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If the period of revolution of an artifi...

If the period of revolution of an artificial satellite above the earth's surface be T and the density of earth be p, then prove that p `T^(2)` is a universal constant. Also calculate the value of this constant. Given G=`6.67xx10^(-11)m^(3) kg^(-1)s^(-2)`

Text Solution

Verified by Experts

The correct Answer is:
1.41
As `T=2pisqrt((R+h)^(3)/(GM))`
`therefore T^(2)=(4pi^(2)(R+h)^(3))/(GM)`
or `M=(4pi^(2)(R+h)^(3))/(GT^(2))`
For the satellite revolving just above the earth.s surface h = 0. So
`M=(4pi^(2)R^(3))/(GT^(2))`
Also, `M=(4)/(3)piR^(3)rho`
`therefore (4)/(3)piR^(3)rho=(4pi^(2)R^(3))/(GT^(2))`
`therefore rhoT^(2)=(3pi)/(G)`, which is a universal constant.
And `rhoT^(2)=(3xx3.14)/(6.67xx10^(-11)m^(3)kg^(-1)s^(-2))`
`=1.41xx10^(11)kgs^(2)m^(-3)`.
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