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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.

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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) n2 = M2e×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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