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Bohr's quantum model successfully accounts for the size of the spectroscopy of the hydrogen atom, which consists of an electron moving around a proton. There are many similarities between the electrostatic forces that bind an electron to a proton and the gravitational forces that maintain the planets in orbit around the Sun, and a satellite in orbit around the Earth. Both forces obey an attractive inverse-square law. It is therefore possible to adapt Bohr's postulates and his calculations to obtain a gravitational analogue of the Bohr model of the hydrogen atom, and this is what you will do in answering this question. In particular, you are asked to set up a simple quantum model to describe a satellite, mass m_p, that travels at speed v in a (circular) orbit, radius R, around the Moon, mass M. You should ignore gravitational forces between the satellite and all other bodies apart from the Moon.

(a) Write down adapted forms of Bohr's four postulates that apply to the satellite orbiting the Moon.

(b) Show that the allowed radii R_n of the orbits of the satellite are given by R_n = n^2*h^2/(4*Pi^2*m_s^2*G*M), where R_n is greater that the Moon's radius and n is a (suitably large) positive integer.

(c) Show that the speed V_n of the satellite in the orbit with radius R_n is given by V_n = 2*Pi*G*M*m_s/(n*h).

(d) Now assume that the satellite has mass 5.0 x 10^4 kg and is in a circular orbit 200 km above the Moon's equatorial surface. The Moon's radius is 1720 km and its mass is 7.2 x 10^22 kg. Calculate the approximate value of the quantum number n for this orbit, and hence calculate the value of the speed of the satellite.

Added Thu, 18 Jun '15

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(a)

1. satellite orbits round the moon under gravitational attraction.

2. orbital angular momentum is quantized. Only certain orbits are possible

3. satellite doesn't lose energy by radiation.

4. satellite can change orbits by radiating or absorbing energy.

(b)

The centripetal force for the satellite orbiting round the moon is given by the gravitational attractive force, i.e. mv^2/r = GMm/r^2

where m and M are the masses of the satellite and moon respectively

r is the radius of the satellite

v is the orbital speed of the satellite

G is the universal gravitational constant

hence, v^2 = GM/r ------------------- (1)

Bohr's postulate says that the angular momentum of the satellite is quantized, equal to a whole number of h/(2.pi), where h is Planck's constant, and pi = 3.14159.....Bohr's postulate says that the angular momentum of the satellite is quantized, equal to a whole number of h/(2.pi), where h is Planck's constant, and pi = 3.14159.....

i.e. m*v*r = nh/(2*Pi), where n is a positive integer

(mvr)^2 = (n^2)(h^2)/(4*pi^2)

use equation (1) for the value of v^2, we have,

m^2*(GM/r)*r^2 = (n^2)(h^2)/(4*pi^2)

r = (n^2)(h^2)/(4*pi^2*m^2*GM) ------------ (3)

substitute the value of r back into equation (1):

v^2 = GM[(4*pi^2*m^2*GM)/n^2h^2)]

i.e. v = 2*pi*GM*m/(n*h) ---------------------- (4)

Given: m = 50000 kg, M = 7.2x10^22 kg, r = 1920000 m

substitutes these values into equation (3) to find n

Then use equation (4) to find v

Added Thu, 18 Jun '15

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