Playground swings exhibit actions of a pendulum and simple harmonic motion. A swing will continue swinging for a long time by using only the energy from its initial push. At the end of each oscillation, the swing weight slows down and stops for an instant and than swings back in the other direction. The maximum potential energy occurs when the swing is at its height, when kinetic enery energy is zero. This potential energy is stored up and than converted and released to kinetic energy when the swing swings down. Kinetic energy is at its maximum value when the swing is at the bottom, and potential energy is zero. In a frictionless environment with no air resistance, swings exhibit conservation of energy.
PE (initial) = KE (final) mgh = 1/2 mv^2
The energy transformed from Potential Energy to Kinetic Energy, which occurs over and over, doesn't effect the total amount of energy - which will stay constant in absense of external forces.
If the swing has no way to convert its energy (via friction) and no way to transfer the energy elsewher (via air resistance) it will continue to swing forever. However, when a swing exists in an environment with friction and air resistance, these forces will cause it to slow down gradually by extracting energy and eventually stop.
One of the most commonly encountered oscillations is pendulums. Swings are classified as pendulums and therefore exhibit simple harmonic motion and oscillations. As a swing swings away from its equilibrium position through and angle X, it rises through a vertical height H given by:
H = L - L cos X
and it gains potential energy
U = mgh
In terms of displacement s (which is equal to LX)
H= s^2 / 2L
Its equation of energy conservation (see above) is equivalent for the hormonic oscillator, with force constant k (k= mg/L). For small angles, the swing vibrates harmonically. Therefore all equations for simple harmonic motion is relevant to the actions of a swing. For example, the period of a swing is:
T= 2 (3.14159) (m/k)^1/2
which equals:
T= 2 (3.14159) (L/g)^1/2
If one was to graph the displacement vs. time for a swing swinging through a small angle, you would find it to be a sine curve with an amplitude (A) and period (T). The velocity of the swing at a given time can be found through:
v = A w cos wt
where w= angular frequency of the swing (w = 2 (3.14159) / T )
The acceleration of the swing at a given time can be found through the equation:
a = - A w^2 sin wt