It is a law of physics that the gravitational force exerted...

It is a law of physics that the gravitational force exerted by the Earth on an object above the Earth's surface varies inversely as the square of its distance from the Earth's center. Thus, an object's weight $w(x)$ is related to its distance $x$ from the Earth's center by a formula of the form $$ w(x)=\frac{k}{x^{2}} $$ where $k$ is a constant of proportionality that depends on the mass of the object. (a) Use this fact and the assumption that the Earth is a sphere of radius $4000 \mathrm{mi}$ to obtain the formula for $w(x)$ in Example $4 .$ (b) Find a formula for the weight $w(x)$ of a satellite that is $x$ mi from the Earth's surface if its weight on Earth is $6000 \mathrm{lb}$. (c) How much work is required to lift the satellite from the surface of the Earth to an orbital position that is 1000 mi high?

It is a law of physics that the gravitational force exerted by
the Earth on an object above the Earth's surface varies inversely as the square of its distance from the Earth's center. Thus, an object's weight $w(x)$ is related to its distance $x$ from the Earth's center by a formula of the form
$$
w(x)=\frac{k}{x^{2}}
$$
where $k$ is a constant of proportionality that depends on the mass of the object.
(a) Use this fact and the assumption that the Earth is a
sphere of radius $4000 \mathrm{mi}$ to obtain the formula for $w(x)$
in Example $4 .$
(b) Find a formula for the weight $w(x)$ of a satellite that is
$x$ mi from the Earth's surface if its weight on Earth is
$6000 \mathrm{lb}$.
(c) How much work is required to lift the satellite from the
surface of the Earth to an orbital position that is 1000
mi high?

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Key Concepts

Determining the Constant of Proportionality

When a relationship between two quantities is given by a proportionality constant, this constant must be calculated using known values. In problems involving gravitational weight, the constant is determined by substituting a known weight at a specific distance (such as at the surface of the Earth) into the inverse square equation.

Calculating Work from a Variable Force

Work done by a variable force is computed by integrating the force function over the displacement. In this problem, because the gravitational force changes with distance, the work required to lift an object is found by integrating the gravitational force function over the path of the object's movement.

Inverse Square Law

The inverse square law is a principle stating that a specified physical quantity or intensity is inversely proportional to the square of the distance from the source. In the context of gravitational forces, it means that the force of gravity decreases as the square of the distance between two bodies increases.

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