We will investigate how to express a motion of a particle in one dimension using simple harmonics.
A simple harmonic function is trigonometric representation of motion of any body that oscillates about its mean position ( Xm ). An object is attached to a spring and pulled down. We will define the one dimensional coordinate axis in the vertical direction as (upward- positive) and (downward-negative):
The mean position of any object hanging down a spring is denoted by the length of the spring stretch under the weight of the object.
If we pull the object down in the ( negative direction ) the object will oscillate about its mean position ( Xm ) i.e up and down the mean position.
While oscillating the object will achive its maximum displacement about its mean position both above and below ( Xo ). This maximum displacement is called the amplitude of oscillation. The Amplitude of oscillation is given to us as follows:
[tex]\text{Amplitude ( X}_o)\text{ = 9 cm}[/tex]All oscillations are measured in terms of time period ( T ) which denotes the time taken to complete one cycle. The time started from when the object was released from ( -Xo ) the obect moved up due to the restoring force in spring and crosses its mean position then reach the maximum displacement ( +Xo ). Then the object reverses direction and moves down under gravity and crosses the mean position ( Xm ) an reach back to its initial start point ( -Xo ). This constitutes a complete cycle. The amount of time to complete one cycle of oscillation takes:
[tex]\text{Time Period ( T ) = 3 seconds}[/tex]The general simple harmonic function is represented by the followin trigonometric function. This function relates the displacement ( X ) of the object at any point in time ( t ) during the oscillating motion:
[tex]X(t)=X_o\cdot\cos \text{ ( }\frac{2\pi}{T}\cdot t)[/tex]We will plug in the respective paramters of a simple harmonic motion i.e ( Amplitue ) and ( Time Period ).
[tex]X(t\text{ ) = -9 cos ( }\frac{2}{3}\pi t\text{ ) }[/tex]
