Forces, Orbits, and Energy

What You Need to Know

The subject material we are now covering in class is difficult to take in all at once. It is essential that you see the material, if you are to appreciate how the astronomer determines how the mechanism of the universe operates. However, it is equally imperative that we not get bogged down in the physics and that we get on to astronomy. This is a brief summary of the salient points of Newton's Laws.

Newton formulated 3 laws of motion:

- The law of inertia. This had been stated earlier by Galileo, among others.
This law can be summarized as: An object at rest tends to remain at rest;
an object in motion tends to remain in uniform linear motion.
In the absence of forces, masses do not accelerate (change their velocity).
Note that velocity is a
*vector*; it incorporates both magnitude and direction. A chance in the direction of motion requires an acceleration, and hence a force. - The rate of change of motion (the acceleration) is proportional to the
force acting on the body.
**F = m a**provides the relation between the strength and direction of the force**F**acting on a mass**m**, and the resulting acceleration**a**. Note that the mass**m**is defined by this relation. - For every action, there is an equal and opposite reaction.

Newton's Laws describe how matter reacts to forces. Gravity is a force. All matter in the universe feels an attractive force to all other matter in the universe. The gravitational force is described by the expression

This equation is a algebraic expression describing the gravitational force
**F _{g}**
between 2 objects with masses

By Newton's second law, a force produces an acceleration, or a change in
velocity. The algebraic expression for this is **F=ma**. The force **F**
is given by the product of the mass of the object **m** and the acceleration
**a**. If any two of these quantities are known, the third can be determined.
Using these expressions,
the gravitational acceleration, the change in velocity of an object with mass
**m** due to the gravitational
force, is described by the expression

If you are **m** and **M**
represents the mass of the Earth, this acceleration **a** represents your
downward acceleration due to gravity.
Your weight is the upward force exerted on the soles of your feet
(if you're standing) by the surface of the earth to prevent you from falling
towards the center of the Earth. The gravitational force down and
your weight (an upwards force) balance and you do not accelerate. You are
in equilibrium. If instead you are actually falling, you have no
weight (although your mass does not change).

Note that there is nothing to prevent me
from using **M** to represent your mass, and **m**
to represent the mass of the
Earth. In that case, the variable **a** would represent the acceleration
of the Earth due to your mass. This is small, but real. This is an example of
Newton's third law - of equal and opposite reactions. The gravitational force
attracting you to the center of the Earth is the same magnitude (but opposite
in direction) to the gravitational force attracting the center of the Earth
to you. While the forces are the same strength, the accelerations are
different by the ratio of the masses (**F = ma**). Your acceleration is
some 10^{21} times that
experienced by the Earth because of your gravity.

An orbit is merely the trajectory followed by a mass under the influence of the gravity of another mass. Gravity and Newton's laws explain orbits. The gravitational force between two bodies causes an acceleration. This can take the form of a change in the speed (the magnitude of the velocity) or in the direction of motion. In a circular orbit (or for any particle travelling is a circular path) the acceleration is given by the expression

This acceleration in orbit is the gravitational acceleration, so we can equate the two expressions. This gives

There is no mysterious force which keeps bodies in orbit. In fact, bodies in orbit are continuously falling. What keeps them in orbit is their sideways velocity. The force of gravity changes the direction of the motion by enough to keep the body going around in a circular orbit. You can look at it in the following way: as the body falls, the object it falls towards keeps moving to the side so the falling body never gets to it. As an aside, an astronaut in orbit is weightless because he (or she) is continuously falling. Weight is the force exerted by the surface of the Earth to counteract gravity. The Earth, the Sun, and the Moon have no weight! Your weight depends on where you are - you weigh less on the top of a mountain than you do in a valley; your mass is not the same as your weight.

Newton's theory, also called Newtonian mechanics, is a theory of masses and how they act under the influence of gravity. Through using this theory we can determine the masses of objects without having to go to them. We can observe the universe; it is not necessary to experiment with it.

Another important concept is that of energy. There are two kinds of energy
we are concerned with now: kinetic energy (abbreviated **K**) and potential
energy (abbreviated **U**).
Kinetic energy is energy of motion, and is evaluated
using the equation

In an orbit, conservation of energy can be used to explain Kepler's second law. When the distance to the Sun decreases, the potential energy decreases as well. This must be compensated for with an increase in kinetic energy, which means the orbital velocity must be fastest when closest to the Sun.

In a circular orbit, the potential energy is given by the expression

The point of going through this exercise is to illustrate the elegance and the power of Newton's theory. It is a true triumph of the human intellect. It permits Earthbound observers to determine the mass of the universe, based on observations of distances (from parallaxes), periods (from timing), and velocities (which we'll talk about when we discuss light). I do not expect you remember these equations, much less be able to derive the results outlined above. However, I do expect you to understand the concepts involved. You need to know Newton's laws. You need to understand the concepts of forces, orbits, and energy. We will use them repeatedly as we take the measure of the universe.