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ANIMATION OF CIRCULAR MOVEMENT - AN EXAMPLE IN PROCESSING

Processing with the java programming language, as well as the javascript variant p5js can be used to create animations of circular motion. In the following, an interactive application will be shown to display circular motion, with the ability to change the angular velocity, radius and angular acceleration. The application can be used for education and a better understanding of the kinematics of the circular movement of a material point. Also, in the following, it will be shown how a simple version of the application for circular movement can be programmed using processing.

Circular(rotational) movement​

Circular motion is movement along a curvilinear path in the shape of a circle, as opposed to rectilinear where the body moves along a straight path. In this article, an animation of the circular movement of a material point will be shown. In contrast to rectilinear movement where the physical quantities that describe the movement are the distance traveled s[m], the speed v[m/s] and the acceleration a[m/ s²], in the case of circular motion we will observe physical quantities such as the angle that overlaps the position vector with the X axis φ[rad], angular velocity ω[rad/s], angular acceleration α[rad/s²]

A circular motion can be:

• Uniform, where the angular acceleration is α=0
• Uniformly accelerated, with some non-zero angular acceleration α <>0

Figure 1: Uniform circular motion of material point M

In uniform circular motion, a material point rotates along a circular path with an angular velocity that does not change over time, i.e. ω=const.

The radius vector R at every moment changes the angle coinciding with the X axis according to the law:

φ=ω * t [work]

The intensity of the linear velocity v is also constant, i.e. v=const , but the direction changes, so there is a change in the velocity vector Δv always pointing towards the center, which means there is therefore an acceleration causing this change.

It is the normal or centripetal acceleration:

a_N = v²/R, or

a_N = R*ω²

Unlike normal acceleration, tangential acceleration is equal to zero in uniform motion, because angular acceleration is:

α = 0

If we know the polar coordinates R and φ the position coordinates of point M, X and Y can be calculated as follows:

X=R*cos(φ)

Y=R*sin(φ)

The projections of the position vector R on the coordinate axes X and Y are shown in figures 1 and 2 with blue thick lines

Figure 2 shows an example of circular motion, where ω = 1.1 rad/s and α = 0, with all displayed sizes . Velocity and normal acceleration vectors are also shown

Figure 2: Uniform circular movement of a material point M - size display

Uniformly accelerated circular motion

In uniformly accelerated circular motion, there is an angular acceleration &alpha ><0. The angular velocity changes during time t according to the following law:

ω= ω₀ + α*t

where ω₀ is the angular velocity at the initial moment

Figure 2 shows an example of uniformly accelerated motion for ω = 0.5rad/s and α = 0.5 rad/s²

Figure 3: Uniformly accelerated circular motion of a material point

In addition to the normal acceleration, there is now a tangential acceleration aT, which always has a tangent direction, and the direction is the same as the velocity if the angular velocity increases over time, and the opposite direction if it decreases.

aT=R*α[m/s²]

Total acceleration is obtained as a vector sum of normal and tangential acceleration, as shown in Figure 3.

From the triangle formed by the vectors aN, aT and a, the intensity of the total acceleration can be determined:

a=√aN²+aT²

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Circular movement - application​

         Below is a circular motion display application. In the blue panel, the starting values ​​can be reduced: radius of the path R, initial angular velocity, angular acceleration using the slider. Below that is a panel with navigation buttons for start, one iteration step, pause, and reset. Also in the same panel there are "check boxes" for selecting the direction of rotation, displaying information, speed and acceleration. It is possible to slow down the animation up to 20 times, via the slow down slider, which is also located on the blue panel.

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Circular motion - example code in processing

The following picture shows a simple animation of circular motion, uniformly accelerated, and then the code in processing.

Figure 4: Uniformly accelerated circular motion-processing

          In the scath "RotationPointAcceleratedBasic" shown in Figure 5, first the objects "pointM" (the point that rotates), "path" (the circular line along which the point rotates), "center", processing class PShape are declared. Also declared are x-axis and y-axis, also objects of class PShape. Also declared are the physical quantities angle(fi), path radius "R", angular velocity "w", angular acceleration "alpha" of type float, and vectors (PVector) of position of the rotating point(M), velocity(v), centripetal acceleration "ac" .
Within the setup method, the scatch size is set to 500*500, the white background, the number of iterations per second, and then the previously declared objects are created.

Figure 5: Uniformly accelerated circular motion-processing-code1

       PShape objects are created using processing's createShape() method, to which a shape, ELLIPSE or LINE is passed as parameters, and the other parameters are, for a circle, the coordinates of the upper left corner of the rectangle in which the ellipse (circle) is written, followed by the width and height the same. In the case of a line, the start and end points must be passed.
The point M is an object of class PVector, and the linear velocity v, a, w, alpha and fi are real numbers. The angular velocity w is initialized using the formula w=v.mag()/R.
The mag() function for a vector returns the intensity of the vector, as explained earlier in the article: Vectors in Processing.

Figure 6: Uniformly accelerated circular motion-processing-code 2

      Below is the draw function, which draws the background (background()), translates the coordinate system to the middle of the canvas to display the animation, and then changes the angular velocity w based on the defined angular acceleration (deceleration) alpha and rotation angle fi .
In the same method, the center, the circular path, the material point and the moving coordinate system, related to the rotating point, are then plotted.
Before drawing the small circle that rotates (point M), it is necessary to rotate the coordinate system by the angle fi, using the rotate(fi) method, and then translate it by the value -R, so that the point is then defined relative to that moving coordinate system with position coordinates (0,0).
The shape(point) method draws a point at coordinates (0,0) because these are the default coordinates, unless specified otherwise (see shape reference).

Figure 7: Uniformly accelerated circular motion-processing-code 3

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