Simple Harmonic Motion

 

THEORY

 

Vibration is the motion of an object back and forth over the same patch of ground.

The most important example of vibration is simple harmonic motion (SHM).

One system that manifests SHM is a mass, m, attached to a spring of spring constant , k. Suppose such a system resides on a horizontal table top. If the mass is pulled a distance x from the equilibrium recall it will experience a force by the spring given by Hooke’s law,

F = - k x.

If we neglect friction this will be the net force on m, and by Newton’s second law,

-k x = m d2x / dt2 OR

 

d2x / dt2 + w2 x = 0 where, w2 º k/m.

and w is called the (angular) frequency of vibration = 2p/T, T = period.

The above equation is called a differential equation because it contains both a function, x(t), as well as derivatives of the function.

We will solve the differential equation two ways:

  1. guessing (in class) we find

    2. x(t) = A cos(wt + d) , where A = amplitude and d = phase constant

     

    The initial conditions [x(0) and v(0)] determine A and d.

  2. using energy conservation and integration.

Recall the elastic potential energy of a stretched spring-mass system is

V = ½ k x2 .

Therefore, the total mechanical energy of the oscillator is

E = ½ m v2 + ½ k x2 Note: when x = A, v = 0 so,

E = ½ k A2

Using v = dx/dt and rearranging we get,

dx/dt = [ w2 ( A2 – x2 ) ]1/2

 

w dt = dx/ (A2 – x2)1/2

Integration of this result gives the same x(t) we found previously. Note: the constant of integration we denote d and this is added to the LHS above.

 

 

 

EXAMPLES

· Simple Pendulum

For this system use Newton’s second law for rotation. One finds the angle obeys the same differential equation as x did for the mass on a spring. Therefore, the system behaves as a SHO. The period is found to be,

 

T = 2p (L/g)1/2 , where L is the length of the pendulum.

 

 

 

 

· SHM and UCM

Consider a particle moving in UCM. It turns out that if you look at the shadow of this particle on the x-axis, the shadow executes SHM!

The angular velocity for the UCM equals the angular frequency of vibration of the shadow.

 

 

 

 

 

 

· Driven Oscillator and Resonance

One can apply an additional force to the mass on a spring. Suppose this additional force (AKA the driving force) has an angular frequency wd .

It can be shown, by Newton’s second law, that the mass undergoes very large amplitude vibrations if it so happens the driving frequency matches the natural frequency of the SHM,

wd = w

This phenomenon is known as resonance. Examples include: breaking a wine glass with sound, the Tacoma Narrows bridge catastrophe, soldiers required to break rank when crossing a bridge, etc. .

 

From Webster’s

RESONANCE º to relate harmoniously.