If inertial mass were equal to for example half of gravitational mass why would things not still fall at same rate? [duplicate]

Question

I have read that it is the equality of inertial and gravitational mass that causes all objects to fall at the same rate irrespective of size. But if this were not true, what effect would we see? Would inertial mass being half g-mass mean an object twice as massive would accelerate twice as fast? If so, why?

EDIT: Here is where it is asserted that the equality is responsible for all objects falling at same rate:

Why do two bodies of different masses fall at the same rate (in the absence of air resistance)?

Answer 1

Objects would be accelerated by gravity at a uniform rate regardless of their mass provided that inertial mass and gravitational mass were proportional. There's no requirement for them to be identical, from that perspective.

Answer 2

Say you have Earth's mass as $M$ and your object's mass and distance from Earth is $m$ and $r$

Using Newtonian gravity we have the Force experiences by the object to be $F_g = \frac{GMm}{r^2}$ where $G$ is a constant.

Now the force experienced by earth due to the object's mass is $F_{g2} = \frac{GMm}{r^2}$

Acceleration of both objects, $F_1 = ma_1 \to \frac{GMm}{r^2} = ma_1$ so $a_1 = \frac{GM}{r^2}$

$F_2 = \frac{GMm}{r^2} = Ma_2 \to a_2 = \frac{Gm}{r^2}$

The acceleration that the earth experiences is negligible in comparison. The acceleration the object experiences is only due to Earth's mass. (Note that for considerably large objects the acceleration due to the object's mass would make it seem like the object is accelerating quicker towards Earth than other objects because Earth is moving towards the object fairly quickly too)

If inertial mass $F = ma$ was different from the gravitational mass you would just have a different acceleration for each object BUT every object would still be accelerated at the same magnitude.

ex: $m_g = \frac{m_i}{2}$ then $F_g = \frac{GMm_g}{r^2} = m_ia$ so that $a = \frac{GM}{2r^2}$ and you'd have halved acceleration.

Answer 3

Assume that Force by each if $W = \frac{GM_gm_g}{R^2}$

Where $m_g$ and $M_g$ are the gravitational masses of both body under test and earth respectively Suppose that $m_i$ is the body inertial mass Assuming $m_i=km_g$

Where k is not 1 but a proportionality constant

Newton 2nd law:

$W = \frac{GM_gm_g}{R^2}=k m_g a$

Where $a$ is the acceleration

==>

$a=\frac{G M_g m_g}{k R^2}$ ==> independent of the both values of $m_i$, and $m_g$, but different than the case when:

k=1

The acceleration is different in its absolute value but independent of the inertial/gravitational mass of the falling body