Our calculations show that g equals 9.78 m/s^2. The value for g varies by location on the Earth for reasons you will learn about later.

In the steps above, the value for G, the gravitational constant, is used to calculate g, with the mass of the Earth given in the problem. However,

ifg and G are both known, then the mass of the Earth can be calculated, a calculation performed by the English physicist Henry Cavendish in

the late 18th century.

Physicists believe G is the same everywhere in the universe, and that it has not changed since the Big Bang some 13 billion years ago. There

is a caveat to this statement: some research indicates the value of G may change when objects are extremely close to each other.

12.3 - Shell theorem

Shell theorem: The force of gravity outside a

sphere can be calculated by treating the sphere’s

mass as if it were concentrated at the center.

Newton’s law of gravitation requires that the distance between two particles be known in

order to calculate the force of gravity between them. But applying this to large bodies

such as planets may seem quite daunting. How can we calculate the force between the

Earth and the Moon? Do we have to determine the forces between all the particles that

compose the Earth and the Moon in order to find the overall gravitational force between

them?

Fortunately, there is an easier way. Newton showed that we can assume the mass of

each body is concentrated at its center.

Newton proved mathematically that a uniform sphere attracts an object outside the

sphere as though all of its mass were concentrated at a point at the sphere’s center.

Scientists call this the shell theorem. (The word “shell” refers to thin shells that together

make up the sphere and which are used to mathematically prove the theorem.)

Consider the groundhog on the Earth’s surface shown to the right. Because the Earth is

approximately spherical and the matter that makes up the planet is distributed in a

spherically symmetrical fashion, the shell theorem can be applied to it. To use Newton’s

law of gravitation, three values are required: the masses of two objects and the distance

between them. The mass of the groundhog is 5.00 kg, and the Earth’s mass is

5.97×10^24 kg. The distance between the groundhog and the center of the Earth is the

Earth’s radius, which averages 6.38×10^6 meters.

In the example problem to the right, we use Newton’s law of gravitation to calculate the

gravitational force exerted on the groundhog by the Earth. The force equals the

groundhog’s weight (mg), as it should.

The shell theorem

Consider sphere’s mass to be

concentrated at center

·r is distance between centers of

spheres

How much gravitational force

does the Earth exert on the

groundhog?

F = 48.9 N

(^224) Copyright 2000-2007 Kinetic Books Co. Chapter 12