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Pendulum Motion Lesson
Shodor > CSERD > Resources > Activities > Pendulum Motion Lesson
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Lesson - Pendulum Motion

Consider a pendulum created by connecting a large mass to a thin stiff rod, suspended from a pivot point in which it is able to move about a full 360 degree circle. Unless the pendulum is pointed directly downward, or balanced perfectly above the pivot point, the gravitational force will result in a torque which pulls the pendulum downwards.


\begin{displaymath}\tau = - m g L sin(\theta)\end{displaymath}

This results in a angular acceleration for the pendulum of


\begin{displaymath}\frac{d^2 \theta}{dt^2} = - g/L * sin(\theta)\end{displaymath}

For small angles $\theta$, the value of $sin(\theta)$ is approximately $\theta$, and with this approximation the equation of motion is given by


\begin{displaymath}\frac{d^2 \theta}{dt^2} = - g/L * \theta\end{displaymath}

The equation whose second derivative is a constant multiplied by the opposite of itself is well known, and the solution is readily given.


\begin{displaymath}\theta(t) = sin((g/L)^{1/2} t)\end{displaymath}

However, this solution was based upon the assumption that the initial displacement was small.

While the original problem may not have a closed form solution, a numerical integration can yield interesting results as to the motion of pendulums at large displacements.

In order to numerically solve the equation of motion without the small angle approximation, we first take our equation which is written in terms of a second derivative, and rewrite it in terms of two equations written in terms of a first derivative. We do this by realizing that the derivative of the angle of the pendulum is just the angular velocity, $\omega$.


\begin{displaymath} \frac{d \theta}{dt} = \omega \end{displaymath}

and we can write the second derivative as the derivative of the first derivative


\begin{displaymath} \frac{d}{dt} \left[ \frac{d \theta}{ dt} \right] = -(g/L) sin(\theta) \end{displaymath}

giving us as our two equations written in terms of first derivatives,


\begin{displaymath} \frac{d \omega}{dt} = -(g/L) sin(\theta) \end{displaymath}

and


\begin{displaymath} \frac{d \theta}{dt} = \omega \end{displaymath}

These two equations make up a system of ordinary differential equations, and these equations, together with the initial position and velocity of the pendulum, make up what is referred to as an initial value problem.

Using the model of pendulum motion provided, modify the parameters of the pendulum by changing the initial condition values of the length and the acceleration due to gravity.

Exercises:

  1. What happens to the path of the pendulum as you increase the initial displacement?
  2. What happens to the velocity of the pendulum as you increase the initial displacement?
  3. What happens to the acceleration of the pendulum as you increase the initial displacement?
  4. How large does the initial displacement have to be before the small angle approximation will fail?
  5. Would a pendulum swing more quickly, more slowly, or the same on the moon?
  6. What is the acceleration at the bottom of the pendulum's swing?
  7. Would a pendulum swing more quickly, more slowly, or the same if the pendulum bob was more massive?
  8. For small angle pendulums, we are told that the initial displacement does not affect the period. How do you feel this statement would be best modified to account for larger angles?
  9. Suppose you have a weight ties to the end of a fishing rod, and start it swinging gently back and forth. If you gently real in the rod, what will happen to the motion of the weight? (Hint, if you real in the rope at constant speed, what is the derivative of the length of the pendulum?)

Also be sure to check out the Pit and the Pendulum model.

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