# Homework Help: Physical Pendulum least period

1. Nov 25, 2007

### ghetto_bird25

1. The problem statement, all variables and given/known data
In Fig. 16-41, a stick of length L = 1.6 m oscillates as a physical pendulum. (a) What value of distance x between the stick's center of mass and its pivot point O gives the least period? (b) What is that least period?
http://edugen.wiley.com/edugen/courses/crs1141/art/qb/qu/c16/Fig15_46.gif

2. Relevant equations
well i'm assuming you have to use the equation for a a physical pendulum which is
T= 2pi $$\sqrt{I/mgh}$$
where I is the rotational inertia, m is the mass, g is gravity, and h is the distance between the axis of rotation and the centre of the pendulum.

3. The attempt at a solution
i tried using that equation but i dont understand the value for I because im assuming if the value for h changes then the other end of the rod changes as well.

2. Nov 25, 2007

### azatkgz

Moment of Inertia of the center of mass is

$$I_0=\frac{mL^2}{12}$$

If it is pivoted distance x from the centre

$$I=I_0+mx^2$$

$$T=2\pi\sqrt{\frac{I}{mgx}}=2\pi\sqrt{\frac{I_0+mx^2}{mgx}}$$

for T minimum we have
$$\frac{dT}{dx}=0$$