Imagine a simple pendulum suspensded from a support which at infinite height from the surface of the earth. The bob of the pendulum is close to the surface of the earth. Show that the period of such a pendulum (of infinite length) is `T = 2 pi sqrt((R_e)/g)`

Explanation

`implies` Here the point of suspension is at infinite height , the small path of the motion of the bob can be considered almost a linear one.
Mass of the bob = m .
Here , the `mgcostheta` component of the gravitational force Fg (=mg) provides the necessary restoring force.
The restoring force ` = -mgcos theta`
(Since the force is restoring negative sign is put)
`=-mg(x/R_e)` (From figure `cos theta=x/R_e` )
`:. F = -kx ( :. (mg)/R_e` = constant) ... (1)
Equation (1) indicates simple harmonic motion.
If m is the mass of bob, its periodic time
`T =2pisqrt(m/k)`
`T = 2pisqrt((m)/(mg//R_e))`
`= 2pisqrt((R_e)/(g))`