Mass-Spring System

This Mass-Spring System calculator computes the period and angular frequency of an oscillating mass-spring system.  As its name suggests, a mass-spring system is simply a mass attached to a spring.  We generally assume that one end of the spring is anchored in place, or attached to a sufficiently massive object that we may assume that it doesn't move.  We ignore the mass of the spring and any damping forces.

The Math / Science

By combining Newton's Second law, the acceleration of Simple Harmonic Motion (SHM), and Hooke's Law, we can derive the equation relating the angular frequency(ω) to the mass (m) and the spring constant (k) ¹.  We shall follow the convention of using "a" for acceleration:

• Newton's Second law: F = m•a  
• Simple Harmonic Motion: `a = -ω^2x`
• Hooke's Law: F = -kx 
  `F = -mω^2x = -kx`
  `k = mω^2`
  `ω = sqrt(k/m)`

We can then find the period (T) associated with this oscillating mass-spring by the definitions of period and angular frequency. We shall use "f" to indicate the frequency (not the angular frequency, they're different). `T = 1/f`, `ω= 2πf`, so `T = (2π)/ω = 2π sqrt(m/k)`.

Qualitative Discussion

There are a number of variables and concepts in this page that a casual reader may not be familiar with, so let's try to discuss what they mean and why they matter.  

Simple Harmonic Motion (SHM) is periodic motion, motion that repeats itself over consistent intervals, that is assumed to ignore damping.  For example, imagine a swinging pendulum.  For small swings, the pendulum obeys the equation we used to describe simple harmonic motion (`a = -ω^2x`) .  In particular, we must ignore any friction or air resistance that would slow the pendulum down (the damping forces).  SHM is most commonly represented by sine and cosine functions, something like `x(t) = x_0 cos(ωt)` in the simpler cases.  By the nature of cosine, this function will satisfy the conditions of SHM: it will repeat itself periodically and it will not slowly die away.  One may derive the equation used to describe SHM (`a = -ω^2x`) by taking the second derivative of x(t) with respect to t.

The period (T) of the system is how long it takes the mass-spring to make one full oscillation.  The frequency (f) is the inverse of the period, `1/T`, and it describes how many oscillations the system undergoes in a second.  For example, a system with a period of 2 seconds will go through half an oscillation every second.

As we discussed before, SHM is commonly represented with sine and cosine functions.  Sine and cosine (we're using radians) repeat themselves every 2π radians.  This means that for every oscillation, the sine or cosine argument must "sweep through" (change by) 2π.  The easiest way to make sure that happens every time is to define the angular frequency (ω) such that `ω = 2πf`.  Therefore, for every oscillation per second, we sweep through 2π radians, so the sine or cosine equation will repeat how we want it to.

Finally, the spring constant (k) is a quality of the spring that describes how "strong" the spring is.  Springs with a higher spring constant are more difficult to compress and extend than springs with a lower spring constant.

Spring Equation Calculators

• Period of an Oscillating Spring: This computes the period of oscillation of a spring based on the spring constant and mass.
• Mass of a Spring: This computes the mass based on the spring constant and the period of oscillation. 
• Angular Frequency of a Spring: This computes the angular frequency based on the spring constant and the mass.
• Spring Constant: This computes a spring's constant based on the mass and period of oscillation.
• Work done on a Spring: This computes the work based on the spring constant and the two positions of a spring.
• Hooke's Law: This computes the force to change the length of a spring based on the spring constant and length of displacement.
• Force to Fully Compress a Spring: This computes the force required to fully compress a spring based on the spring's physical attributes including the Young's Modulus, wire diameter, length of spring, number of windings, Poisson ratio, and outer diameter of the spring.

1. ^ Walker, Jearl, David Halliday, and Robert Resnick. "Simple Harmonic Motion." Fundamentals of Physics. 8E ed. Hoboken, NJ: Wiley, 2008. 389-90. Print.