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Question
Assuming the earth to be a sphere of uniform mass density, how much would a body weigh half way down to the centre of the earth if it weighed 250 N on the surface?
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Solution 1
Weight of a body of mass m at the Earth’s surface, W = mg = 250 N
Body of mass m is located at depth, d = `1/2R_e`
Where
`R_e` = Radius of Earth
Acceleration due to gravity at depth g (d) is given by the relation:
`g = (1-d/R_e)g`
`=(1-(R_e)/(2xxR_e))g = 1/2 g`
Weight of the body at depth d,
W' = mg
`= mxx1/2g = 1/2 mg = 1/2 W`
`=1/2 xx 250 = 125 N`
Solution 2
As`g_d = g(1-d/R) => mg_d = mg(1-d/R)`
Here `d= R/2`
`:. mg_d = (250)xx (1-"R/2"/R) = 250 xx 1/2 = 125 N`
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