Simple Pendulum

The objective in this section is to visualize the motion of a simple pendulum which consist of a small mass m suspended by a light inextensible cord of length L from a fixed support. If we apply the law of the conservation of mechanical energy to analyze the motion of the mass m, we obtain the following initial value problem described by the differential equation

+

subject to the initial conditions and . g is the acceleration of gravity and the angular displacement is measured from the vertical. If we account for the frictional resistance of the surrounding medium which is proportional to the instantaneous velocity, c , the result is the differential equation:

+ c +

This equation has no closed-form solution for , so we find a numerical solution using the Runge-Kutta-Fehlberg fourth-fifth method. We define the general nonlinear pendulum eqation by:

> deq:=(c,L)->diff(theta(t),t$2)+c*diff(theta(t),t)+9.8/L*sin(theta(t))=0:

With the damping coefficient , L = 5 m and the initial conditions , we get:

> sol:=dsolve({deq(0.2,5),theta(0)=Pi/3,D(theta)(0) = 0},theta(t),numeric,startinit=true);

For t = 5 we get:

> sol(5);

The graphs of and is shown in Figure 2.

> with(plots):

> plt1:=odeplot(sol,[t,theta(t)],0..10):

> plt2:=odeplot(sol,[t,diff(theta(t),t)],0..10,color=blue):

> display(plt1,plt2);

Figure 2 Angular and angular velocity position for the simple pendulum with damping

The pseudoperiod of time for the first complete damped oscillation is about four times the amount of time required for to decrase from to .

> alpha:=u->subs(sol(u),theta(t)):

> T[1]=4*fsolve('alpha(u)'=0,u=0..2);

Figure 3 animates damped motion of the pendulum and the simultaneous position of the mass on the angular position curve.

• Pendulum (L, ,c) create an animation of the motion of a simple pendulum. The angular posistion is controlled by the numerical solution of the differential equation describing the pendulums motion. The initial angular velocity is assumed zero. L: length of the pendulum; : initial angular position in degrees; c: damping coefficient.

> Pendulum(5,60,0.2,scaling=constrained,axes=normal,tickmarks=[4,4]);

Figure 3 Damped motion of a simple pendulum.

t and (in degree) are displayed in the cyan window