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# Lecture Outlines

Chapter 12

## Physics, 3rd Edition

James S. Walker

2007 Pearson Prentice Hall This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted. The work and materials from it should never be made available to students except by instructors using the accompanying text in their classes. All recipients of this work are expected to abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors who rely on these materials.

Chapter 12
Gravity

Units of Chapter 12
Newtons Law of Universal Gravitation Gravitational Attraction of Spherical Bodies Keplers Laws of Orbital Motion Gravitational Potential Energy Energy Conservation Tides

## 12-1 Newtons Law of Universal Gravitation

Newtons insight: The force accelerating an apple downward is the same force that keeps the Moon in its orbit. Hence, Universal Gravitation.

## 12-1 Newtons Law of Universal Gravitation

The gravitational force is always attractive, and points along the line connecting the two masses:

## 12-1 Newtons Law of Universal Gravitation

G is a very small number; this means that the force of gravity is negligible unless there is a very large mass involved (such as the Earth).

If an object is being acted upon by several different gravitational forces, the net force on it is the vector sum of the individual forces. This is called the principle of superposition.

## 12-2 Gravitational Attraction of Spherical Bodies

Gravitational force between a point mass and a sphere: the force is the same as if all the mass of the sphere were concentrated at its center.

## 12-2 Gravitational Attraction of Spherical Bodies

What about the gravitational force on objects at the surface of the Earth? The center of the Earth is one Earth radius away, so this is the distance we use:

Therefore,

## 12-2 Gravitational Attraction of Spherical Bodies

The acceleration of gravity decreases slowly with altitude:

## 12-2 Gravitational Attraction of Spherical Bodies

Once the altitude becomes comparable to the radius of the Earth, the decrease in the acceleration of gravity is much larger:

## 12-2 Gravitational Attraction of Spherical Bodies

The Cavendish experiment allows us to measure the universal gravitation constant:

## 12-2 Gravitational Attraction of Spherical Bodies

Even though the gravitational force is very small, the mirror allows measurement of tiny deflections.

Measuring G also allowed the mass of the Earth to be calculated, as the local acceleration of gravity and the radius of the Earth were known.

## 12-3 Keplers Laws of Orbital Motion

Johannes Kepler made detailed studies of the apparent motions of the planets over many years, and was able to formulate three empirical laws:

1. Planets follow elliptical orbits, with the Sun at one focus of the ellipse.

## 12-3 Keplers Laws of Orbital Motion

2. As a planet moves in its orbit, it sweeps out an equal amount of area in an equal amount of time.

## 12-3 Keplers Laws of Orbital Motion

3. The period, T, of a planet increases as its mean distance from the Sun, r, raised to the 3/2 power.

This can be shown to be a consequence of the inverse square form of the gravitational force.

## 12-3 Keplers Laws of Orbital Motion

A geosynchronous satellite is one whose orbital period is equal to one day. If such a satellite is orbiting above the equator, it will be in a fixed position with respect to the ground.

These satellites are used for communications and and weather forecasting.

## 12-3 Keplers Laws of Orbital Motion

GPS satellites are not in geosynchronous orbits; their orbit period is 12 hours. Triangulation of signals from several satellites allows precise location of objects on Earth.

## 12-3 Keplers Laws of Orbital Motion

Keplers laws also give us an insight into possible orbital maneuvers.

## 12-4 Gravitational Potential Energy

Gravitational potential energy of an object of mass m a distance r from the Earths center:

## 12-4 Gravitational Potential Energy

Very close to the Earths surface, the gravitational potential increases linearly with altitude:

Gravitational potential energy, just like all other forms of energy, is a scalar. It therefore has no components; just a sign.

## 12-5 Energy Conservation

Total mechanical energy of an object of mass m a distance r from the center of the Earth:

This confirms what we already know as an object approaches the Earth, it moves faster and faster.

## 12-5 Energy Conservation

Another way of visualizing the gravitational potential well:

## 12-5 Energy Conservation

Escape speed: the initial upward speed a projectile must have in order to escape from the Earths gravity

## 12-5 Energy Conservation

Speed of a projectile as it leaves the Earth, for various launch speeds

## 12-5 Energy Conservation

Black holes: If an object is sufficiently massive and sufficiently small, the escape speed will equal or exceed the speed of light light itself will not be able to escape the surface. This is a black hole.

## 12-5 Energy Conservation

Light will be bent by any gravitational field; this can be seen when we view a distant galaxy beyond a closer galaxy cluster. This is called gravitational lensing, and many examples have been found.

12-6 Tides
Usually we can treat planets, moons, and stars as though they were point objects, but in fact they are not. When two large objects exert gravitational forces on each other, the force on the near side is larger than the force on the far side, because the near side is closer to the other object. This difference in gravitational force across an object due to its size is called a tidal force.

12-6 Tides
This figure illustrates a general tidal force on the left, and the result of lunar tidal forces on the Earth on the right.

12-6 Tides
Tidal forces can result in orbital locking, where the moon always has the same face towards the planet as does Earths Moon.

If a moon gets too close to a large planet, the tidal forces can be strong enough to tear the moon apart. This occurs inside the Roche limit; closer to the planet we have rings, not moons.

Summary of Chapter 12
Force of gravity between two point masses:

## G is the universal gravitational constant:

In calculating gravitational forces, spherically symmetric bodies can be replaced by point masses.

Summary of Chapter 12
Acceleration of gravity:

Keplers laws:

## 1. Planetary orbits are ellipses, Sun at one focus

2. Planets sweep out equal area in equal time 3. Square of orbital period is proportional to cube of distance from Sun

Summary of Chapter 12
Orbital period:

## Gravitational potential energy:

U is a scalar, and goes to zero as the masses become infinitely far apart

Summary of Chapter 12
Total mechanical energy:

Escape speed:

Tidal forces are due to the variations in gravitational force across an extended body
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