4.1 Development of Force Concept

Learning Objectives

By the end of this section, you will be able to:

Understand the definition of force.

Dynamics is the study of the forces that cause objects and systems to move. To understand this, we need a working definition of force. Our intuitive definition of force—that is, a push or a pull—is a good place to start. We know that a push or pull has both magnitude and direction (therefore, it is a vector quantity) and can vary considerably in each regard. For example, a cannon exerts a strong force on a cannonball that is launched into the air. In contrast, Earth exerts only a tiny downward pull on a flea. Our everyday experiences also give us a good idea of how multiple forces add. If two people push in different directions on a third person, as illustrated in Figure 4.3, we might expect the total force to be in the direction shown. Since force is a vector, it adds just like other vectors, as illustrated in Figure 4.3(a) for two ice skaters. Forces, like other vectors, are represented by arrows and can be added using the familiar head-to-tail method or by trigonometric methods. These ideas were developed in Two-Dimensional Kinematics.

(a) Overhead view of two ice skaters pushing on a third. One skater pushes with a force F two, represented by an arrow pointing up, and a second skater pushes with a force F one, represented by an arrow pointing from left to right. Vector F one and vector F two are along the arms of the two skaters acting on the third skater. A vector diagram is shown in the form of a right triangle in which the base is vector F one pointing east and perpendicular is shown by vector F two pointing north. The resultant vector is shown by the hypotenuse pointing northeast. (b) Free-body diagram showing only the forces acting on the skater. — **Figure 4.3** Part (a) shows an overhead view of two ice skaters pushing on a third. Forces are vectors and add like other vectors, so the total force on the third skater is in the direction shown. In part (b), we see a free-body diagram representing the forces acting on the third skater.

Figure 4.3(b) is our first example of a free-body diagram, which is a technique used to illustrate all the external forces acting on a body. The body is represented by a single isolated point (or free body), and only those forces acting on the body from the outside (external forces) are shown. (These forces are the only ones shown, because only external forces acting on the body affect its motion. We can ignore any internal forces within the body.) Free-body diagrams are very useful in analyzing forces acting on a system and are employed extensively in the study and application of Newton’s laws of motion.

A more quantitative definition of force can be based on some standard force, just as distance is measured in units relative to a standard distance. One possibility is to stretch a spring a certain fixed distance, as illustrated in Figure 4.4, and use the force it exerts to pull itself back to its relaxed shape—called a restoring force—as a standard. The magnitude of all other forces can be stated as multiples of this standard unit of force. Many other possibilities exist for standard forces. (One that we will encounter in Magnetism is the magnetic force between two wires carrying electric current.) Some alternative definitions of force will be given later in this chapter.

(a) A spring of length x, fixed at one end, is shown in horizontal position. (b) The same spring is shown pulled by a person by a distance of delta x. The restoring force F restore is represented by an arrow pointing left toward the position where the spring is fixed. (c) A spring balance containing a spring stretched a distance delta x is shown. The restoring force is represented by an arrow F restore pointing toward the left in the direction opposite to the elongation of the spring. — **Figure 4.4** The force exerted by a stretched spring can be used as a standard unit of force. (a) This spring has a length $x$ when undistorted. (b) When stretched a distance $Δ x$ , the spring exerts a restoring force, $F_{restore}$ , which is reproducible. (c) A spring scale is one device that uses a spring to measure force. The force $F_{restore}$ is exerted on whatever is attached to the hook. Here $F_{restore}$ has a magnitude of 6 units in the force standard being employed.

Note

Take-Home Experiment: Force Standards

To investigate force standards and cause and effect, get two identical rubber bands. Hang one rubber band vertically on a hook. Find a small household item that could be attached to the rubber band using a paper clip, and use this item as a weight to investigate the stretch of the rubber band. Measure the amount of stretch produced in the rubber band with one, two, and four of these (identical) items suspended from the rubber band. What is the relationship between the number of items and the amount of stretch? How large a stretch would you expect for the same number of items suspended from two rubber bands? What happens to the amount of stretch of the rubber band (with the weights attached) if the weights are also pushed to the side with a pencil?

Test Prep for AP Courses

The diagram looks like a solid black oval race track with 16 equally-spaced short perpendicular hash marks crossing the track. The oval is longer than it is tall and the top and bottom parts of the track are horizontal and parallel to the bottom of the page. To complete the oval, the race track starts to curve in a half-circle starting from the second perpendicular hash mark to the right of the top center hash mark. The curve continues for four perpendicular hash marks and the horizontal bottom part of the track starts two perpendicular hash marks to the right of the center bottom hash mark. The half-circle is mirrored on the left side of the track. On the right side of the oval is an arrow curving around the track and pointing up with the text “Direction of Cars’ Motion.” There is one solid line above the track and one to the right outside of the track. Both lines are indicated by the lowercase letter d. One line starts at the first hash mark’s location on a horizontally straight bit of track in the upper right side and indicates that the size of the line goes for 4 additional hash marks. The second line starts at the end of the horizontal stretch on the upper left of the track and curves around for 4 additional hash marks. — **Figure 4.5**

The figure above represents a racetrack with semicircular sections connected by straight sections. Each section has length d, and markers along the track are spaced d/4 apart. Two people drive cars counterclockwise around the track, as shown. Car X goes around the curves at constant speed v_c, increases speed at constant acceleration for half of each straight section to reach a maximum speed of 2v_c, then brakes at constant acceleration for the other half of each straight section to return to speed v_c. Car Y also goes around the curves at constant speed v_c, increases its speed at constant acceleration for one-fourth of each straight section to reach the same maximum speed 2v_c, stays at that speed for half of each straight section, then brakes at constant acceleration for the remaining fourth of each straight section to return to speed v_c.

(a) On the figures below, draw an arrow showing the direction of the net force on each of the cars at the positions noted by the dots. If the net force is zero at any position, label the dot with 0.

There are two dashed oval tracks representative of the larger oval track shown in figure 04_M1_track_img earlier. Both tracks have the same 16 equally spaced perpendicular hash marks shown in the earlier figure but there are dashes around the tracks instead of a solid line. Between each of the perpendicular dashes is one smaller dash at the perpendicular dash and three additional dashes. Centered above the figure on the left is the text Car X and over the figure on the left is Car Y. There are 6 black dots positioned on each of the Car X and Car Y tracks. The position of the six dots on the Car X track on the left are as follows: The first dot is on dashed line on the perpendicular hash mark at the very center left of the track. Moving to the right past three perpendicular hash marks the second dot is on the top horizontal line on the second of four small dashes before the top center perpendicular hash mark. The third dot is one perpendicular hash mark to the right of the center perpendicular hash mark. The fourth dot is two perpendicular hash marks from the third dot and one perpendicular hash mark above the right center perpendicular hash mark. The fifth dot is on the bottom horizontal line of the track and about one and one-third perpendicular hash marks to the right of the center bottom perpendicular hash mark. The sixth dot is on the bottom horizontal line about one and two-third perpendicular hash marks to the left of the center bottom perpendicular dash. — **Figure 4.6**

The position of the six dots on the Car Y track on the right are as follows:

The first dot on the left center of the track is at the same position as it is on the Car X track.
The second dot is just slight to the right of the Car X dot (less than a dash) past three perpendicular hash marks moving to the right.
The third dot is about one and two-thirds perpendicular hash marks to the right of the center top perpendicular has mark.
The fourth dot is in the same position as the Car X figure (one perpendicular hash mark above the center right perpendicular hash mark).
The fifth dot is about one and two-third perpendicular hash marks to the right of the center bottom perpendicular hash mark.
The sixth dot is in the same position as the Car Y dot (one and two third perpendicular hash marks to the left of the center bottom hash mark).

(b)

i. Indicate which car, if either, completes one trip around the track in less time, and justify your answer qualitatively without using equations.

ii. Justify your answer about which car, if either, completes one trip around the track in less time quantitatively with appropriate equations.

Two identical diagrams feature oval-shaped dashed paths with alternating plus and minus signs. Each path includes black dots with arrows indicating specific vector directions at points along the loop. — **Figure 4.7**

Car X is shown on the left, and Car Y is shown on the right.

Car X takes longer to accelerate and does not spend any time traveling at top speed. Car Y accelerates over a shorter time and spends time going at top speed. So Car Y must cover the straightaways in a shorter time. Curves take the same time, so Car Y must overall take a shorter time.

ii.

The only difference in the calculations for the time of one segment of linear acceleration is the difference in distances. That shows that Car X takes longer to accelerate. The equation $\frac{d}{4 v_{c}} = t_{c}$ corresponds to Car Y traveling for a time at top speed.

Substituting $a = \frac{v_{c}}{t_{1}}$ into the displacement equation in part (b) ii gives $D = \frac{3}{2} v_{c} t_{1}$ . This shows that a car takes less time to reach its maximum speed when it accelerates over a shorter distance. Therefore, Car Y reaches its maximum speed more quickly, and spends more time at its maximum speed than Car X does, as argued in part (b) i.

Which of the following is an example of a body exerting a force on itself?

a person standing up from a seated position
a car accelerating while driving
both of the above
none of the above

A hawk accelerates as it glides in the air. Does the force causing the acceleration come from the hawk itself? Explain.

A body cannot exert a force on itself. The hawk may accelerate as a result of several forces. The hawk may accelerate toward Earth as a result of the force due to gravity. The hawk may accelerate as a result of the additional force exerted on it by wind. The hawk may accelerate as a result of orienting its body to create less air resistance, thus increasing the net force forward.

What causes the force that moves a boat forward when someone rows it?

The force is caused by the rower’s arms.
The force is caused by an interaction between the oars and gravity.
The force is caused by an interaction between the oars and the water the boat is traveling in.
The force is caused by friction.

Section Summary

Dynamics is the study of how forces affect the motion of objects.
Force is a push or pull that can be defined in terms of various standards, and it is a vector having both magnitude and direction.
External forces are any outside forces that act on a body. A free-body diagram is a drawing of all external forces acting on a body.

Conceptual Questions

Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly.

What properties do forces have that allow us to classify them as vectors?

Adapted from College Physics 2e by OpenStax (openstax.org), licensed under CC BY-NC-SA 4.0. Changes were made. License: CC-BY-NC-SA-4.0.