There are two astronauts somewhere in space, 3 meters apart, and not moving relative to each other. Astronaut Lucy has a total mass of 100 kg including her suit. Astronaut Ricky has a total mass of 140 kg including his suit. Question 1 – Is there a physical attractive force between them? Question 2 – What are the variables which affect the strength of that force? Question 3 – How large is that force? Those questions will be answered in this post.
Newton’s Law of Universal Gravitation
Isaac Newton published this law of gravity July 5, 1687, in his monumental work Philosophiæ Naturalis Principia Mathematica “Mathematical Principles of Natural Philosophy”, often referred to as the Principia. It is widely regarded as one of the most important published works of science.
The Principia addressed massive objects in motion under different situations. It covered motions of celestial bodies and projectiles here on Earth. It also posed problems of motions where several forces acted at once.
Question 1
The answer to the first question is yes. There is an attractive force between their masses. There is attractive force between all masses regardless of size or location. The force is always attractive. It is always present. The force of gravity cannot be shielded against or turned off. The force the first mass exerts upon the second is exactly equal to, but opposite in direction, to the force the second mass exerts upon the first. For example, the force the Earth exerts upon your body’s mass is equal to the force your body exerts upon the Earth’s mass. Your body is forced toward Earth while Earth is forced toward your body equally.
Question 2
There are only two variables which affect the strength of the attractive force of gravity. The strength is affected by the quantities of the two masses which attract each other. The strength is directly proportional to the product of the two masses (mass 1 x mass 2). Two masses of 10 kg and 20 kg have a product of 200. Two masses of 30 kg and 40 kg have a product of 1200. If they are the same distance apart as the first pair, their attractive force is 6x as much as the first.
The strength of the force is also affected by the distance apart (r) of the centers of the two masses. The force of attraction between the masses decreases as the distance between them increases. We can this an inverse relationship. Specifically, the force grows weaker in what is called an inverse square relationship. If two masses are moved twice as far apart, center to center, the attractive force is one quarter as much (1/4). If they are moved three times as far apart, the force is one ninth as much (1/9). Ten times as far apart gives a force of one hundredth (1/100). It never goes to zero. This is known as the inverse square law and is a common relationship in physics.
You can play with the mass, distance, and force relationship in this online simulator. I used it to get values for the graph above.
The only two factors which can vary and affect the strength of the force of gravity are the masses and their separation distance. Force of attraction is directly proportional to the product of the two masses but inversely proportional to the distance between them squared. It is very simple. Notice this is not an equality statement.
It was not until 1797-98 that an experimental apparatus of sufficient sensitivity was used to actually measure the force of attraction between lead balls in a laboratory. It was performed by Henry Cavendish in England. The device suspended two lead balls of 0.73 kg about 25 cm from two larger lead balls of 158 kg.
The twist, or torsion, of the wire supporting the smaller balls allowed the force of attraction to be determined. The wooden rod which connected the two smaller masses was about 6 ft long.
The Cavendish results later allowed the proportional statement above to be expressed as an equation with the introduction of a constant of proportionality value G. This equation allowed force values to be calculated between any two masses at any known distance apart.
Question 3
We will use the previous equation to calculate the attractive force between the astronauts.
There are many varied units used for quantities in physics. Careful attention to them here yields a force in row three of kg m/sec2. This is also called a Newton of force. For reference, a Newton of force is about equal to the weight of an average apple. The astronauts are attracted to each other with a force of a tenth of a millionth of a Newton. Not much. But, it does exist. Over time, the astronauts would show evidence of gravitating toward each other and eventually touch.
The same equation above could be used with values of your mass, the Earth’s mass, and the radius of the Earth. The calculation would yield a force of several hundred Newtons because the Earth’s mass is so large. The value would then be called your weight on the surface of the Earth.
Consider this question. Astronauts in the space station appear to be weightless as they float around the rooms. But, are they weightless? On Earth they would weigh several hundred Newtons. In orbit they are only a small additional distance from the center of the Earth. So, the distance in the equation above is Earth radius + 6% more. That would make the force calculation less, but not by much. The astronauts in the station are not weightless. Not even close. What is going on? Maybe a future post is needed on gravitation and orbits.
How I See It 
From what I’ve been reading, to get prepared for what you will be posting ahead, Einstein deduced that “free-fall” is actually inertial motion. Objects in free-fall do not accelerate downward (e.g. toward the earth or other massive body) but rather experience weightless and no acceleration. In “inertia”, bodies (and light) obey Newton’s first law, moving at constant velocity in “straight lines”. Analogously, in a curved spacetime the world line of an inertial particle or pulse of light is as straight as possible (in space and time). Such a world line is called a “geodesic” and from the point of view of the inertial frame is a “straight line”. A “geodesic” line can also bend, however. In general relativity, objects in “free-fall” follow geodesics of spacetime, and what we perceive as the force of gravity is instead a result of our being unable to follow those geodesics of spacetime, because the mechanical resistance of matter prevents us from doing so. This “law” of attraction (I don’t know if it’s a law), when you say:
“The astronauts are attracted to each other with a force of a tenth of a millionth of a Newton. Not much. But, it does exist. Over time, the astronauts would show evidence of gravitating toward each other and eventually touch.”
I still don’t understand this too well, are you saying that there is still a gravitational field that pulls them closer because they are objects of comparable mass?
Every mass attract every other mass. Period.
The more massive one or both of the bodies, the stronger the attraction.
The closer they are, the stronger the attraction.
They don’t need to be of comparable mass. Replace one of the astronauts with a penny. They will attract each other. But with less force. Which will move the most in response to the force? The penny. But they both would move toward the other.
More confused now? 😦
No Jim, I love it. You inspire me to go back to pick up another degree, not in physics but in biology. You truly inspire me to learn. Thank you.
Learning is good for us. 🙂
I was interested in the “geodesic” line. Is this the actual “curvature” that diminishes the supposed “free fall”?
Hello Maria,
A geodesic is the shortest line between two points. In Euclidean spaces it is a straight line. On a sphere it would be some portion of a great circle (think longitudinal lines). The idea can easily be extended to more complex surfaces in multiple dimensions.
Hi Jim!
Nicely done!
Hi palantir. Thanks for taking that question. I had gone to bed. Gravity got the best of me and took me down.
shortest line — I should have said the shortest path
Despite my engineering education, I long struggled to understand gravity. How, I wondered, could such a “force” exist? As you say, Jim, its reach is effectively infinite. Also, it cannot be shielded. (Wouldn’t that be marvelous if it could?) Given these qualities, I wondered about the speed of such a force. It seems to me that that too must be infinite, i.e., not limited to the speed of light as one might imagine. [Interestingly, the Wikipedia article on gravity states that a Chinese experiment measuring the effects of the moon on tides indicates that gravity is limited to light speed. I have to question this.]
In his general theory of relativity, Einstein also reckoned with this puzzle (thanks for the company, Al) and decided that gravity was not actually a “force”, but rather
After years of puzzlement I decided that this actually makes sense because it explains the conundrums of distance and speed. It’s not a force, it’s a manifestation of the reality that “space and time do not exist separately from one another but rather as a duality that is termed, “spacetime”. Because of its exponential nature, the time dilation effects are only apparent near light speed, which is why Newton’s theory can explain and predict it as though it were a force, something that you have explained very well.
You make some excellent points about gravity. I intentionally limited this post to the classical behavior described by Newton. It accurately describes the vast majority of the actions and motions we ever encounter. I also didn’t attempt to explain the nature of why it exists. That is where Big Al comes into the picture.
The more deeply we understand the natural world, the more we seem to let go of the time honored classical explanations from Newton, et al. They are still useful as a basic tool of how things work. But, they don’t quite answer the questions of why they work that way.
Thanks for your great additions.
Gravity is such a wonderful force in nature … yet, we can’t hold it, touch it, taste it, etc.
Nicely explained. The distinction between mass and weight could also make a good topic for a post or folded into a discussion on orbits. I remember calculating the radius of a geostationary orbit around earth while in high school and the pleasant realization that you can just figure that out yourself in very little time.
It is a ‘pleasant realization’ indeed. Not a mystery. 🙂
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