10.1 The Universal Forces
Though we feel and see forces in many ways, e.g. as exerted by springs and strings or frictional forces as well as gravity, all forces that we have considered arise due to gravity and due to the nature and structure of the materials that usually oppose gravity.
Consider the mass hanging from a wire at the front of the lecture hall. As we know, the forces on the mass are balanced: the downward force of gravity is balanced by the upward tension of the wire which has something to do with the strength of the material from which it is made. The friction of the air and the contact of the wire with a hook in the ceiling are due to the nature of these materials. The nature and structure of materials are manifestations of the chemical forces due to the electric and magnetic (electromagnetic) interactions of the molecules, atoms and electrons. Within the atoms, the nuclei are subject to nuclear forces.
But how do the forces arise? The wire is evident even if the force it exerts is not visible. However, there is no obvious connection between the earth and the mass: the force seems to be carried invisibly. Moreover, the force of gravity seems to have no limits: we know that the gravitational attraction of the sun, the moon, all the planets and presumably every star in the galaxy, and so on affect the motion of the earth and moon. The tides are caused predominantly by the gravitational attraction of the oceans and earth to the moon and sun.
Gravitational forces and electrical forces in classical physics are described by so-called Universal laws that have an inverse square dependence on the distance between two objects. The force exerted by one object on another depends on its properties of mass and electric charge. Magnetic forces depend on electric charge, but also on the relative motion of the objects. The view of modern physics produces results consistent with the classical approximation, but electric and magnetic forces are unified and quantum in nature, and gravitational forces are attributed to the effect of mass on the geometry of space and time. Einstein's General Theory of Relativity is an example of such an explanation of gravity.
Electric forces are dominant in the binding together of atoms, made up of positively charged nuclei and negatively charged electrons. Gravity holds the sun, the solar system, and the galaxy together. But a moments thought about the electrical forces in the nucleus of an atom leads to the realization that there must be forces even stronger than the electric forces. This nuclear binding force is called the strong force. One more force completes the set of fundamental forces: the so-called weak nuclear force. This force leads to the radioactive decay of the neutron and other radioactive nuclei.
The modern view of physics is that these fundamental forces govern all physical processes. Modern quantum theory shows that electromagnetic, weak and strong nuclear interactions are all similar in structure differing only in the kind of "charge" leading to the force (e.g. electric charge leads to electromagnetic forces, "weak charge" leads to weak interactions, and "strong charge," also called color charge leads to strong interactions. The electric charge of an ordinary object is always an integer multiple of a fundamental unit of charge e, which in SI units of Coulombs is 1.6 x 10^(-19) C. Electric charge
comes in two types referred to as positive (+) and negative (-). The charge of the
electron is -e and of the proton is +e.
Negatively charged objects have more electrons than protons, and positively
charged objects have fewer electrons than protons.
Quarks are particles with charge
and 
, for example the top quark has charge +2/3 and the
anti-top quark has charge -2/3. The bottom quark has charge -1/3, and the
anti-down quark has charge +1/3. Electric forces are attractive and repulsive. Gravitational charge, i.e. mass, has SI units of kilograms, but gravitational forces are attractive only. There are also two kinds of weak charge, but three kinds of color charge referred to as RED, GREEN, and BLUE. The combination of all three makes a color neutral object in the same way that the combination of the two kinds of electric charge make an electrically neutral object. The fundamental forces are listed below.
Table
10.2 Inverse Square Laws
Classical formulation of the electrical and gravitational forces provide that the force exerted by one object on another is proportional to the product of the electric charge or mass and inversley proportional to the square of the distance between them. Inverse square behavior is characteristic of any physical quantity that depends on the areal density, that is number per unit area, of something. When we study the nature of forces, we will see that that something is called flux and is proportional to the strength of a source, namely the charge or the mass of the object exerting the force. Such an areal density is like an intensity, and it should be familiar that the intensity of light from a source, such as a candle or a light bulb falls off with distance, in fact as the square of the distance. Another example using light: the intensity of a beam of light on a card becomes smaller as the card is tilted with respect to the direction of propogation of the light beam. This is the main cause of our seasons: as the Earth tilts on its axis, the intensity of sunlight in the Northern Hemisphere changes and more or less heat is absorbed by the Earth.
10.3 Gravitational and Electrical Forces
Two objects with masses 
and 
exert gravitational forces
on one-another and two objects with electrical charge 
and 
exert electrical forces. These forces act as if all the mass or charge
was concentrated at the center and directed toward (or possibly
away from, for electrical forces) the center of the mass exerting the force.
The magnitudes of the forces on body 1 due to body 2
are given by:
There is one difference which should be noted: the gravitational force seems to be only an attractive force, that is body 2 exerts a gravitational force on body 1 which is directed toward it. This is not the case for the electrical force which is attractive if the charges are of opposite sign and repulsive if the charges have the same sign. The mass and charge of the electron and proton are
m
kg; m
kg
q
coul; 
coul
The gravitational force between two objects was expressed by Newton and the value of the constant G has been determined by experimenters such as Cavendish. G is considered constant and universal, that is it does not change and its value in our solar system is identical to its value everywhere else in the universe. This cannot be proven to be absolutely true and is always subject to test. In fact we know that k is quite similar in other star systems and galaxies by observing atomic radiations which depend on its value.
The formula for gravitational force shows that an object exerts a force that is proportional to its mass. In this case, the mass acts just like electrical charge, as a source of the force. Recall our definition of mass, as a quantitative measure of a body's inertia. The source of gravitational force is an additional role for mass to play and sometimes a distinction is made between inertial mass, which appears in
and gravitational mass which appears
in 
We do know, through modern experiments, that
these two forms of mass are very nearly if not exactly equal and so we
shall simply use the term mass in both cases.
10.4 Numerical Exercises
1. What are the gravitational and electrical forces in the
hydrogen atom where the electron and proton are separated by an average distance
m?
The total force is dominated by the electric force and may seem quite small, but consider this question: what is the centripetal acceleration of the electron as it moves around the proton in a circular orbit?
So
Use the gravitational force on 1 kg (or any mass) at the surface of the earth, 6380 km from the center, to determine the earth's mass.
Newton confirmed the "universality" of his law of gravity by comparing the
centripetal
acceleration of the moon to that of an object at the Earth's surface. In both
cases, the acceleration is, to a good approximation, due only to gravity.
What is the centripetal acceleration of the moon, given that its mean
distance from the earth is 
(center to center) and mean orbital
period is T=27.3 days?
The acceleration at the surface of the earth, 
from
the center is 
, which multiplied by 
is 
.
10.5 Weight
Recall that we consider weight an ill defined quantity. It depends on the device used to measure it as well as on where the device is located. Generally, the weight of an object is determined by a scale that is stationary on the surface of the earth. One thing we left out before, but that we'll now include is the fact that the Earth is rotating and therefore that you, I, and any object on the scale is in fact accelerating. The centripetal acceleration is
Let's draw a Free Body Diagram:
.
My "weight" as read by the scale is
and v is found by the circumference of the circle of
latitude at angle 
with radius given by 
. At Ann Arbor
with latitude 
, that circumference is
Earth (
) and the period of rotation (one day).
You'll find that 
and 
, i.e. it is downward.
At the equator 
has its maximum value
(
), and at the poles, it is zero.
Thus the effect of the centripetal acceleration due to the rotating Earth
is to reduce the "weight" of an object by up to about 0.4% of the 
.
A related result is the acceleration of a freely falling object near the surface
of the Earth. Of course the sum of the forces is 
only, and the
total acceleration is therefore equal in magnitude and directon to vec g.
But part of this acceleration is centripetal, i.e. at the equator 
, which slightly reduces the rate at which the vertical velocity component
increases. For the moon or any object in orbit, the entire acceleration is
centripetal.