Unit 13: Gravitation

13.1 - Newton's Law of Gravitation

Any particle in the universe attracts any other particle with a gravitational force whose magnitude is:

F = G\frac{m_1m_2}{r^2}

where $m_1$ and $m_2$ are the masses of the particles, $r$ is their separation, and $G$ is the gravitational constant. With standard SI units (N, m, and kg), the gravitational constant is $6.67\times10^{-11} N \cdot m^2 / kg^2$ .

Shell Theorem

Although Newton’s law of gravitation applies strictly to particles, we can also apply it to real objects as long as the sizes of the objects are small relative to the distance between them. The Moon and Earth are far enough apart so that, to a good approximation, we can treat them both as particles—but what about an apple and Earth? From the point of view of the apple, the broad and level Earth, stretching out to the horizon beneath the apple, certainly does not look like a particle.

Newton solved the apple–Earth problem with the shell theorem:

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A uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell’s mass were concentrated at its center.

13.2 - Gravitation and the Principle of Superposition

Gravitational forces obey the principle of superposition; that is, if $n$ particles interact, the net force $\vec F_{1,net}$ on a particle labeled particle 1 is the sum of the forces on it from all the other particles taken one at a time:

\vec F_{1, net} = \sum_{i=2}^{n} \vec F_{1i}

in which the sum is a vector sum of the forces on particle 1 from particles 2, 3, . . . , $n$ .

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This is a very fancy way of saying that you can add all the forces to get a net force.

The gravitational force on a particle from an extended body is found by first dividing the body into units of differential mass $dm$ , each of which produces a differential force $d \vec F$ on the particle, and then integrating over all those units to find the sum of those forces:

\vec F_1 = \int d \vec F

13.3 - Gravitation Near Earth's Surface

The gravitational acceleration $a_g$ of a particle (of mass $m$ ) is due solely to the gravitational force acting on it. When the particle is at distance $r$ from the center of a uniform, spherical body of mass $M$ , the magnitude $F$ of the gravitational force on the particle can be found using Newton's Law of Gravitation. Thus, by Newton’s second law,

F_g = ma_g

and thus,

a_g = \frac{GM}{r^2}

Because Earth’s mass is not distributed uniformly, because the planet is not perfectly spherical, and because it rotates, the actual free-fall acceleration $\vec g$ of a particle near Earth differs slightly from the gravitational acceleration $\vec a_g$ , and the particle’s weight (equal to $mg$ ) differs (very slightly) from the magnitude of the gravitational force on it.

13.4 - Gravitation Inside Earth

A uniform shell of matter exerts no net gravitational force on a particle located inside it. Therefore, an object at the center of the earth would experience no net gravitational force, as all parts of the earth would be pulling on it equally.

The gravitational force on a particle inside a uniform solid sphere, at a distance $r$ from the center, is due only to mass $M_{ins}$ in an “inside sphere” with that radius $r$ :

M_{ins} = \frac{4}{3} \pi r^3 \rho = \frac{M}{R^3}r^3

where $\rho$ is the solid sphere's density, $R$ is its radius, and $M$ is its mass. We can assign this inside mass to be that of a particle at the center of the solid sphere and then apply Newton’s law of gravitation for particles. We find that the magnitude of the force acting on mass $m$ is:

F = \frac{GmM}{R^3}r

13.5 - Gravitational Potential Energy

The gravitational potential energy $U(r)$ of a system of two particles, with masses $M$ and $m$ and separated by a distance $r$ , is the negative of the work that would be done by the gravitational force of either particle acting on the other if the separation between the particles were changed from infinite (very large) to $r$ . This energy is the gravitational potential energy:

U = -\frac{GMm}{r}

Systems of Several Particles

If a system contains more than two particles, its total gravitational potential energy $U$ is the sum of the terms representing the potential energies of all the pairs. Here's an example for three particles, with masses $m_1$ , $m_2$ , and $m_3$ :

U = -(\frac{Gm_1m_2}{r_{12}}+\frac{Gm_1m_3}{r_{13}}+\frac{Gm_2m_3}{r_{23}})

Escaping the Gravitational Pull

An object will escape the gravitational pull of an astronomical body of mass $M$ and radius $R$ (that is, it will reach an infinite distance) if the object’s speed near the body’s surface is at least equal to the escape speed, given by:

v = \sqrt{\frac{2GM}{R}}

Common Escape Speeds

13.6 - Planets and Satellites: Kepler's Laws

The motion of satellites, both natural and artificial, is governed by Kepler's laws:

The Law of Orbits: All planets move in elliptical orbits with the Sun at one focus.

The Law of Areas: A line joining any planet to the Sun sweeps out equal areas in equal time intervals. (This statement is equivalent to conservation of angular momentum.)

The Law of Periods: The square of the period $T$ of any planet is proportional to the cube of the semimajor axis $a$ of its orbit. For circular orbits with radius $r$ :
$T^2 = (\frac{4\pi^2}{GM})r^3$
where $M$ is the mass of the attracting body—the Sun in the case of the solar system. For elliptical planetary orbits, the semimajor axis $a$ is substituted for $r$ .

13.7 - Satellites: Orbits and Energy

When a planet or satellite with mass m moves in a circular orbit with radius $r$ , its potential energy $U$ and kinetic energy $K$ are given by

U = -\frac{GMm}{r} \textrm{ and } K = \frac{GMm}{2r}

The mechanical energy $E = K + U$ is then

E = -\frac{Gmm}{2r}

For an elliptical orbit of semimajor axis $a$ ,

E = -\frac{GMm}{2a}

13.8 - Einstein and Gravitation

Einstein pointed out that gravitation and acceleration are equivalent. This principle of equivalence led him to a theory of gravitation (the general theory of relativity) that explains gravitational effects in terms of a curvature of space.