**Determination of height of a mountain by simple pendulum**

We can determine the height of a mountain or a tall building from earth’s surface by a simple pendulum [Figure]. For this, we need to measure the time periods of oscillation at the bottom and at the top of the mountain. Let the two time periods be T and T_{1} and if g and g_{1} be the values of the acceleration due to gravity at the bottom and at the top of the mountain respectively, then,

**T _{1}/T = √(g/g_{1})**

Nov from Newton’s law of gravitation, at the bottom of the mountain,

**g = GM/R ^{2}** … … (1)

and at the top of the mountain,

**g _{1} = GM/(R+h)^{2}** … … (2)

Here, M = mass of the earth, R = radius of the earth and h = height of the Mountain.

Dividing equation (1) by equation (2) we get,

**g/g _{1} = (R + h)^{2} / R^{2} = (H x h/R)^{2}**

or, (R + h) / R = √(g/g_{1}) = T_{1}/T

or, 1 + h/R = T_{1}/T

or, h/R = T_{1}/T – 1

or, **h = R (T _{1}/T – 1)** … … … (3)

In equation, (3), T, T and T_{1} is known, so h can be find out.