Ø Definition of a Simple Pendulum

        Small object (bob) suspended from the end of  a lightweight cord.

Ø Under the following assumptions, the simple pendulum resenbles SHM

         (1)  The cord does not stretch

         (2) The mass of the cord is negligivle compared to that of the bob

        (3) Negligible friction

Equation for a Simple Pendulum

For the following derivation, I use up as the positive direction

Ø The resoring force on the bob, F, is equal to the component of the weight, mg, that is tangent to the arc.

Ø Figure from the preliminaries of this chapter of the manual

Ø To visualize, enclose the 3 red arrows at the bottom in a box (remember the vectors in an earlier lesson)

Ø F With a little analysis of the angles we arrive at

        F = -mgsinq                   Negative because of the above sign convention

For small angles, sin of the angle is approximately equal to the angle, so we can write

F  @ -mgq

From the definition of radian measure

The arc length, x, is equal to the radius, L, times the angle, or

x = Lq    ð   q = x/L

Then F = -mgx/L

From this, the equations for the period, T, and frequency f, of a simple pendulum are derived

T = 2p(L/g)^1/2

f = 1/2p(g/L)^1/2

NOTE: These equations are in the handout under Newtonian Machanics