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Undamped and Undriven Pendulum

Damped and Driven Pendulum

Linearized Equations of Motion

The Equation of Motion
of an undamped and undriven pendulum

geometry of forcesAccording to Newton's laws the inertia force FI (i.e., mass times acceleration) has to be equal to the applied force. In our case, the applied force is the restoring force FR caused by gravity G. From the geometry of the problem (see figure), it is clear that

FR = -G sinphi  = - mg  sinphi,

where m is the mass of the pendulum and g is the acceleration of gravity. Note that the negative sign is caused by the fact that the restoring force FR wants to bring the pendulum back to equilibrium (i.e., phi = 0).

Next, we have to express the inertia force FI in terms of the angle phi. Assuming a rigid pendulum (i.e., its length l is fixed), the mass can move only on a circle with radius l. The position (i.e., the spatial coordinate) along this circle is given by lphi. Note that the angle phi is measured in radians (i.e., 180° corresponds to pi). The acceleration is therefore given by l d2phi/dt2. Thus, from Newton's law we get
ml d2phi/dt2 = -mg sinphi.

Dividing by ml and moving the term on the right-hand side to the left-hand side leads to the equation of motion of an undamped and undriven pendulum


d2phi/dt2 + omega02 sinphi = 0,



 omega0 = (g/l)1/2.

Additional comments:

QUESTIONS worth to think about:
  1. What are the stationary equilibra of the pendulum (i.e., the solutions of (1) which are constant in time)? Which of them are stable and which are unstable?
  2. How large is the component of the force parallel to l?







© 1998 Franz-Josef Elmer,  elmer@ubaclu.unibas.ch last modified Sunday, July 19, 1998.