Show that a pendulum undergoes simple harmonic motion (SHM). State your assumptions. The pendulum is made up of a light inextensible string, attached to a ceiling at one end and with a particle of mass m attached to the other end.
Begin with a diagram of the system, and definition of directions. Vertically up and clockwise rotations are positive. It must be recalled that in SHM force is proportional to displacement from equqilibrium. The key assumptions to make are:
-
the string is taught throughout the motion of the pendulum,
-
the string doesn't break thtroughout the motion of the pendulum,
-
the initial angle of displacement from vertical is small,
-
there is no drag.
Take the angular displacement from veritcal to be x, and look at the forces on the particle. Assumptions 1) and 2) imply that there is no motion parrallel to the string, and hence the tension in the string must be equal magnitude to the weight of the mass parallel to the string. Hence the resultant force must act perpendicular to the direction of the string. Using trigonometry, this force (F) is: -mgsin(x). where g is the acceleration due to gravity. Now, in the small angle limit sin(x) ~ x so F=-mgsin(x) becomes F~-mgx. Since x is displacement from equilibrium, the system undergoes SHM.
