Question

In the following four periods

(i) Time of revolution of a satellite just above the earth’s surface $\left( T _ { s t } \right)$

(ii) Period of oscillation of mass inside the tunnel bored along the diameter of the earth $\left( T _ { m a } \right)$

(iii) Period of simple pendulum having a length equal to the earth’s radius in a uniform field of 9.8 N/kg $\left( T _ { s p } \right)$

(iv) Period of an infinite length simple pendulum in the earth’s real gravitational field $\left( T _ { i s } \right)$

• A. $T _ { s t } > T _ { m a }$
• B. $T _ { m a } > T _ { s t }$
• C. $T _ { s p } < T _ { i s }$
• D. $T _ { s t } = T _ { m a } = T _ { s p } = T _ { i s }$

Answer 1

Answer: (C)

$T _ { s p } < T _ { i s }$

Answer 2

(i) $T _ { s t } = 2 \pi \sqrt { \frac { ( R + h ) ^ { 3 } } { G M } }$ $= 2 \pi \sqrt { \frac { R } { g } }$

[As h <<R and $\left. G M = g R ^ { 2 } \right]$

(ii) $T _ { m a } = 2 \pi \sqrt { \frac { R } { g } }$

(iii) $T _ { s p } = 2 \pi \sqrt { \frac { 1 } { g \left( \frac { 1 } { l } + \frac { 1 } { R } \right) } } = 2 \pi \sqrt { \frac { R } { 2 g } }$ [As l = R]

(iv) $T _ { i s } = 2 \pi \sqrt { \frac { R } { g } }$