Simple Pendulum

• Simple pendulum
• Period of the small oscillations of a pendulum
• Determination of the acceleration due to gravity by means of the simple pendulum

A simple pendulum is a pendulum consisting of a single small (or point) mass attached to a wire of negligible weight. A “physical” pendulum has extended size and is a generalization of the simple pendulum. An example would be a bar rotating around a fixed axis. A simple pendulum can be treated as a special case of a physical pendulum whose mass m is hanging from a string of length L and fixed at a pivot point P. When displaced to an initial angle and released, the pendulum will swing back and forth with periodic motion.

What sort of experiment could we perform with our system? What physical principle could we investigate? Let us try to build a chronometer!

Let us measure this time with our stop-watch in case the oscillations around the equilibrium position can be reasonably considered small (θ expressed in radians far smaller than 1); to do this, let us measure the time the pendulum takes to oscillate 20 times; now let us repeat the previous step for L=1 m; as we experimentally observe the period of oscillation doubles becoming about 2 seconds; notice that going on in this way we would observe the time doubled as L switched to 4L; that means that the period of oscillation of our system depends on time according to the square root of L; formally we can write.

LgLgHow would our period of oscillation change repeating our experiment on the moon? Curiously it would increase! The way it would change is well described as follows aside from a multiplicative constant, where g is the acceleration of gravity our system would perceive in a hypothetical trip on the moon.

With the assumption of small angles, the frequency and period of the pendulum are independent of the amplitude of the initial angular displacement. All simple pendulums should have the same period regardless of their initial angle (and regardless of their masses).

By using a clock, it is possible to verify this equation and to deduce the acceleration due to g. Measure the time the pendulum takes to perform 50 oscillations (this is done to reduce the error in the measured oscillation period T).