Simple Pendular Motion

A simple pendulum is a mass m attached to the end of a rod of length l meters. The other end of the rod is attached to a fixed point. When the mass is raised radians from equilibrium and released, the function which describes the angular motion in radians is called pendular motion. The units of t are seconds. We will develop a second order differential equation governing pendular motion functions by studying the forces that act on m in a simple pendulum. Move m to see the forces on m. You can also adjust l using the slider.
To develop the second order differential equation that governs motion of m, as is standard practice in Newtonian physics, we'll express the force acting on m in two ways and then equate the two expressions. First, we'll express the force acting on m with Newton's second law, F=ma (force equals mass times acceleration). The acceleration, a, of the mass m is the second derivative of s, the arc length of circular path travelled by m. From geometry, we know that the circular arc s is the length of the rod, l (the radius of the circular arc), times the angle of displacement from the plumb line, . Therefore: Substituting this into F=ma yields the following expression of the force acting on m: On the other hand, we can also calculate the force of gravity acting on m. Based on inspection of the applet above, the force of gravity acting on m is the component of gravity that is tangent to s (The component perpendicular to s is in balance with support provided by the rod of length l). As can be seen in the applet above, the component of the force of gravity tangent to s is . Therefore we can express the force of gravity acting on m as: The force is negative because positive s (and ) is in the opposite direction. Finally, we can equate these two expressions of the forces acting on m to obtain this second order differential equation:  Dividing both sides by ml yields the standard form of the second order differential equation governing undamped pendular motion: We can add damping by adding an additional term on the right, proportional to the rate of change of . Just as in damped spring motion and air resistance, damping is proportional to velocity. Thus the second order equation governing damped pendular motion is You may also see it written as: or if all the terms are collected on the left: Any function which satisfies the above equation is said to be a model of pendular motion. For small initial displacements, , and so the above can be estimated using traditional second order algebraic methods for solving second order equations with constant coefficients. However if initial displacement is larger than about a half a radian, this approximation is very poor, and alternate methods should be used instead. There are algebraic methods to solve this non linear equation, but it's easier to estimate solutions with a numerical method. To use numerical methods, convert the second order equation to a system of two first order equations. The most common conversion is to use the substitutions and . With these, note that . Furthermore differentiating , yields . Putting it all together yields the first order system form of the second order equation governing pendular motion: The following applet illustrates 100 iterations of the Runge Kutta 4 numerical methods applied to this first order system of differential equations. In the left pane, the x coordinates of the dots are the numerical estimates of ; the y coordinates of the dots are estimates of . In the right pane, these coordinates are transformed into a visualization of a pendulum. On the left you can adjust:
  • l, the length of the rod (in meters) 
  • the damping coefficient
  • stepsize, the amount of time that advances between estimates
  • The initial displacement of (in radians; drag the point Initial to adjust)
On the right you can animate the solution, or manually advance by dragging the slider. At the outset, the initial displacement is 1 radian. The length of the rod is 30 centimeters. There is no damping. The stepsize is 0.02. You can and should change all of these values to explore a variety of different pendular motions.