http://www.ck12.org Chapter 10. Periodic Motion
to the displacement of the bob. It dips below thex−axis rather than being along thex−axis, and the restoring force
is therefore not proportional to the displacement of the bob. However, if the angleθis made small enough as in the
case of the green bob, then the restoring force will, indeed, be approximately proportional to the displacement,x.
FIGURE 10.10
Under the conditions stated above, we can set up an approximate proportionality when the angle between the
pendulum string and the vertical is small (less than 15 degrees).
InFigure10.11 we assume thatmgis almost equal to the tensionTin the string. The bob is hanging almost
vertically. We can form one triangle composed of the weightmgof the bob and the component of weightFx„ and
another triangle composed of the displacementxof the bob and the lengthLof the pendulum. The triangles are
almost similar, somamgx= ̇xL.
Note that the symbol ̇=stands for approximately.
FIGURE 10.11
Uniform circular motion, which we are approximating, holds near the bottom of the swing of the pendulum, thus
ax=^4 π
2
T^2 x.
Substituting^4 π
2
T^2 xforaxin the approximate proportionality above and eliminating the massm, we arrive at
4 π^2 x
T^2 g =
x
L→T(^2) = 4 π^2 L
g →T=^2 π
√
L
g.The period of the simple pendulum,T= 2 π
√
L
gis therefore proportional to the square root of the length of the