Simplistic question about mass and gravity

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Discussion Overview

The discussion revolves around the relationship between mass and gravity, specifically in the context of a hypothetical planet formed from the asteroid belt with a mass of approximately four Earth masses. Participants explore how this mass would affect the planet's surface gravity and the weight of an object on that planet, while considering various factors such as density and radius.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question whether a planet with four Earth masses would have a gravity four times that of Earth, suggesting that the relationship is not straightforward.
  • One participant notes that surface gravity depends on both mass and the distance from the center of the planet, referencing the formula for gravitational force.
  • Another participant argues that the asteroid belt would need to be significantly denser than Earth to achieve four times the surface gravity, which they believe is unlikely.
  • Some participants discuss the implications of density on the radius of the hypothetical planet, indicating that if density were the same, the radius would increase, affecting the gravity calculation.
  • There is a mention of the weight of a 100 kg person on this hypothetical planet, with some participants attempting to clarify the relationship between mass, gravity, and weight without seeking specific equations.
  • One participant challenges the initial claim about the mass of the asteroid belt, citing a different estimate from Wikipedia that suggests it is much less than four Earth masses.
  • Another participant emphasizes that without knowing the radius of the planet, one cannot definitively answer the question about gravity.
  • There is a discussion about the potential variations in surface gravity depending on the density of the planet.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether a planet with four Earth masses would have four times the gravity of Earth, as multiple competing views remain regarding the effects of density and radius on gravity calculations.

Contextual Notes

There are limitations in the discussion regarding assumptions about density and radius, which are not fully resolved. The calculations and estimates presented depend on these factors, and the discussion reflects uncertainty about the implications of the asteroid belt's mass.

Noisy Rhysling
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Noisy Rhysling said:
I've mention that the asteroid belt, if consolidated, would create a planet with about 4 Earth masses. I went to http://sciencing.com/relationship-gravity-mass-planets-stars-8487902.html and still don't get what I'm wondering.

Would a planet with four Earth mass have a gravity four times that of Earth?
I will assume you are speaking of the acceleration of gravity. Let's compare...
Gravity on Earth-- 9.8 m/s^2
Gravity on Jupiter-- 24.9 m/s^2
EDIT: forgot to put in, Jupiter is over 300x more massive than Earth, so I don't think there is a simple proportion
 
So, the mythical fifth planet would have a gravity that's much closer to Earth's. I figured that. But what would the "weight" of a 100 kg. person be on that planet?
 
Noisy Rhysling said:
So, the mythical fifth planet would have a gravity that's much closer to Earth's. I figured that. But what would the "weight" of a 100 kg. person be on that planet?
The mass of that person (100 kg) X the acceleration of gravity would give you their weight.
 
But I'm mathgnostic.
 
Surface gravity is a function of two things:
1. How much mass is contained within the object.
2. How far the surface is from the center.

If you know these, the surface gravity is simply ##Gm/r^2## (feel free to calculate this for the Earth, and you'll get roughly ##9.8m/s^2##). So if you have a small mass, you can still have a high surface gravity if the planet is very dense, so that it has a small radius.

But I don't think the numbers work out here: the asteroid belt is tiny, at only about 0.05% the mass of the Earth today. It would need to be something like 4,000 times as dense as the Earth to have a four times the surface gravity. That's just not happening.

More realistically, what they might have meant is that if Jupiter had not formed first, then what is today the asteroid belt might have formed into a rocky planet significantly larger than the Earth, with quite a bit more gravity. But Jupiter did form further out, which disrupted the orbits of most of the dust and rocks that would have made up that planet, so that their orbits destabilized and they ended up striking other planets.
 
Okay, so it wouldn't have four times the gravity of Earth.
 
Noisy Rhysling said:
Okay, so it wouldn't have four times the gravity of Earth.
Correct.

And the weight equation you are looking for (mathgnostically) is simply F=ma, which is really W=ma where W is weight, and the a is just the acceleration of gravity on that planet.
 
I'm not actual looking for an equation.
 
  • #10
Noisy Rhysling said:
I'm not actual looking for an equation.
Then I guess we're good here! :smile:
 
  • #11
Comeback City said:
Then I guess we're good here! :smile:
Well, I noted that I was mathgnostic. Maybe I'm doing a poor job of communicating. Let's try again. If a 100 kg man was on a planet with four times the mass of Earth, what would his weight be there?
 
  • #12
F = GMm/r2
ma = GMm/r2
a = GM/r2

W = ma
W = mGM/r2

W... weight
m... mass of body (100 kg in this case)
G... gravitational constant
M... mass of planet (4 Earth masses)
r... distance separating the two masses

You can calculate the weight with this.
 
  • #13
Noisy Rhysling said:
I've mention that the asteroid belt, if consolidated, would create a planet with about 4 Earth masses. I went to http://sciencing.com/relationship-gravity-mass-planets-stars-8487902.html and still don't get what I'm wondering.

I'm not quite sure where you got this, but it is waaay off. According to Wikipedia, "The mass of all the objects of the asteroid belt, lying between the orbits of Mars and Jupiter, is estimated to be about 2.8–3.2×10^21 kg, or about 4% of the mass of the Moon. "
 
  • #14
I wish you fair winds and folllowing seas.
 
  • #15
Noisy Rhysling said:
Would a planet with four Earth mass have a gravity four times that of Earth?

There's not enough information here to answer the question. You would also have to know the radius of the planet; just saying it has four times the mass of the Earth does not determine the radius.
 
  • #16
Noisy Rhysling said:
Well, I noted that I was mathgnostic. Maybe I'm doing a poor job of communicating. Let's try again. If a 100 kg man was on a planet with four times the mass of Earth, what would his weight be there?
Depends upon the density.

If the density were the same, then the radius of a planet with four times the mass would be the cube root of four times Earth's radius, or about 1.6 times the Earth's radius. Since the equation for surface gravity is proportional to ##m/r^2##, the surface gravity would be 4 / (1.6)^2 = 1.6 times Earth's gravity.

If the density were different, the surface gravity could be very different.
 
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