Why is Gravitational Force Proportional to Mass Product?

[dinesh2002k]

why is that the gravitational force proportional to product of masses why not sum of masses or something else .........

 

[PeroK]

why is that the gravitational force proportional to product of masses why not sum of masses or something else .........

The sum of masses would be inconsistent with the data. In particular, the acceleration would depend on the mass of both objects.

 

[dinesh2002k]

Sir , how do you say that the sum of masses would be inconsistent .I also request you to justify what makes multiplication of the masses correct.

 

[Ibix]

$F=ma$, and we observe near Earth that the acceleration does not change whatever the mass of object we drop (once we correct for air resistance, anyway). What does that tell you about the gravitational force?

 

[dinesh2002k]

That states that the gravitational force completely depends on the mass of the object, given that the complete observation is carried out close to the earth surface (acceleration due to gravity =9.8 m/sec^2 (constant)).

 

[Ibix]

So the force is proportional to the mass, and not to the mass plus something, right?

Let's consider that gravitational force is proportional to $m+M$. Our gravitational force law near Earth's surface would be $F=k(m+M)$ where $k$ is a constant we don't know. What would the acceleration due to gravity be under this law? Is it independent of $m$?

 

[Drakkith]

why is that the gravitational force proportional to product of masses why not sum of masses or something else .........

The ultimate, but perhaps unsatisfactory answer, is that gravitational force depends on the product of the masses because that's just the way the universe works. One consequence of this is that if you change the mass of one of the two objects, the force changes proportionally. That is, if you double the mass of one object, the force also doubles. This would NOT be the case if the force was proportional to the sum of the masses.

This would mean that, in cases where one object vastly out masses the other, such as stars vs planets, doubling the mass of the planet would add only negligible force while doubling the inertia. Two planets (or other objects much less massive than the star) of nonequal mass placed at the same distances from the star might experience wildly different orbits since their accelerations would be different. One might spiral outwards to infinity, or spiral inwards until it impacts the star.

 

[dinesh2002k]

So the force is proportional to the mass, and not to the mass plus something, right?

Let's consider that gravitational force is proportional to $m+M$. Our gravitational force law near Earth's surface would be $F=k(m+M)$ where $k$ is a constant we don't know. What would the acceleration due to gravity be under this law? Is it independent of $m$?

Sir can i know what this M mean (mass + something)

 

[dinesh2002k]

The ultimate, but perhaps unsatisfactory answer, is that gravitational force depends on the product of the masses because that's just the way the universe works. One consequence of this is that if you change the mass of one of the two objects, the force changes proportionally. That is, if you double the mass of one object, the force also doubles. This would NOT be the case if the force was proportional to the sum of the masses.

This would mean that, in cases where one object vastly out masses the other, such as stars vs planets, doubling the mass of the planet would add only negligible force while doubling the inertia. Two planets (or other objects much less massive than the star) of nonequal mass placed at the same distances from the star might experience wildly different orbits since their accelerations would be different. One might spiral outwards to infinity, or spiral inwards until it impacts the star.

Thank you sir.

 

[A.T.]

why is that the gravitational force proportional to product of masses why not sum of masses or something else .........

Have you tried adding 1kg and 2kg to the mass of the Earth? Is the ratio of those sums consistent with the ratio of weight forces we measure for 1kg and 2kg test masses?

 

[vanhees71]

Sir , how do you say that the sum of masses would be inconsistent .I also request you to justify what makes multiplication of the masses correct.

The gravitational force (or rather interaction) is a fundamental notion in contemporary physics, i.e., the mathematical laws describing it correctly cannot be derived from any other more fundamental laws.

Within Newtonian mechanics you describe it as an instantaneous interaction between massive bodies. The feature that distinguishes the gravitational interaction from any other interaction is that this force on a body is proportional to its mass. Now Newton's 3rd Law, which is assumed to be generally valid for all kinds of interactions, tells you that the gravitational force must be proportional to both particle masses, i.e.,
$$\vec{F}_{12}=-\vec{F}_{21} \propto m_1 m_2.$$
Then another empirical fact is that the force is along the straight line connecting the two particles and its magnitude is falling with $1/r_{12}^2$, where $r_{12}$ is the distance between the bodies. Thus the gravitational force between two (point-like) bodies is
$$\vec{F}_{12}=-\frac{G m_1 m_2}{|\vec{r}_1-\vec{r}_2|^2} \frac{\vec{r}_1-\vec{r}_2}{|\vec{r}_1-\vec{r}_2|},$$
where $G$ is Newton's gravitational constant. The minus-sign is due to the fact that the gravitational interaction is always attractive.

 

[Dale]

why is that the gravitational force proportional to product of masses why not sum of masses or something else .........

We see experimentally that if you double either mass you double the force. The product of the masses does that. The sum of the masses does not.

 

[dinesh2002k]

Have you tried adding 1kg and 2kg to the mass of the Earth? Is the ratio of those sums consistent with the ratio of weight forces we measure for 1kg and 2kg test masses?

Mass of earth being 5.97 × 10^24 kg , an added mass of 1kg or 2kg doesn't make a big difference in the gravitational forces ,I think , Sir.

 

[dinesh2002k]

We see experimentally that if you double either mass you double the force. The product of the masses does that. The sum of the masses does not.

Thank you sir

 

[A.T.]

Mass of earth being 5.97 × 10^24 kg , an added mass of 1kg or 2kg doesn't make a big difference in the gravitational forces ,I think , Sir.

Exactly. But we observe that the force on 2kg is twice the force on 1kg, which is consistent with a product, not a sum of the masses.

 

[dinesh2002k]

Exactly. But we observe that the force on 2kg is twice the force on 1kg, which is consistent with a product, not a sum of the masses.

Understood.Thank you for response and guidance, Sir.