Question

Amit buys a few grams of gold at the poles as per the instruction of one of his friends. He hands over the same when he meets him at the equator. Will the friend agree with the weight of gold bought? If not, why?

Answer 1

Step 1: Find the weight of a mass at the poles of the earth.

Assume that, we have a mass $m$ placed on the pole of the earth.

We know that the axis of rotation of the earth passes through the poles. So, the only force acting on the assumed mass is the gravitational force.

So, the weight of the mass $m$ at the pole of the earth can be given by:

$mg=\frac{G{M}_{e}m}{R^2}...1$

Where, $G$ is the gravitational constant, $M_e$ is the mass of the earth, and $R$ is the radius of the earth.

Step 2: Find the weight of a mass at the equator of the earth.

Assume that, we have a mass $m$ placed on the equator of the earth.

We know that the axis of rotation of the earth passes through the poles. So, the force acting on the assumed mass is the gravitational force and the centripetal force of the earth.

So, the weight of the mass $m$ at the equator of the earth can be given by:

$mg=\frac{G{M}_{e}m}{R^2}-M_e{\omega }_e^{2}R...2$

Where, $G$ is the gravitational constant, $M_e$ is the mass of the earth, $R$ is the radius of the earth, and ${\omega }_e$ is the angular velocity of the earth.

So, from the equation $1$ and equation $2$.

It is clear that the weight of objects of the same mass is more at the poles and less at the equators.

Therefore, Amit's friend will not agree with the weight of gold.