6-16-98
In the absence of any resistance forces (like friction and air resistance), most simple harmonic motions would go on unchanged forever. In reality, this doesn't happen, because there are resistance forces.
Damped harmonic motion - harmonic motion in which energy is steadily removed from the system.
There are three kinds of damping:
Also known as forced harmonic motion, this is harmonic motion in which the system is given a periodic push. A perfect example is a person on a swing.
How the system behaves depends on how the frequency of the driving force compares to the natural frequency of oscillation of the system.
The most efficient way to transfer energy from the driver to the system is to match the frequency of the driving force to the natural frequency of the system, such as you do when pushing someone on a swing. This is known as resonance. At resonance, relatively small driving forces can build up to large-amplitude oscillations, just because energy is continually being injected into the system at just the right frequency.
A 0.123 kg block sits on a plane inclined at 20°. The block is pushed back against a spring (k = 23.4 N / m), compressing the spring by 0.345 m. When the block is let go, it is accelerated up the incline by the spring. The coefficient of kinetic friction between the block and the incline is 0.220.
How far up the incline does the block go?
Attack this problem using work and energy. The initial energy (stored in the spring) is equal to the final energy (gravitational PE) plus whatever gets lost to friction. Writing this as an equation gives:
Energy before = energy after + energy lost to friction
A free-body diagram tells us that:
Substituting this in to 
transforms the energy equation to:
Solving this for d, the distance the block travels up the slope, gives:
Plugging in all the numbers gives:
