Reading Assignment: Arny pp. 69-73, Chpt. 2

**The Law of Inertia**: If the sum of the forces on an object add to zero then the object's velocity (speed + direction) will remain constant. This is sometimes stated as, "An object in motion tends to stay in motion, and an object rest tends to stay at rest." The idea here is simple, do nothing to an object and it will keep doing what it has been doing all along. The idea of inertia is that an object will resist a change in its motion (either making it move or trying to stop it from moving). Note that since neither speed nor direction of motion changes that we always have motion in a straight line when there are no forces.**F = ma**: This law states that the acceleration (change in velocity, which can be a change in speed, direction, or both) experienced by an object is directly proportional to the amount of force exerted on it. The constant of proportionality is the object's mass. This is how mass is essentially defined. Sometimes it's refered to as inertial mass.**Action and Reaction**: This law states that when two bodies interact, they create equal and opposite forces on each other. (Examples: Two gravitating bodies exerting the same gravitational force on each other, two skateboarders pushing on one another, a ball bouncing off the wall, astronaut throwing a hammer, etc...)

(DEMO)

The force that keeps planets, moons, and whatnot in orbits is gravity, the natural attraction of all matter to other matter. We can visualize how to put something in orbit by imagining firing a canonball from a very high mountain. The greater the initial velocity we give the ball the farther down range it will go before falling to the Earth. If we get it at just the right speed it will continue to fall toward the Earth, but the Earth's curvature will continue to curve away from it and it will make a full circle and eventually hit the back end of the canon.

So an object in orbit is indeed experiencing gravity. For example the Space Shuttle when it is in orbit around the Earth is always falling toward the Earth, but it is also heading tangentially at just the right speed to stay in a circle (or ellipse). So it is always under the influence of gravity. The astronauts appear weightless because they and the Shuttle are all falling together at the same rate.

IMPORTANT: If m_{2} << m_{1} then we may ignore m_{2} and then the relation is just

This fact can be used to measure the *mass* of the Sun. For all planets in the Solar System we may write

- Energy may neither be created or destroyed, but it can change form

**Kinetic Energy**: the energy of motion

for any object with mass, m, and velocity, v, its kinetic energy, KE, is given byKE = 1/2 mv ^{2}**Potential Energy**: the amount of work that can potentially be done

This has no simple definition, it is defined in a mathematical way. For an object, with mass, m, under the gravitational attraction of another with mass, M, and a distance, r, away the potential energy, PE, isPE = - GMm/r

E is a constant and so KE and PE can only change in such a way that as one becomes smaller the other becomes larger by the same amount. In other words we can convert potential energy into kinetic energy and vice versa.

When you throw a ball up into the air, it initially has no Potential Energy, but has all Kinetic Energy. As the ball ascends it slows down, losing Kinetic Energy but gaining Potential Energy (the distance between the ball and Earth is increasing). When it reaches the top of its trajectory it has no velocity and therefore no Kinetic Energy, therefore it now has *only* Potential Energy. As the ball falls back to Earth, the situation reverses; Potential Energy converting to Kinetic Energy. All the while the value of E stays fixed.

- If the value of E is negative, then the system is considered
**Bound** - If the value of E is zero, then the system is just
*barely***Unbound** - If the value of E is positive, the the system is
**Unbound**

That means if you have mass, m, and are standing on the surface of a planet with Mass, M, and Radius, R, then the force of gravity acting on you is

Now notice also that it is true by Newton's 2nd Law that

(NOTE: the force you feel due to gravity is your *weight*)

**Escape Velocity**

If you wish to escape from the surface of a gravitating object there is a special velocity you must have in order to accomplish this. It's called the escape velocity. The best way to understand this is to think in terms of energy. In order to be able to escape from a planet you want to have an initial velocity that will make your total mechanical energy be unbound. Recall that the total mechanical energy is written as

An interesting idea that was raised not long after Newton's time was that if an object were to have just enough mass enclosed in a small enough radius the escape velocity would be greater than the speed of light, and then not even light could escape the surface of this object. It was then known as a "Black Star". Today this is rather similar to the idea of a Black Hole.

The conic section that an orbit will exhibit depends on its total energy. If the total energy of the system is negative then the orbit is bound and is an ellipse. If the energy is exactly equal to zero then it is unbound and follows a parabolic shape. If the energy is positive, then the orbit is unbound and follows a hyperbola. This is summed up in the following table...

Energy | Bound/Unbound | Shape | Eccentricity |

E < 0, minimum | Bound | Circle | e = 0 |

E < 0 | Bound | Ellipse | 0 < e < 1 |

E = 0 | Unbound | Parabola | e = 1 |

E > 0 | Unbound | Hyperbola | e > 1 |

Return to Class Notes Page