial velocity would emerge after some time on the other side in Australia.  Assuming that density of the Earth is uniform (which is not correct), calculate how long it would take for an object to pass through such a tunnel. 

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Problem:

At larger distances from the center of the spiral galaxies, the rotation curve appears to be flat (the orbital speed is independent of the distance from the center).  This behavior is attributed to the peculiar distribution of the “dark mater”​.  Assume that the distribution of dark matter is independent of θ and Φ and calculate the radial density distribution of dark matter, which produces such a flat rotation curve.

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