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 $9.8 \,N/kg (T_{sp})$
(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 $GM= gR^2$]
$ \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 $l= \infty$]