Name: Gravitational Potential
Uploaded: 2019-07-19
Duration: 4 min 53 s
Description: As we move an object up against gravity, we do work on it, and that work gets stored as the potential energy of the object. Let's understand the math behind...

Question

Derive an expression for variation in gravitational acceleration of the Earth at with latitude.

Accepted Answer

The latitude of a point is the angle Φ between the equatorial plane and the line joining that point to the centre of the earth. The latitude of the equator is 0° and that of poles is 90°.

Now, consider a body of mass ‘m’ at a point P with latitude ‘Φ’ on the surface of the earth. Let ‘gΦ’ be the acceleration due to gravity at point P

Due to the rotational motion of the earth about its axis, the body at P experiences a centrifugal force which is given by $m r ω^{2}$ . Now resolve this centrifugal force into two rectangular components. Its component along the radius of the earth is $m r ω^{2} \cos Φ .$

Now the body is acted upon by two forces its weight mg acting towards the centre of the earth and the component $m r ω^{2} \cos Φ$ acting radially outward. The difference between the two forces gives the weight of that body at that point is given by

$N o w \cos Φ = O ’ P / O P = r / R m g_{ϕ} = m g - m r ω^{2} \cos Φ \dots \dots \dots \dots . . (1) ∴ r = R \cos Φ$

Substituting in Equation (1)

$m g_{ϕ} = m g - m (R \cos Φ) ω^{2} \cos Φ ∴ g_{ϕ} = g - R ω^{2} \cos^{2} Φ$

This is an expression for acceleration due to gravity at a point P on the surface of the earth having latitude Φ.

Derive an expression for variation in gravitational acceleration of the Earth at with latitude.

Important Questions on Gravitation

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