How to Solve Gravitational Force Problems

What goes up, must come down -- unless you can reach escape velocity.
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The discoveries of Sir Isaac Newton revolutionized our understanding of the natural world. Of all his many contributions, one of the most far-reaching was his theory of gravity. Although gravity is the weakest of the four major forces, it's also one that plays an immense role in our everyday lives -- because as weak as it is, the mass of the Earth is so great that its pull on us is very strong. We can calculate the force of gravitational attraction between two objects using Newton's equations.

    Write down Newton's equation for gravitational force, F = G (M x m) / r squared, where M is the mass of one object, m is the mass of the other object, and r is the distance between the centers of the two masses. If you're standing on Earth's surface, for example, r is the distance from Earth's center to you (or to your center, to be more precise, but typically there's no need for that level of accuracy). G is a universal constant. It is a very small number: 6.67 x 10 ^ -11 newton meters squared per kilograms squared. The units at the end of the constant cancel with the units in the equation so that your answer is always in newtons, the standard unit of force.

    Determine the distance between the centers of the two objects. if you're working a quiz problem, you'll probably be given this information. If you're doing a calculation for an object at or near Earth's surface, you can use the average radius of Earth, 6,371 kilometers, and add the height of the object above the ground.

    Determine the masses of the two objects. If Earth is one of the two objects, its mass is 5.9736 x 10 ^ 24 kilograms -- an extremely large number.

    Plug these numbers into the equation. Let's say, for instance, that your weight is 80 kilograms and you are standing on Earth's surface. If you plugged all the numbers above into the equation, you would have the following:

    Force = ( (6.67 x 10 ^ -11 newton meters squared per kilograms squared) * (5.9736 x 10^24 kilograms) * (80 kilograms) ) / (6371 x 10 ^ 3 meters) squared = 785.3 newtons. Multiply your answer in newtons by 0.224809 to get 177 pounds -- which is in fact how much you weigh. Note that weight is just a measurement of force, so when we say pounds we're really talking about how much force Earth exerts on you, which varies depending on your mass.

    Notice something interesting? Not only does Earth exert a force on you, but you also exert a force on the Earth. Remember Newton's equation for force, however:

    Force = mass x acceleration

    If you divide the force you exert on Earth (785.3 newtons in our example) by the mass of Earth, you get the acceleration of Earth due to your gravitational pull. The mass of the Earth is so huge that this acceleration is ridiculously small -- in fact, for all practical intents and purposes, it's negligible. If you divide your mass of 785.3 newtons by your mass of 80 kilograms, however, you get 9.81 meters per second squared -- a very substantial acceleration.

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