phy1020.DVI

Figure 8.4: A physical pendulum. The object has massMand is suspended from pointP;his the distance
betweenPand the center of mass.

you can form a physical pendulum by suspending a
meter stick from one end and allowing to swing back and forth.
In a physical pendulum of massM, there is a forceMgacting on the center of mass. Suppose the body
is suspended from a point that is a distancehfrom the center of mass (Fig. 8.4). Then there is a weight force
Mgacting on the center of mass of the body, which creates a torqueMghsinabout the pivot point. Then
by the rotational version of Newton’s second law,

DI ̨ (8.14)

MghsinDI ̨; (8.15)

whereIis the moment of inertia of the body when rotated about its pivot point, and ̨is the angular accel-
eration. Like the simple plane pendulum, this is a difficult equation to solve for.t/, but it becomes much
easier to solve if we restrict the problem to small oscillations.Ifis small, we can make the approximation
sin, and we have

MghI ̨: (8.16)

It can be shown, using the theory of differential equations, that this equation has solution

.t/D 0 cos.!tCı/; (8.17)

where 0 is the (angular) amplitude of the motion (in radians),!D

p
Mgh=Iis the angular frequency of
the motion (rad/s), andıis an arbitrary integration constant (seconds).
The periodTof the motion (the time required for one complete back-and-forth cycle) is given by

TD

2

!

; (8.18)

or

TD2

s

I

Mgh

: (8.19)

(See Appendix Q for a table of moments of inertia.)