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Why would we weigh less on equator as the weighing machine measures the force by which we push ground and that should not change if are on equator or poles?

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Short answer: centrifugal force may not be a real force but centripetal force certainly is.

Long answer: A weighing machine (which I am assuming is equivalent to a spring balance) is not measuring the force with which the Earth attracts an object. It is actually measuring the reaction force that the ground (or, to be exact, the spring) exerts in the object.

Normally we assume that these two forces are equal and opposite, so we refer to the magnitude of either of them as an object's "weight". But this is only approximately correct because it assumes that the object is in equilibrium i.e. it is moving at constant velocity relative to an inertial frame of reference.

However, an object sitting on the equator is not in equilibrium, because it is rotating about the Earth's axis. So there must be a net centripetal force $mr\omega ^2$ acting on the object (where $m$ is the mass of the object, $r$ is the radius of the Earth and $w$ is the rate of rotation of the Earth about its axis). In other words

$\text{Attraction due to gravity }-\text{ Reaction force exerted by spring }=mr\omega^2$

or

$mg - \text{ Apparent weight} = mr\omega^2$

If we want to work in a frame of reference in which the object is stationary (which is not an inertial frame of reference) then we can re-arrange this equation to give:

$\text{Apparent weight} = mg - mr\omega^2$

and then we "explain" the $mr\omega^2$ term as being due to a "centrifugal force".

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