The Equation of Motion
of an undamped and undriven pendulum

According 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 sin  = - mg  sin, 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.,  = 0). Next, we have to express the inertia force FI in terms of the angle . 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 l. Note that the angle is measured in radians (i.e., 180° corresponds to ). The acceleration is therefore given by l d2/dt2. Thus, from Newton's law we get ml d2/dt2 = -mg sin. 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 (1) d2/dt2 + 02 sin = 0, where (2)  0 = (g/l)1/2.

Additional comments:

• The mass m of the pendulum does not appear anymore in the equation of motion. Galileo Galilei (1564-1642) was the first who discovered this effect. Maybe you know his legendary experiment of dropping two balls of the same size but of different mass from the tower of Pisa, where both balls had reached the ground simultaneously. From the viewpoint of Newton's laws, there is no reason that the inertial mass (i.e., the m in F_I) has to be the same as the gravitational mass (i.e., the m in F_R). It was the ingenious idea of Albert Einstein (1879-1955) to take this equality not as accidentally but as a deep principle of nature. From this equivalence of gravitational and inertial mass, he developed a new understanding of gravity which led to his general theory of relativity.
• Although the equation of motion is derived only for a mathematical pendulum (where all the mass is concentrated in one point), it is also true for a physical pendulum with distributed mass. In this case, the parameter l is some effective length which is smaller than the distance between the center of mass and the rotation axis.
• The equation of motion is a second-order differential equation (due to the second derivative of the angle ). In order to get a unique solution, one needs two real numbers, e.g. the angle and the angular velocity at a specific time. Both variables define uniquely the state of the undriven pendulum.
• The equation of motion is nonlinear because the second term depends nonlinearly on the angle .

QUESTIONS worth to think about:

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? How large is the component of the force parallel to l?

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