Question

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

• 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

Answer: (C)

$(i)T_{st}=2\pi \sqrt{\frac{{\left(R+h\right)}^3}{GM}}=2\pi \sqrt{\frac{R}{g}}$

[As $h<<R$ and $GM=g{R}^2$]

$\left(ii\right)T_{ma}=2\pi \sqrt{\frac{R}{g}}$

$\left(iii\right)T_{sp}=2\pi \sqrt{\frac{1}{g\left(\frac{1}{l}+\frac{1}{R}\right)}}$

$=2\pi \sqrt{\frac{R}{2g}}$ [As $l=R$]

$\left(iv\right)T_{is}=2\pi \sqrt{\frac{R}{g}}$ [As $l=\infty$]