If
the acceleration is proportional to the position (with a negative constant
of proportionality), a general solution for the position as a function of
time is possible.If
the acceleration is proportional to the position (with a negative constant
of proportionality), a general solution for the position as a function of
time is possible.
This relation is usually is expressed in equation form with the constant on the acceleration side of the equation. Anticipating the result, the constant of proportionality is written as (minus) omega squared.
The equation is solved with a sine or a cosine function of time (t), as shown. This can be verified by calculating the time derivatives of the position and inserting the result into the original equation.
The parameter, omega, is characteristic of the system. The constant A is called the amplitude of the motion. The motion in this situation is called simple harmonic motion (SHM).
Because the original equation uses the second derivative of position, there will be two constants of integration in the complete solution. Physically, the two constants determine the maximum value of position and the part of the cycle which occurs at t=0. The latter information is called the "phase" of the motion.
One way to write a complete solution is to combine the sine and cosine solutions with two independent amplitudes, as shown. The sum of sine and cosine is also sinusoidal. The relative size of the amplitudes A and B determines the phase. A and B are chosen to make the solution match the initial conditions for the physical system.
The more intuitive complete solution addresses the phase more directly. The part of the cycle that occurs at t=0 is determined by the angle alpha (or beta for the cosine solution). Alpha is called the "phase angle" for the motion. (Beta differs from alpha by pi/2 .) A and alpha are chose to make the solution match the initial conditions for the physical system.