^{1,a)}and Matthew J. Moelter

^{1,b)}

### Abstract

We have found that incorporating computer programming into introductory physics requires problems suited for numerical treatment while still maintaining ties with the analytical themes in a typical introductory-level university physics course. In this paper, we discuss a numerical adaptation of a system commonly encountered in the introductory physics curriculum: the dynamics of an object constrained to move along a curved path. A numerical analysis of this problem that includes a computer animation can provide many insights and pedagogical avenues not possible with the usual analytical treatment. We present two approaches for computing the instantaneous kinematic variables of an object constrained to move along a path described by a mathematical function. The first is a pedagogical approach, appropriate for introductory students in the calculus-based sequence. The second is a more generalized approach, suitable for simulations of more complex scenarios.

The authors thank R. Echols, T. Gutierrez, and K. Saunders for helpful discussions and T.B.'s introductory physics students, particularly T. W. Weber, during the 2008-2010 terms for their enthusiasm. The authors also thank an anonymous reviewer for efforts in implementing our theory, and encouraging us to further address numerical errors inherent in the simple numerical techniques used in this work.

I. INTRODUCTION

II. NEWTON'S LAWS FOR AN OBJECT CONSTRAINED TO A CURVE

III. EFFECTS OF THE CONSTRAINT: POSITION, VELOCITY, AND ACCELERATION

IV. A COMPUTATION-FRIENDLY INCLINED WIRE

V. INTRODUCTORY PHYSICS PEDAGOGY

A. Examination of the normal force

B. Assessment of and response to numerical results

VI. GENERALIZED TREATMENT OF AN OBJECT CONSTRAINED TO A CURVE

VII. EXAMPLE: THE TRISECTRIX OF MACLAURIN

VIII. CONCLUSIONS

IX. ADDITIONAL PROBLEMS

### Key Topics

- Kinematics
- 27.0
- Physics education
- 9.0
- Computer software
- 6.0
- Textbooks
- 6.0
- Calculus
- 4.0

## Figures

Examples of the difficulties with a path having a flat-to-incline transition, as indicated in the dotted region. In (A) there is a discontinuity in the curve, which is difficult to treat numerically. In (B) the transition is smoother but there is a non-constant slope, which is difficult to treat analytically.

Examples of the difficulties with a path having a flat-to-incline transition, as indicated in the dotted region. In (A) there is a discontinuity in the curve, which is difficult to treat numerically. In (B) the transition is smoother but there is a non-constant slope, which is difficult to treat analytically.

An object (bead) constrained to an arbitrary curve (shown solid). The normal force ( ) and weight ( ) are shown. The analytical solution uses a *y*-axis instantaneously parallel to . The numerical solution is better found taking the *y*-axis anti-parallel to .

An object (bead) constrained to an arbitrary curve (shown solid). The normal force ( ) and weight ( ) are shown. The analytical solution uses a *y*-axis instantaneously parallel to . The numerical solution is better found taking the *y*-axis anti-parallel to .

A plot of the function . The two horizontal sections connected by a smooth transition make it a useful candidate for studying an object constrained to a sloped wire.

A plot of the function . The two horizontal sections connected by a smooth transition make it a useful candidate for studying an object constrained to a sloped wire.

Pseudocode for simulating the time evolution of an object (of mass *m*) constrained to move along the curve given by the function . Our experience shows that, similar to the code shown in Ref. ^{ 4 } , this algorithm is quite useable in an introductory physics course.

Pseudocode for simulating the time evolution of an object (of mass *m*) constrained to move along the curve given by the function . Our experience shows that, similar to the code shown in Ref. ^{ 4 } , this algorithm is quite useable in an introductory physics course.

Instantaneous normal force ( ), velocity ( ), and acceleration ( ) vectors displayed for an object constrained to a wire described by . These vectors were obtained using kg, N/kg, m, m, m/s, and, from Eq. (4) , m/s (enhanced online). [URL: http://dx.doi.org/10.1119/1.4773561.1]doi: 10.1119/1.4773561.1.

Instantaneous normal force ( ), velocity ( ), and acceleration ( ) vectors displayed for an object constrained to a wire described by . These vectors were obtained using kg, N/kg, m, m, m/s, and, from Eq. (4) , m/s (enhanced online). [URL: http://dx.doi.org/10.1119/1.4773561.1]doi: 10.1119/1.4773561.1.

Total (solid), kinetic (dashed), and potential (dotted) energies as a function of horizontal position for an object on the wire (see Fig. 3 ). The parameter values are the same as in Fig. 5 , with the object traversing the curve from left to right. The arrows indicate 16 J of initial total mechanical energy. The slight increase in total energy is typical of a numerical integration that uses simple Euler steps.

Total (solid), kinetic (dashed), and potential (dotted) energies as a function of horizontal position for an object on the wire (see Fig. 3 ). The parameter values are the same as in Fig. 5 , with the object traversing the curve from left to right. The arrows indicate 16 J of initial total mechanical energy. The slight increase in total energy is typical of a numerical integration that uses simple Euler steps.

Summary of the motion of a bead constrained to move along a wire in the shape shown in Fig. 3 . The normal force magnitude (top) and direction (center) and the vertical position (bottom) are all plotted as a function of horizontal position. In the bottom plot, the solid curve shows the exact result and the dots are the numerical results. The labelled vertical lines are guides to the eye at interesting pedagogical points (see text). All plots were obtained using the same parameter values as in Fig. 5 .

Summary of the motion of a bead constrained to move along a wire in the shape shown in Fig. 3 . The normal force magnitude (top) and direction (center) and the vertical position (bottom) are all plotted as a function of horizontal position. In the bottom plot, the solid curve shows the exact result and the dots are the numerical results. The labelled vertical lines are guides to the eye at interesting pedagogical points (see text). All plots were obtained using the same parameter values as in Fig. 5 .

The Trisectrix of MacLauren, an example of a multi-valued function resembling a “loop-the-loop” roller coaster track. This curve is described by Eqs. (13) and (14) and is plotted using m and .

The Trisectrix of MacLauren, an example of a multi-valued function resembling a “loop-the-loop” roller coaster track. This curve is described by Eqs. (13) and (14) and is plotted using m and .

Normal force ( ), velocity ( ), and acceleration ( ) vectors rendered at three different times for an object constrained to a loop-the-loop curve, modeled by a Trisectrix of MacLaurin. We use Eqs. (13) and (14) with parameters m and . For integration of the object's motion we used (see text), m/s, and *m* = 1 kg (enhanced online). [URL: http://dx.doi.org/10.1119/1.4773561.2]doi: 10.1119/1.4773561.2.

Normal force ( ), velocity ( ), and acceleration ( ) vectors rendered at three different times for an object constrained to a loop-the-loop curve, modeled by a Trisectrix of MacLaurin. We use Eqs. (13) and (14) with parameters m and . For integration of the object's motion we used (see text), m/s, and *m* = 1 kg (enhanced online). [URL: http://dx.doi.org/10.1119/1.4773561.2]doi: 10.1119/1.4773561.2.

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