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Question:

The relative uncertainty in the period of a satellite orbiting around the earth is 10⁻⁶. If the relative uncertainty in the radius of the orbit is negligible, the relative uncertainty in the mass of the earth is :

10⁻⁶

3×10⁻⁶

6×10⁻⁶

2×10⁻⁶

Solution:

The correct option is C
2×10⁻⁶
Time period,T=2π√(r³/GM)
Squaring om both sides,T²=4(π)²r³/GM
Taking log,ln(T²)=ln(4(π)²r³/GM)
ln(T²)=ln(4(π)²r³G)-ln(M)
2ln(T)=ln(K)-ln(M)
2ΔT/T=ΔM/M
ΔM/M=2×10⁻⁶