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February 14, 2013 / Science in Suburbia

The science of physical attraction

First off, this post is probably not about what you think it’s about.

Second off, even if you are alone on Valentine’s Day, take comfort in the fact that it doesn’t mean that no one is attracted to you. In fact, everyone is attracted to you! Every THING is attracted to you.

Gravitationally, that is.

We’re used to defining gravity as the force that holds us down on Earth, but there’s actually a gravitational force between any two objects that have mass. There’s an attractive force between you and Earth, you and the moon, you and your good-looking neighbor…so why are we always pulled toward Earth and not our neighbors?

The answer can be explained mathematically.

OLYMPUS DIGITAL CAMERA

This is Newton’s law of universal gravitation. (For those of you who are pedantic about your vector notation, just mentally insert the necessary magnitude signs.) That big G is an experimentally determined constant- it’s always the same, so for our purposes we can just ignore it.

The gravitational force between two objects depends on three things: the mass of the first object, the mass of the second object, and the distance between the two objects.

Since both masses are in the numerator of the fraction, the gravitational force is directly proportional to these masses. If either mass increases, the force increases by the same factor; if either mass decreases, the force decreases by the same factor. For example, if the gravitational force between two objects is 20N and the mass of one object is doubled, then the force doubles to 40N. If the force between two objects is 10N and the mass of one object is halved, then the force is halved to 5N. If the force between two objects is 30N, the mass of one object is doubled, and the mass of the other object is halved, then the force is doubled and then halved; these factors cancel each other out and the force remains constant at 30N.

The square of the distance between the objects is in the denominator of the fraction, so the gravitational force is inversely proportional to the square of the distance. The “inversely proportional” part means that if the distance decreases, then the force increases (and vice versa); the “square” part is the detail that many high school physics students struggle with. (The reasoning behind the inverse square property comes up a lot in physics and is pretty cool, so it definitely merits its own post- for now we’ll just stick to the numbers.) If gravitational force was inversely proportional to the distance itself, when that distance doubled, the force would be cut in half. However, since we have an inverse square relationship, when the distance is increased by a factor, then the force is decreased by the square of that factor. If the force between two objects is 20N and the distance between them is multiplied by 2, the force is divided by 4 to become 5N. If the force between two objects is 20N and the distance between them is divided by 3, then the force is multiplied by 9 to become 180N.

When studying this formula, most (and by “most” I mean “virtually all”) homework and test questions ask you to figure out what will happen to the gravitational force between two objects based on specified changes to the masses and distance.

  • What will happen to the gravitational force between objects A and B if the mass of object A is doubled, the mass of object B is tripled, and the distance between them is quadrupled? (Answer at the end of this post.)

So now back to our original question: why are we always attracted to Earth instead of our good-looking neighbors?

The force between you and your neighbor will be the product of your masses (and that capital-G constant) divided by the square of the distance between the two of you. The force between you and Earth will be the product of your mass and Earth’s mass (and that capital-G constant) divided by the square of Earth’s radius. (In physics, we can frequently pretend that all of an object’s mass is concentrated at its center of mass, which in this case is Earth’s center; the distance between you and Earth’s center is Earth’s radius, which is about 6,378km.)

Your neighbor’s mass is probably 55-80kg.

Earth’s mass is 5.972*10^24kg. That’s 5,972,000,000,000,000,000,000,000kg. Even though we know that gravitational force increases when distance decreases, and you’re much closer to your neighbor than you are to Earth’s center, there’s no overcoming a mass like that (“dat mass” for those of you who are well-versed in internet memes). You’re attracted to everything in a giant gravitational tug-of-war, but Earth always wins.

Coming up soon on SiS: we’ll use the formula we discussed today to see why, barring air resistance, everything really does fall toward Earth at the same rate.

Oh, and the answer to our practice question above: the force is multiplied by 3/8.

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