What is the acceleration of a swing as it passes the lowest point and as it turns at the highest point? What are the forces acting? These were a couple of the questions students were asked to discuss in small groups during their first week at university, as part of a tutorial session. On one occasion, two students were unable to reconcile their different viewpoints without teacher intervention. One of them emphasized that the swing moves fastest at the bottom, and concluded that the acceleration must be zero. The other student claimed that there must be a force, since you feel heavier at the bottom. They noted the contradiction, but failed to recognize that acceleration is the derivative of velocity, not the derivative of speed: For the lowest point, the speed is maximum, but the direction of motion changes. These students had certainly been taught all the elements of physics needed to calculate the force and acceleration, but forgot to make the connection on their own. A small hint from the teacher, reminding them about centripetal acceleration, was sufficient.

This paper gives examples of how the forces acting on a person in a swing can be visualized using simple equipment (Fig. 1), as well as with electronic data, e.g., from a smartphone taken along in the swing. This can be a way to strengthen students’ understanding of the relation between force and acceleration—Newton’s second law.^{1}

## Force and acceleration in pendulum motion

What forces act on a playground swing? To a child in a swing, it is obvious that the forces vary during the motion, leading to experiences of alternating between feeling heavier and lighter than normal. To a physics teacher, it may be just as obvious that it is sufficient to consider the tangential forces to obtain an equation for the motion, since the distance *L* from the suspension point remains constant, as the swing moves along a circular arc. However, many students in introductory physics courses have an incomplete understanding of force and acceleration in a pendulum. When asked to help fellow students resolve the paradox of acceleration in the bottom of the swing, they may resort to reformulating the paradox: “There is no acceleration, but there is still a force”—violating Newton’s second law.

Most textbooks discuss pendulum motion primarily in the context of harmonic motion, where the angular acceleration is in focus. Radial forces are discussed only briefly, if at all. Textbooks rarely deal with the experiences of the human body. If the concept of acceleration would be introduced through Newton’s second law in the form ** a** =

**/**

*F**m*rather than through mathematics, it could be accessible already in grade school, long before studying kinematics in one dimension.

Schwarz^{2} describes having taught the pendulum so many times that, after a while, she took it for granted. This could be seen as following a “teaching ritual.”^{3,4} It took her some time playing with simulations in “Interactive physics” to discover that her way of thinking about the acceleration at the lowest point was flawed.

Figure 2 shows examples of free-body diagrams for a pendulum at the turning point, where the centripetal acceleration is zero, and at the lowest point, where the tangential acceleration is zero. For all other parts of the pendulum motion, the total acceleration has both centripetal and tangential components.

## Well-known conceptual problems

Conceptual problems with forces in pendulum motion are common and well documented—and not limited only to students. Shaffer and McDermott^{5} found that less than 15% of teaching assistants and less than 5% of preservice teachers drew correct free-body diagrams for the forces at the bottom of a swing. Most stated that the string tension is equal to the weight of the child and seat. Some explicitly stated that the net force is zero and thus the acceleration is zero, not realizing that neither quantity can be zero if the velocity is changing. For forces at the turning point, the results were only slightly better.

Reif and Allen^{6} found that even some experts who “had recently taught the subject of acceleration in introductory physics courses” gave incorrect answers about acceleration during pendulum motion. They also found that students often invoked inconsistent knowledge elements, causing them to run into paradoxes that they were often unable to resolve even if they recognized them.

Santos-Benito and Gras-Marti^{7} have analyzed textbook drawings of forces in pendulum motion and discovered several mistakes that may lead students to believe that the acceleration is zero at the bottom of the swing, where the velocity is maximum. Comments on this point made by students sometimes go unnoticed (e.g., in Ref. 8). Heron^{9} and Kryjevskaia et al.^{10} describe how responses are often given using fast intuitive processes, while forgetting to apply the slower, more deliberate analytical process required for unusual problems.

## Calculating the acceleration at the lowest point

*calculate*the acceleration at the bottom of the swing, they would have been less likely to claim that the acceleration is zero. The students have certainly learned about centripetal acceleration,

*a*

_{c}=

*v*

^{2}/

*r*. They would also know how to calculate the change in potential energy between the highest point at an angle

*θ*

_{0}and the lowest point, and use it to find the kinetic energy in the lowest point:

*mv*

^{2}/2 =

*mgL*(1 − cos

*θ*

_{0}) for a pendulum with length

*L*. Combining these expressions using

*r*=

*L*gives

The students’ responses reveal that their mental connection between force and acceleration is not sufficiently well established to challenge the paradoxes arising when they apply fragmentary elements of knowledge that may not be applicable in a given situation.

Schuster and Undreiu^{11} have studied the complex cognitive process used by an expert considering the problem of acceleration in different parts of a pendulum motion. Their paper describes “the zigzag thinking path and interplay of reasoning modes, knowledge elements cases and schemas involved.”

## The forces experienced by your own body

What we feel in a playground swing must be related to real forces acting on and within our bodies. During physics courses for preschool teachers, we have found that they accept during discussions that there must be an acceleration in the lowest point of a swing, as it changes from moving down to moving up again on the other side. (As part of the course, they had been taught that a change in motion requires a force.) Newton’s laws apply also to human bodies, even if that aspect is not always prominent in textbooks, where people more typically exert forces by pushing or pulling other bodies.

There is sometimes a more or less explicit reluctance in textbooks to deal with the forces acting on an accelerating human, who is obviously not an inertial system—the textbooks prefer avoiding discussing inertial forces arising within noninertial systems. However, it is important to notice that interaction between the swing and our body is independent of our choice of mathematical description and whether we write Newton’s second law in one of the more conventional forms ** F** =

*m*

**or**

*a***=**

*a***/**

*F**m*or apply d’Alembert’s principle and write it as

**−**

*F**m*

**= 0. Even if they are conceptually different, they are mathematically equivalent.**

*a*The experience of changing weight is more likely to be discussed by textbooks and teachers who apply the operational definition of weight, *m*(** g** −

**), rather than the more common gravitational definition of weight,**

*a**m*

*g**.*The operational definition is closely related to the principle of equivalence between inertial and gravitational mass. It agrees with the intuitive notion of heaviness/lightness in different types of motion. Using this approach, which acknowledges the forces experienced by the human body, the forces in accelerated motion have been successfully taught to middle-school students.

^{12}Textbook choices of definition of weight have been reviewed by Taibu and collaborators.

^{13–15}

The experiments below focus on the force from the swing acting on a person. It should be noted that the body does not feel gravity, but only the forces counteracting it or causing acceleration.

## Visualizing the forces on the body

Figure 1 shows a student holding a Slinky while swinging. The length of the Slinky changes with the forces from the swing during the motion, shorter when you feel lighter and longer when you feel heavier.^{16} This experiment can be enjoyed by children—from preschool and up. The Slinky provides an illustration of what is measured by inertial accelerometer sensors: they measure the vector ** a** −

**in their own coordinate system.**

*g*Today, accelerometers are easily accessible through smartphone apps, such as Physics Toolbox^{17} or Phyphox.^{18} The Play part of Physics Toolbox includes a series of exercises to get familiar with what is measured by the sensors.^{19} [In one-dimensional motion, the acceleration can be obtained by subtracting the readings at rest, whereas in three dimensions, the “linear acceleration” provided by the apps builds on “motion tracking,”^{20} which requires both 3D accelerometer and 3D rotation (gyro) data.]

During the discussions presented in Ref. 21, the students first took for granted that the acceleration would be zero, both at the lowest point of a pendulum and in the related question of forces at the lowest point of a roller coaster loop, referring to the speed being maximum. However, the students quickly realized the errors of their ways of thinking after viewing a spiral toy in pendulum motion (similar to the Slinky in Fig. 1 and in Ref. 16). They were also challenged with the question “what if,” which made one of the students exclaim, and show with hand gestures, that, without acceleration, the roller coaster train would continue along a straight line after the lowest point. The group then discussed how the treatment of pendulum motion in their textbook focused only on the angular acceleration.

## Forces at the turning point

Another question found to be challenging concerns the forces as the swing turns after reaching the highest point. I have met new students who claim they have been taught that the acceleration at the turning point is straight down, just for an instant—i.e., in a direction it cannot move (unless the swing turns at a 90° angle). Still, the students must have learned in high school how to obtain the period for a pendulum, looking into the tangential acceleration. Other students have claimed that the acceleration is zero at the turning point when the velocity is zero—another example of how students often “run into contradictory fragments of knowledge, unable to resolve the paradox,” as discussed by Reif.^{22}

An illustration of the forces of a pendulum reaching the highest point is to bring a bottle partially filled with cordial, coffee, or other colored liquid to the playground. The bottle can be held on a swing seat^{16} or placed at the bottom of a large basket swing (like the one in Fig. 3). This question of what happens to the liquid can also be used in a large lecture-hall setting by letting the bottle swing from a string. The question works well for peer instruction.^{23}

Obviously, if there were no acceleration at the turning point, the liquid surface would be horizontal. “Voting” before performing the demonstration of letting the bottle go sometimes reveals that even physicists may be unfamiliar with this type of question—and the liquid is not influenced by the voting. It may qualify as a question where children are more likely to give correct answers than physicists. Do try the experiment in front of a mirror before attempting the demonstration! The behavior of the liquid is discussed, e.g., in Refs. 24–26.

## Measurements using a smartphone accelerometer

Figure 4 shows accelerometer data from a smartphone resting at the bottom of a playground swing, with coordinate axes reorienting during the motion as indicated in Fig. 3.

A first observation is that the values in the direction of motion are very small, as noted already in Ref. 16. This is a reminder that the accelerometer does not measure acceleration but the vector ** G** =

**−**

*a***. Since the acceleration in the direction of motion is given by the tangential component of the acceleration of gravity, the vector**

*g***−**

*a***has no component in the direction of motion [if the distance**

*g**r*between the sensor and the axis differs from the pendulum length

*L*—or the radius of gyration for a physical pendulum—there will be a small tangential component

*g*(

*r*/

*L*− 1) sin

*θ*].

*ϕ*= 20° in this case) of these two coordinate axes around the third axis (Fig. 5). The values for the

*Z*-axis pointing to the suspension point and the

*Y-*axis pointing to the side can be expressed as

This transformation results in the data shown in Fig. 6. (An alternative way to use the raw data is to calculate the total force as the absolute value of the vector ** G** as measured by the smartphone sensors. For the data shown in Fig. 4, the graph obtained in this way is essentially indistinguishable from the one obtained by rotating the coordinate system—the non-radial components are very much smaller than the radial component of the force.)

## Quantitative estimates

The force from the string on the pendulum bob with mass *m* is lowest at the turning point, and given by *mg* cos *θ*_{0}. The graph in Fig. 6 shows cos *θ*_{0} ≈ 0.8, giving an estimate of the largest angle *θ*_{0} ≈ 37°. The largest value for the force from the string is given by *mg*(3 − 2 cos *θ*_{0}) for a mathematical pendulum. This expression is obtained by using the centripetal acceleration given in Eq. (1)—the centripetal acceleration that is often forgotten in student discussions and responses.

It can be noted that the theoretical value of the deviation from the value *g* in the accelerometer graph is twice as large for the lowest point and with an opposite sign compared to the deviation at the highest point, where the radial force is smallest. This holds for a mathematical pendulum. (For a physical pendulum, the expression holds for a sensor at the radius of gyration, which takes the moment of inertia into account.)

## Discussion

The human body is a sensitive detector of acceleration and changes in acceleration. Rather than excluding the experiences of the body from the physics classroom, we can choose to help students pay attention to the sensory signals and relate them to velocity changes. Visual tools, from toys to smartphones, add additional representations of the forces. We can remind students that Newton’s laws apply also to human bodies and encourage them to use the experiences of their own body as a resource for physics understanding.^{28} The force changes that students experience during acceleration are real, whether in an elevator, car, or amusement ride, or bouncing on a trampoline. Students who have been encouraged and required to make the connection between mathematical descriptions and the experiences of their body would not forget that the increased force they feel as they pass the lowest point of the arc of a swing is needed to keep
them moving along the circular arc.

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**Prof. em. Ann-Marie (Mårtensson-)Pendrill** *has a background in computational atomic physics at the University of Gothenburg, Sweden. She is a fellow of the APS and of the UK Institute of Physics. For many years, she used playground and amusement-park examples in her teaching, including teacher professional development in support of large physics days in the Swedish amusement parks Liseberg and Gröna Lund. During the years 2009–2019, she was the director of the Swedish National Resource Center for Physics Education at Lund University.*