i. Identify and make note of the axes labels: 

  

      a.  If the axes labels are potential energy vs time, identify stable and unstable equilibrium locations.  If displaced from a stable equilibrium location, oscillations will occur.

   

            i.  If the stable equilibrium locations have the same shape as a quadratic funtion, then the oscillations about this equilibrium can be approximated as simple harmonic oscillations.

  

      b.  If the axes labels are displacemnt vs time, identify the amplitude, and the period directly from the graph.  The angular frequency can then be determined from the wavelength $\omega = \frac{2 \, \pi}{T}$ .

   

             i.  Look at the initial conditions (at $t = 0$ ) to identify if the oscillation can be written as a sine or cosine function.



      b.  For mathematical problems:

            i.  If the mathematical equation is a displacement as a function of time:

      a. Identify the amplidue from the given equation. 

      b. Identify the angular frequency from the equation; relate this angular frequency to the period of oscillation.

            ii.  If the mathemaical equation is a force as a function of displacement:

      a.  Identify if the force is proportional to the displacement.  If so, check if the force acts in the opposite direction of displacement (there will be a negative sign in the equation).  If both conditions are met, then this force is describing a SHO.

  

      b.  If there are any velocity dependent forces that also act in the opposite direction of displacement along with a linear displacement force that acts in the opposite direction of motion, then the equation is describing a damped oscillation.

      c.  For pictorial problems:

            i.  Visualize the scenario. Mentally try to understand what the object(s) is(are) doing and what forces are acting on it (them).

            ii.  Use Newton's 2nd law to write an equation of motion for the system.

 

            iii.  Determine if there is a restoring force that is proportional to displacement.  If this is the only force, then the motion is SHM.

            iv.  Determine if there is a damping force along with the restoring force.  If this is the case, then the motion is damped harmonic motion.

            v.  Rewrite your Newton's law translation into a form of the equation of motion.   $ \left( a_x \rightarrow \frac{\Delta^2 x}{\Delta t^2} \right)$.  With this new form, identify the angular frequency, and if damped, identify the time constant.

3. Evaluate your answer, make sure units are correct and the results are within reason.

• The argument that goes into a sine or cosine (any trig fucntion really), does not have units.  The evaluation of this argument also does not have units.  For example, with $sin(\omega \,t)$, the argument is "$\omega \, t$" which when multiplied together the units cancel, and when we take the sin of that value it returns a unitless value.  
• All oscillations are not simple harmonic oscillations.  Simple harmonic oscillations are a special type of oscillation where the restoring force is linear with respect to the displacement from equilibrium; this is the same as saying that the potential energy plot vs displacement is quadratic. 
• The amplitude is measured from the equilibrium position, it is not the difference between the max and min values.
• Although we are using circular motion variables and quantities (e.g. period, angular frequency, etc...), the underlying physical phenomena we are describing need not be circular in nature; for example a mass on a spring is linear motion yet we can still describe this motion with quantities such as period and angular frequency.

• Sine and cosine are very similar functions - they only differ by a phase. If you look at a plot of sine versus cosine, you can see that if you shift one of the plots by $\frac{\pi}{2}$, they become the same function.
• The slope of a position versus time plot is its velocity; the slope of a velocity versus time plot is its acceleration. The slope of a sine function is cosine. So if you have a function that describes the position of an object that begins at its max amplitude $x = x_max sin(\omega t)$ , you quickly know that the function to describe its velocity is $v_x = v_max cos( \omega t)$. Understanding the relationship in terms of slopes instead of just memorizing off an equation sheet will help you answer qualitative questions on exams.
• Sine and cosine functions come up over and over, so its worth spending some time practicing now to get familiar with them. In particular, it is often helpful to understand the limits - that $cos(0) = cos(2\pi) = 1$, etc. Some find the unit circle to be an incredibly helpful visual tool for remembering these limits - ignore all of the $\frac{\sqrt{2}}{2}$ stuff, and just memorize the values of sin and cosine at $0, \frac{\pi}{2},\pi, \frac{3\pi}{2}$, and $2\pi$.

A spring mass system that oscilates back and forth. There are several  important positions that the objects oscilates between. The max and min position points are related to the positions whre the veloicity is zero at the turning points. In addition, at the equilibrium position the velocity is maximum.

$E=K+U=Const.$

$\nu_{max} = \omega x_{max}$

$a_{max} = \omega^{2} x_{max}$

$\omega =  2 \pi f = \frac{2 \pi}{T}$

$x(t) = \pm x_{max}\frac{\sin \text{or}}{\cos} (\omega t)$

$\nu (t) =\pm \nu_{max} \frac{\sin \text{or}}{\cos} (\omega t)$

$a (t) =\pm a_{max} \frac{\sin \text{or}}{\cos} (\omega t)$

Energy ($E$), Velocity ($V$), Angular frequency ($\omega$), Distance ($x$), Amplitude ($A$), Spring Constant ($k$),

Mass ($m$), Gravity ($g$), Length ($l$), Period ($T$)

The position, velocity, and acceleration graphs for simple harmonic oscillators behave similarly to non oscillatory motion graphs. The relationships between position, velocity, acceleration still hold. The slope of the position graph corresponds to the value of the velocity at corresponding points in time. In turn, the slope of velocity corresponds to the values of acceleration at corresponding points in time.

 Simple harmonic oscillators are systems whose motion is described by a linear restoring force.