Determination of Height of a Mountain by Simple Pendulum

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 T₁ and if g and g₁ be the values of the acceleration due to gravity at the bottom and at the top of the mountain respectively, then,

T₁/T = √(g/g₁)

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

g = GM/R² … … (1)

and at the top of the mountain,

g₁ = 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/g₁ = (R + h)² / R² = (H x h/R)²

or, (R + h) / R = √(g/g₁) = T₁/T

or, 1 + h/R = T₁/T

or, h/R = T₁/T – 1

or, h = R (T₁/T – 1) … … … (3)

In equation, (3), T, T and T₁ is known, so h can be find out.