Uniform Circular Motion and Circular Motion: Animating Displacement, Velocity, and Acceleration by Frank J. Attanucci DESCRIPTION: In this worksheet, we use the graphing and animation capabilities of Maple to illustrate the concepts of (instantaneous) velocity and acceleration. restart:with(plots):with(Student[LinearAlgebra]):

Uniform circular motion is motion around a circular path at a constant speed. Suppose an object moves once around a circular path of radius 10 every 6 seconds. Then the position of its tip on the circle is given by P:=<10*cos(2*Pi*t/6) | 10*sin(2*Pi*t/6)>; where 0 <= t. At t = 0, the object is at the point (10, 0). Viewed from above, the circular path followed by the object is plot([P[1],P[2],t=0..2*Pi],scaling=constrained,color=black,thickness=2); For future use, I name the above plot. Circle:=plot([P[1],P[2],t=0..2*Pi],scaling=constrained,color=black,thickness=2):

To animate the (changing) displacement vector r, we define A:=<0 | 0>; Then, we have r:=s*(P-A)+A; where 0 <= t and 0 <= s <= 1. Here, t is the parameter of animation and s is the object parameter. Displacement:=animate([r[1],r[2],s=0..1],t=0..12,frames=73,color=red,thickness=2): Here is the display of the circle and the animated displacement vector. display(Circle,Displacement,title="ANIMATING THE CHANGING DISPLACEMENT");

Recall the definition: If r = r(t) is displacement, then v = dr/dt is velocity. In particular, when r = <f(t), g(t)>, then v = <f'(t), g'(t)>, i.e., to find the derviative of a vector-valued function, you take the derivatives of each of the component functions! v:=<diff(r[1],t) | diff(r[2],t)>; But to relocate the above velocity vector so that its initial point lies on the circle at P, we must add V:=v+P; where 0 <= s <= 1 and 0 <= t <= 12. Here we name the animation of the (changing) velocity Velocity:=animate([V[1],V[2],s=0..1],t=0..12,frames=73,color=blue,thickness=2): Here we display the circle, displacement, and velocity vectors together. display(Circle,Displacement,Velocity,title="ANIMATING THE CHANGING\nDISPLACEMENT and VELOCITY");

Recall the definition: If v = v(t) is velocity, then a = dv/dt is acceleration. In particular, when r = <f(t), g(t)>, then v = <f'(t), g'(t)> and, hence, a = <f''(t), g''(t)>, i.e., you just take the second derivative of each of the component functions (of displacement)! a:=<diff(r[1],t$2) | diff(r[2],t$2)>; To relocate the above acceleration vector so that its initial point lies on the circle at P, we must add A:=a+P; Here we name the animation of the (changing) acceleration. Acceleration:=animate([A[1],A[2],s=0..1],t=0..12,frames=73,color=red,thickness=2): I now display the circle, velocity, and acceleration vectors together. display(Circle,Velocity,Acceleration,title="ANIMATING THE CHANGING\nVELOCITY and ACCELERATION"); NOTE: In uniform circular motion, the magnitude of the acceleration vector remains constant and it always points in the direction opposite the displacement vector (i.e., towards the center of the circle). For this reason, it is called "centripetal acceleration."

In the above example of uniform circular motion, the object moved around the circular path at a constant speed (completing 2 revolutions in 12 seconds). Now we consider the case where the object speeds up as it completes 2 revolutions in 12 seconds. For example, this will happen if the position of its tip on the circle is given by P:=<10*cos(4*Pi*t^2/144) | 10*sin(4*Pi*t^2/144)>; where 0 <= t. Again at t = 0, the object is at the point (10, 0).

To animate the (changing) displacement vector r, we define A:=<0 | 0>; r:=s*(P-A)+A; where 0 <= t and 0 <= s <= 1. Here, t is the parameter of animation and s is the object parameter. Displacement:=animate([r[1],r[2],s=0..1],t=0..12,frames=73,color=green,thickness=2): Here is the display of the circle and the animated displacement vector. display(Circle,Displacement,title="ANIMATING THE CHANGING DISPLACEMENT");

As before v:=<diff(r[1],t) | diff(r[2],t)>; And to relocate the above velocity vector so that its initial point lies on the circle at P, we must add V:=v+P; where 0 <= s <= 1 and 0 <= t <= 12. Here we name the animation of the (changing) velocity Velocity:=animate([V[1],V[2],s=0..1],t=0..12,frames=73,color=blue,thickness=2): Here we display the circle, displacement, and velocity vectors together. display(Circle,Displacement,Velocity,title="ANIMATING THE CHANGING\nDISPLACEMENT and VELOCITY");

Again, as before a:=<diff(r[1],t$2) | diff(r[2],t$2)>; And to relocate the above acceleration vector so that its initial point lies on the circle at P, we must add A:=a+P; Here we name the animation of the (changing) acceleration. Acceleration:=animate([A[1],A[2],s=0..1],t=0..12,frames=73,color=red,thickness=2): I now display the circle, displacement, velocity, and acceleration vectors together. display(Circle,Displacement,Velocity,Acceleration,title="ANIMATING THE CHANGING\nDISPLACEMENT, VELOCITY, and ACCELERATION");