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Question
How will you ‘weigh the sun’, that is estimate its mass? The mean orbital radius of the earth around the sun is 1.5 × 108 km.
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Solution 1
Orbital radius of the Earth around the Sun, r = 1.5 × 1011 m
Time taken by the Earth to complete one revolution around the Sun,
T = 1 year = 365.25 days
= 365.25 × 24 × 60 × 60 s
Universal gravitational constant, G = 6.67 × 10–11 Nm2 kg–2
Thus, mass of the Sun can be calculated using the relation,
`M = (4pi^2r^3)/(GT^2)`
= `(4xx(3.14)^2 xx (1.5xx10^11)^3)/(6.67xx10^(11)xx(365.25xx24xx60xx60))^2`
=`(133.24xx10)/(6.64xx10^(4)) = 2.0xx10^(30)` kg
Hence, the mass of the Sun is 2 × 1030 kg.
Solution 2
The mean orbital radius of the Earth around the Sun
R = 1.5 x 108 km = 1.5 x 1011 m
Time period, T = 365.25 x 24 x 60 x 60 s
Let the mass of the Sun be M and that of Earth be m.
According to law of gravitation
`F= G(Mm)/RA^2` ...(i)
Centripetal force
`F = (mv^2)/R = m.R.omega^2` ...(ii)
`(GMm)/R^2 = m.R.omega^2`
=`(mR.4pi^2)/T^2`
`:. M = (4pi^2R^3)/(G.T^2)` [∵ `omega = (2pi)/T`]
= `(4xx(3.14)^2xx(1.5xx10^11)^3)/(6.67xx10^(-11)xx(365.25 xx 24 xx 60xx60)^2)`
= `2.009 xx 10^(30) kg = 2.0 xx 10^(30) kg`
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