$$\require{cancel}$$

# 6.4: Balanced Forces

• Wendell Potter and David Webb et al.
• Physics at UC Davis

# When the Net Force is Zero

A fundamental aspect of the Galilean model and the way "forces work" is that when we identify all contact and long-range forces acting on a particular object, and if the net force turns out to be zero (the forces balance), then all of those forces acting together have no effect on the object. It is as if there are no forces acting on the object. That is, there is no effect on the object due to its interaction with all the other Objects. If there is no effect of the interactions then there will not be an change occurring to or with the object. Whatever the object " was doing" , it will continue to do. Nothing will change. We need t be more specific here about what we mean by "was doing" and what we mean by change".

When we are standing on the floor in an airport terminal and then one hour later when standing in the aisle of a jet plane going 500 miles per hour with no turbulence, we experience the same sensations on our body. This illustrates a fundamental aspect of the Galilean model of space and time. If the net force is zero on us when standing on the floor of the airport terminal, and the net force on us is also zero when on the jet traveling at a constant speed with respect to the ground, the effect is the same: no change. We continue to do exactly what we were doing.

From our perspective, we were standing still on the ground and standing still on the airplane. But from a person’s perspective at the airport, we were moving at a constant 500 miles per hour when “standing still” on the plane. Evidently, being motionless and moving with a constant velocity are the same thing with respect to how forces work. So, "no change" really means not changing velocity, which can certainly remain continuing with zero velocity, if that is what we were doing. Because forces work this way, we can turn moving at a constant velocity to being motionless by switching to reference frame that is moving with the same velocity with respect to the original reference frame.

When a zero net force works this way—no change in velocity if the net force is zero— technically we say we are in an inertial reference frame. For ou purposes it means the reference frame is not accelerating with respect to an inertial reference frame. For most purposes, the surface of the earth can be considered a inertial reference frame,even though the Earth is rotating at one revolution per day. The resulting acceleration is sufficiently small so that we can usually ignore it.

We can summarize the previous discussion regarding balanced forces (net force equal to zero) in the following way:

$If \sum F = 0,\thinspace then \: \Delta v = 0$

This relationship is traditionally referred to as Newton’s First Law. If for a particular object $$\sum F=0$$ then there is no "change in the motion" magnitude or direction of that object. No "change in the motion" mean that the velocity does not change $$\: \Delta$$v is zero .

A traditional statement of the First Law is, “An object at rest will remain at rest and an object in motion will continue to move in a straight line at constant speed unless acted upon by an unbalanced force.”

# Distinction Between the First and Third Laws

It is sometimes easy to confuse the first and third laws, especially when there are two forces acting on an object. The pairs of forces that balance each other in the First Law act on the same object, so they cannot possibly be Third Law force pairs, even though they are equal and point in opposite directions.

A more complete force diagram for object A of the figure showing two blocks on a table is shown on the figure 6.4.1. Two objects interact with object A: object B by a contact force and the Earth by long-range gravitational force. Object A is at rest, so the two forces that act on it are equal in magnitude and opposite in direction, by virtue of the First Law, not the Third Law. (Also note that we are following the convention of placing on the force diagram for object A only the forces that can be explicitly labeled as acting on A.)

Figure 6.4.1

Newton's First Law is concerned about the situation when all the forces acting on an object are balanced. There are two complementary ways of sing the First Law. If we know that there is no change in motion, then we know that the forces.impulses acting on the object must be balanced. If we know all but one of the forces/impulses, we can solve for the the magnitude and direction of the unknown force/impulse. The second way to use the First Law is to add up the known forces to see if they balance. If they do, than there can be no resulting change in motion of the object.

# Two Fundamental Forces

The electric and gravitational forces are said to be long-range forces. (They are long range compared to the atom-atom or molecule-molecule forces we studied in Part 1 in Chapter 3 on the particle model of matter. Unlike those atom-atom forces, the electric and gravitational forces, although decreasing with distance, still exist–and have profound effects– even at large separation distances.) The magnitude of the electric and gravitational force between two particles both decrease as 1/r2. We often refer to these two forces as inverse-square law forces. In both cases, electric and gravitational, the force acts along the direction of the vector from one particle to the other. In addition, each force depends on a fundamental property of matter: electric charge and gravitational mass, respectively.

These two forces are the manifestations of two of the four fundamental interactions: gravitational, electromagnetic, weak nuclear, and strong nuclear.

The electrical force and gravitational force are similar in many ways but there is one string difference: the gravitational force is always attractive, while the electric force may be either attractive or repulsive.

Figure 6.4.2

There are two kinds of electric charge, which for historical reasons, are referred to as positive and negative. The electric force is repulsive between like charges (either both positive or both negative) and attractive between unlike charges (one positive and one negative).

Figure 6.4.3

The magnitudes of the forces between two particles (or chunks of matter) are proportional to the amount of charge (or mass) possessed by each particle and inversely proportional to the square of their separation.

$Electric ~Force$

$F_E = k\frac{q_1q_2}{r^2}.$

$Gravitational ~Force$

$F_G = G\frac{m_1m_2}{r^2}.$

The constants k and G have fixed values that are dependent on the unit system employed and are characteristic of these two fundamental interactions.

For electric forces, if the charges have the same sign, then the force at either charge points away from the other charge. If the charges have opposite signs then the force at each charge points toward the other charge. In both cases, the magnitude of the forces F1 on 2 and F2 on 1 are equal, due to Newton's Third Law.

The gravitational force is always attractive. The magnitude of the forces F1 on 2 and F2 on 1 are equal.

Our theory (or model) of the electric and gravitational interaction contains two universal constants, G and k. In addition, the mass and charge of the electron and the proton are also universal constants. Masses and charges of objects are not functions of position. It is the forces between them that depend on their positions.

The strength of these two universal forces are determined experimentally. They are some of the “givens” that make our universe what it is. (Whether they could be something else in another universe is an interesting question that some physicists ponder.) For the electric force, the value of the constant k is actually set to a specific value and the size of the coulomb is what is actually determined experimentally:

$k = c^2 \times 10^{-7} \frac{N . m^2}{Coulomb^2} = 9 \times 10^9 \frac{N . m^2}{Coulomb^2} .$

Here c is the speed of light.

$c= 2.998 \times 10^8 \frac{m}{s},$

and the coulomb is the SI unit of electric charge. The experimentally determined electric charge on a proton is $$e= +1.602 \times 10^{-19}$$ coulombs, and the electric charge on an electron is -e. Note carefully that the coulomb is a very large amount of charge compared to the charge on single atomic size particles. Note: the symbol "q" is generally used to represent an arbitrary charge and the symbol “e” is used to indicate the charge on one electron or proton.

In the case of gravitation, the universal gravitational constant has the experimentally determined value,

$G= 6.672 \times 10^{-11} \frac{N.m^2}{kg^2}$.

Of all the universal constants, G is known with the least precision; it is hard to “weigh the Earth.”