16.2   Simple Harmonic Motion

This physlet shows a typical Simple Harmonic Motion (SHM) of a mass-spring system.

 

Drag the block to a stretch (or compress) the spring and click play button. You will see that the motion of this block is periodic and furthermore its displacement from equilibriun as a function of time varies sinusoidally as shown in the graph below.

Any motion that repeats itself with a certain period is called periodic or harmonic motion.

 
 

   

The number of oscillations per second is an important property of a harmonic motion and is called the frequency . Its S.I. unit is or .


The period of oscillations , is the time to complete one oscillation and is related to the frequency by

The S.I. unit for the period is second.


Amplitude, Angular Frequency and Phase of SHM

For an SHM, only, the displacement of the particle from the origin, , is given by:

where is the amplitude of the motion (maximum displacement), is the angular frequency and is the phase constant (or phase angle). The quantity called the phase and has unit of radian.

The angular frequency is related to the frequency by and has S.I. unit of .

The following applet demonstrate the concepts amplitude, angular frequency and phase constant.

 
Different:

To begin, click on the button above and you will be presented with two spring-block systems with different amplitude but the other two factors (frequency and phase) constant. Then use the VCR buttons below the physlets to play. Similarly click on other animations ( and ).

The two blocks moving in a circle shows an analogous circular motion of two blocks having the same angular speed as the angular frequency of the SHM. The amplitude is the radius of the circle and phase constant is the initial angle.
 

 

The Velocity and Acceleration of SHM

The velocity of a particle moving in simple harmonic motion is given by:

The maximum velocity of the particle is . It is clear that the particle has maximum speed at the origin . The velocity of the particle is zero at .

The acceleration of the particle moving in simple harmonic motion is given by:

Or

The maximum acceleration of the particle is . It is clear that the particle has maximum acceleration when it is at . The acceleration of the particle is zero at x = 0.

Important: Note that using Newton’s second law in this case we can write . Whenever the force acting on a particle is proportional to the negative of the displacement, then the motion is said to be simple harmonic motion.

Demos/Animations Examples/Checkpoints Interactive Problems
     
     
     
cp16.2.1 ex16.2.1 ex16.2.2
     
     
     
     
     
am = animation ,dm = demo, cp = checkpoint, ex = example, ip = interactive problem