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Question

The each revolves round the sun due to gravit ational attraction. Suppose that the sun and the earth are point particles with their existing mas ses and that Bohr's quantization rule for angular momentum is vaild case of gravitation. 

(a) Calculate the minimum  radius the earth can have for its orbit. 

(b) What is value of the principal quantum num ber n for the present radius ? Mass of the earth  =  6.0×1024  kg, Mass of the sun = 2.0×1030  kg, earth-sun distance =  1.5×1011 m.


Solution

Mass of Earth = Me=6.0×1024  kg

Mass of Sun =  M8=2.0×1030   kg

Earth- Sun distance = 1.5×1011  m

Since        mvr =  nh2π

or,      m2v2r2=n2h22π2.....(1)

 GMeM8r2=Mev2r

or    v2=GM8r              ...............(2)

Dividing (1) and (2)

We get     M2er=n2h24π2GM8

for n = 1, 

r =  h2402GM8M2e

= (6.6×1024)24×(3.14)2×6.67×1011×(6×1024)×2×1030

2.29×10138m

2.3×10138m

(b)   n2M2e×r×4×π2×G×M8h2

 (6×1024)2×1.5×1011×4×(3.14)2×6.67×1011×2×1030(6.6×1034)2

2.5×1074

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