Gravity between two unequal masses. Do both masses move?

Question

I've been watching videos about gravity and I have a question
My understanding is that mass have gravity and gravity is a force which attract other object with mass. For example, I jump up and the Earth's gravity pulls me down.

1. So my question is, is it always the case that the smaller mass that move towards the bigger mass?

2. Does the bigger mass EVER move towards the smaller mass?

3. If two objects with same mass are left in a vacuum, they meet in the middle point of the distance, right?

4. so what if one of the object has little bit more mass? i would assume the bigger mass would still move towards the middle point (but bit shorter)

5. If the above is true, can we technically move the Earth by us (human population) jumping indefinitely?

Though, since Earth's mass is 5.972x10^24 and the mass for human population would be around 4.9x10^11 (assuming 70kg avg weight for 7 billion people), it would have a minimal effect but given that we would jump infinity, we can technically move it, I think?

Answer 1

In all cases, the two objects move towards one another. In fact they experience exactly the same gravitational force. However, because acceleration equals force over mass $$\mathbf{a} = \frac{\mathbf{F}}{m}$$

that equal forces causes the heavier object to accelerate much less than the lighter one. But technically, the Earth does move towards you very slightly when you jump. However, it first moves slightly away from you because in order to jump you have to push it. By the time you land it returns to its original position.

The two objects will meet at their centre of gravity. That is to say if, for example, one mass is twice as big as the other, the meeting point will be one quarter of the way from the heavy mass to the light one. In general it is the point where

$$m_1 * r_1 = m_2 * r_2$$

We can't move the Earth by jumping indefinitely because the push-away from the jump exactly cancels the pull-towards from gravity. There is no net motion. This is the same reason you can't move a boat by sitting inside and kicking the walls.

Answer 2

Does the bigger mass EVER move towards the smaller mass?

Yes.

$F = KMm/r^2$

$M*a_{M}=F$

$m*a_{m}=F$

As you see the smaller the mass the higher the acceleration and in consequence the higher the traveled distance in a given time t.

If the above is true, can we technically move the Earth by us(human population) jumping indefinitely?

No.

Each time we jump, assuming all of us are in the same place and jump in a synchronized way, the earth will go a bit in the opposite direction but it comes back and when we land again the earth will be in the same position where it was before we jumped.

For the full explanation you have to add, to the equations above, the conservation of momentum and solve the system.

$M*v_{M} = m*v_{m}$ (conservation of momentum for the moment we jump)

You can also use the already demonstrated fact that the center of mass, for a system of N masses, does not move in time if no external force exist. In the case of the earth - people system, there is no external force and no matter where we are and how we jump the center of mass of all people + earth will always stay in the same place.

Answer 3

There is a mutual attraction from gravity, and we generally only consider the smaller object here on earth because the earth is so massive, the acceleration of the earth is negligible. This is because $a = F/m$, and with equal $F$ between the two objects, the acceleration will scale as $a\propto 1/m$. For the earth, this leaves $a$ ridiculously small, but technically non-zero.

There are some good example cases of both objects moving toward each other, and one particular case is the Pluto-Charon system. In this case, Pluto is more massive that Charon, but both objects are in mutual orbit and constantly "falling" toward one another. This can be observed based on the fact that both objects orbit a point outside of either mass as seen below (publicly available image from wikipedia):

Now, these objects have angular momentum, so they will never meet, but I think it is a good example of how a small mass affects a larger mass under the influence of gravity.

Answer 4

yes, the earth will accelerate towards you , however the Earth's acceleration will be so small for all practical purposes that you usually do not consider it. Earth's acceleration is small because the mutual forces between you and the earth are the same, but the masses are different, so this results in different accelerations (remember: $F=ma$). Now if you could put all humans in on place and jump, will this accelerate earth's in a noticeable way?Assume there are 7 billion humans, each averaging $100~\text{kgm}\;,$ the total mass will be $7 \space 10^{11}~\text{kg}\; .$ The earth's acceleration will be proportional $GM_\text{humankind}/R_\text{earth }^2$ which is approximately $0.007~\text{m}/\text{s}^2$, which is small but still measurable.