Question

In the following four periods
(i) Time of revolution of a satellite just above the earth’s surface $\left(T_{st}\right)$
(ii) Period of oscillation of mass inside the tunnel bored along the diameter of the earth $\left(T_{ma}\right)$
(iii) Period of simple pendulum having a length equal to the earth’s radius in a uniform field of 9.8 N/k $\left(T_{sp}\right)$
(iv) Period of an infinite length simple pendulum in the earth’s real gravitational field $\left(T_{is}\right)$

• A. $T_{st}>T_{ma}$
• B. $T_{ma}>T_{st}$
• C. $T_{sp}<T_{is}$
• D. $T_{st}=T_{ma}=T_{sp}=T_{is}$

Answer 1

The correct option is C $T_{sp}<T_{is}$

$\left(i\right)T_{st}=2\pi \sqrt{\frac{(R+h{)}^3}{GM}}=2\pi \sqrt{\frac{R}{g}}[Ash<<RandGM=R^2]\left(ii\right)T_{ma}=2\pi \sqrt{\frac{R}{g}}\left(iii\right)T_{sp}=2\pi \sqrt{\frac{1}{g(\frac{1}{l}+\frac{1}{R})}}=2\pi \sqrt{\frac{R}{2g}}[Asl=R]\left(iv\right)T_{is}=2\pi \sqrt{\frac{R}{g}}[Asl=\infty ]$