In this part of the experiment you will study
simple harmonic motion (
) using
a simple pendulum. Examples of
are found in many systems: oscillations of any system
around a stable equilibrium point
can usually be well-approximated in terms of .
Consider a simple pendulum consisting of a mass m
suspended by a string of length l as shown below.
Assume that the string is at an angle
with respect to the vertical.
Figure 2.7: Simple Pendulum
The net torque acting on the mass, taking the point where the string is suspended as our origin, is given by
The moment of inertia I of the system is
The instantaneous angular acceleration is therefore given by
If the angle
is small (say less than 15
radians), then
,
where
is measured in radians.
We can then rearrange the above differential equation for
to obtain:
This is the equation for a simple harmonic oscillator
(
).
We can solve this equation by guessing the form of the solution,
then substituting back into the equation.
Guided by our knowledge of a mass on a spring
(another system which obeys the equation),
we anticipate a solution where
the angle is a sinusoidal function of time.
The most general such solution is given by
We can differentiate this equation twice to obtain
then substitute the result into the equation. Doing this, we find that
The values of
and 
depend on the initial conditions, they correspond
to the amplitude of the oscillation
(the maximum angle reached)
and the phase of the oscillations
(essentially depending on when we choose for the time t=0
relative to the motion of the pendulum).