^{1,a)}

### Abstract

This paper examines the motion of a simple system of connected rigid bodies with no net angular momentum. Although it is possible for such a system to move in a way that preserves its orientation, we consider the conditions where orientation of the system changes even in the absence of an external torque. A general formalism is developed for calculating the change of orientation of an articulated figure with two degrees of freedom. This formalism is then used to show that the orientation changes in the absence of an external torque are possible; in particuler it is applied to a simple figure made of three equivalent rods and to a primitive model of the human body. To illustrate how these systems evolve over time, the differential equations governing their movement are solved numerically, and animations of the moving figures are created.

The author would like to acknowledge the support from Physics Curriculum and Instruction, the company for which the author created the “Physics of Sports” software package, a project that inspired the writing of this article.

I. INTRODUCTION

II. THE MODEL

III. APPLICATION TO A SPECIFIC EXAMPLE

IV. COMPUTER ANIMATION OF THE ARTICULATED FIGURE

V. APPLICATION TO A HUMAN BODY MODEL

VI. CONCLUSION

### Key Topics

- Angular momentum
- 17.0
- Phase space methods
- 10.0
- Torque
- 6.0
- Aircraft
- 3.0
- Number theory
- 2.0

## Figures

The three solid objects that make up our model are joined together by the vectors. is embedded in the body of the object and points to the joint that is shared with the object.

The three solid objects that make up our model are joined together by the vectors. is embedded in the body of the object and points to the joint that is shared with the object.

(Left) The articulated figure with each element’s angle defined with respect to the positive direction. (Right) The same figure with the center element defined with respect to the positive direction, but with the other two elements defined with respect to the center element.

(Left) The articulated figure with each element’s angle defined with respect to the positive direction. (Right) The same figure with the center element defined with respect to the positive direction, but with the other two elements defined with respect to the center element.

The contour in space determines how the angles and behave over time. Because the path is closed, the angles will return to their original values once the path is complete. The absolute angle will be changed by as found from Eq. (11).

The contour in space determines how the angles and behave over time. Because the path is closed, the angles will return to their original values once the path is complete. The absolute angle will be changed by as found from Eq. (11).

Nine frames of the animated articulated figure shown in Fig. 2. The values of and are shown in each frame. To clarify how the figure’s orientation is changing, an arrow shows the orientation of the piece.

Nine frames of the animated articulated figure shown in Fig. 2. The values of and are shown in each frame. To clarify how the figure’s orientation is changing, an arrow shows the orientation of the piece.

An example of the general model described in Sec. II. The two degrees of freedom are in the motion of the shoulder and hip joints. The three “pieces” of the body are more complex than the three equivalent rods used to construct the simple model in Sec. III.

An example of the general model described in Sec. II. The two degrees of freedom are in the motion of the shoulder and hip joints. The three “pieces” of the body are more complex than the three equivalent rods used to construct the simple model in Sec. III.

Nine frames of the human model performing a 1.2 s long dive. The center of mass of the model is made to move in a parabolic trajectory, as would be the case in a real dive. The angular momentum of the body is zero, but the body experiences a net backward rotation.

Nine frames of the human model performing a 1.2 s long dive. The center of mass of the model is made to move in a parabolic trajectory, as would be the case in a real dive. The angular momentum of the body is zero, but the body experiences a net backward rotation.

## Tables

Values used for the human body model in Fig. 5, taken from Ref. 7. To simplify the model, the data for the feet and calves are combined into one body part.

Values used for the human body model in Fig. 5, taken from Ref. 7. To simplify the model, the data for the feet and calves are combined into one body part.

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