The diagram below illustrates the oscillation of a simple pendulum:.
.
.
As it moves from its mean position, X, to P, as its equilibrium has been displaced. It then returns to X and then moves to Q, returning to X again to complete one oscillation. .
The angular velocity vector is defined as the derivative of the angular position as a function of time. For an object rotating about an axis, every point on the object on the object has the same angular velocity. The tangential velocity of any point is proportional to its distance from the axis of rotation. Angular velocity has the units rad/s.
.
Angular velocity is the rate of change of angular displacement and can be described by the relationship:.
.
.
Thus we say that the relationship between frequency and angular velocity is defined as:.
=2 F = 2 T.
Conservation of energy.
The energy associated with motion is called kinetic energy. The kinetic energy [K.E = ½ mv ] depends on the mass of the object and its speed. A pendulum possesses this energy. The energy stored in a system by virtue of its position or configuration is called potential energy. A pendulum also possesses this type of energy. .
When you pull the pendulum to a certain height, you turn some of the chemical energy of your body into potential energy of the pendulum system. This is most noticeable when the pendulum is a grown man on a swing. Compared to his young child, the father or bob has more mass than the child. Because of this extra mass, it takes more energy to get the father or the bob with more mass to a given height. As the pendulum swings back and forth, energy transforms between kinetic and potential. .
.
At P and Q, of the swing, the pendulum's potential energy, is maximum, and kinetic energy is zero.
Kinetic energy is maximum at X (when the pendulum is in equilibrium), of the swing, where potential energy is smallest. .
Total mechanical energy is constant throughout and thus we find that;.
