Motion in One Dimension
5.1 What Is a Force?
Chapter 5
Forces in One Dimension
What determines how far a bungee-jumper falls before he starts moving upward? In this chapter you acquire the tools to answer this, sometimes critical, question.
We have introduced a structured approach to find the motion of a object from its acceleration and the initial conditions (see Fig.5.1). But how do we find the acceleration? We could measure it directly, as we did with an accelerometer, but this is not satisfactory. Physics is not only about describing what is happening, but rather about explaining and predicting motion. In order to determine the motion, we need to be able topredictthe acceleration of an object.
In this chapter we will show you that the acceleration of an object is related to the forces acting on the object. In order to predict the motion, we need to:
• Find what forces are acting on an object.
• Introduce quantitative models for the forces—we need numbers for the forces in order to have numbers for the acceleration.
• Determine the acceleration from the forces using Newton’s second law of motion.
• “Solve” the motion from the differential equations of motion and the initial con- ditions.
We will address these points in detail: First we show how to identify the forces acting on an object. Then we introduce Newton’s second law that relates forces to acceleration. Finally, we introduce models for some of the most common forces in the macroscopic world.
84 5 Forces in One Dimension
Identify
What object is moving?
How is the position,x(t), measured? (Origin and axes of coordinate sys- tem).
Find initial conditions, x(t0) andv(t0).
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Model Find the forces acting on the object.
Introduce models for the forces.
Apply Newton’s second law of motion to find the acceleration, a = a(x, v, t).
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Solve
Solve the equation:
d2x dt2=a(x, v, t), with the initial conditions x(t0) =x0andv(t0) = v0using analytical or nu- merical techniques.
The solution gives the po- sition and velocity as a function of time, x(t), andv(t).
→
Analyse Check validity ofx(t) and v(t).
Usex(t) andv(t) the an- swer questions posed.
Evaluate the answers.
Fig. 5.1 The structured problem solving approach
(c) (a)
(b)
(d)
before
after
F
G
(e)(f)
Fig. 5.2 Illustration ofatwo hands pulling on a rubber band,ba rubber band attached to a wall,c a spring attached to a wall,da rope attached to a wall,ea book above a table,fa book on a table, gdeformation of surface bump,hmagnification of bump
In addition, the rubber band stretches. The harder you pull, the longer the rubber band becomes.
If you instead tie the rubber band to the wall, the rubber band will again elongate as you pull. But now it is not a person experiencing the pull, it is the wall. A force may indeed act on the wall as well as on a person. If we pull harder, the rubber band elongates further, and we expect the force on the wall to become larger. This suggests that the elongation of the band is a reasonable way to measure the magnitude of the force, and this is indeed the usual way to define a force: by prescribing how we can measure it. We can measure forces by how they deform rubber bands.
Now, there is nothing special about a rubber band. We could replace the rubber band by a spring or any other material. As you pull on the spring, the spring elongates.
If the spring is stiff, it elongates less than the rubber band, but it still elongates
5.1 What Is a Force? 85 somewhat. A rope may be even stiffer, and would deform even less, but a careful measurement would show that also a rope elongates when pulled.
We are nearing a definition of a force. We could define a force as an interaction—a pull or a push on an object—that can be measured by the deformation of a spring. In this case the magnitude of the force increases with the deformation of the spring. This definition is not altogether satisfactory, but it illustrates a particular type of force—
what we call acontact force. Contact forces occur where an object is in contact with other objects.
What about a book lying on a table, are there any forces acting on the book? The book is not moving, so we may be tempted to say no. Unfortunately, this would be wrong. When we pulled on the wall with the spring, the wall was not moving, but there was still a force acting on it. What about the book—where are the forces acting on the book? First, there is one force we have not discussed so far, the force of gravity.
This is one of the fundamental forces in nature: There are gravitation forces between any two objects pulling the objects toward each other. There is a gravitational force from the Earth on the book, which pulls the book downward.
What is stopping the book from moving? The table! But how? We cannot see any deformation as we could for the rubber band. This is only because you do not look carefully enough. If you zoomed in on the contact between the book and the table using a microscope, you would see that the surface of the table and the surface of the book are not flat, but rough. Small surface irregularities can be seen along the surfaces. When the book is placed on the table, these small irregularities deform (see Fig.5.2). Each irregularity acts as a small spring, and when the irregularities are deformed, that deformation is related to the contact force between the two objects.
The sum of the forces from all of these small springs is the force from the table on the book.
If we zoom further in on the contact between one surface irregularity and the table, we realize that the contact force really is a sum of electromagnetic forces between the atoms on the surface of the book and the atoms on the surface of the table. The atoms are never in actual contact, but as the book and the table are pressed toward each other, electromagnetic forces will act from the table on the book. The electromagnetic force has been shown to be part of the electromagnetic and the weak nuclear force, which is one of three fundamental forces. The other two are gravity and the strong nuclear force, which is responsible for the interactions between subatomic particles and for the interactions in the nucleus. These are the three main forces in nature, and all forces are reducible to these forces.
In most cases, we will study objects that consist of many atoms. In practice, we cannot find the sum of the forces from all the individual atoms to find the magnitude of the force, but we will instead develop simplified models for the macroscopic forces we encounter. We will call such modelsforce models, and they will be our main tools for determining forces on macroscopic objects.
86 5 Forces in One Dimension