Spring Motion

Spring motion belongs to a subset of periodic motion known as Simple Harmonic Motion (SHM), where the restoring force is directly proportional to the object's displacement magnitude and acts towards the object's equilibrium position. The proportionality factor between restoring force and displacement plays a key role in determining the time period of motion. It is commonly denoted by \(\omega^2\)

For a spring, \(\omega^2\) is given by \(\frac{k}{m}\). Where 'k' is known as spring constant. It is the magnitude of restoring force developed for unit displacement of the edge of the spring from its equilibrium position. In essence, it quantifies the 'stiffness' of the spring. 'm' is mass of the spring.

The time period of SHM is universally given by \( \frac{2\pi}{\omega} \). For a spring, \( T = 2\pi \sqrt{\frac{m}{k}} \). Frequency of vibration is given by \( \frac{1}{T} \). For a spring, \( f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \). Hence, a lighter but stiffer spring oscillates faster than a heavier but looser spring. In the below interactive simulation, increase 'k' or decrease 'm' and observe what happens to the frequency of oscillation:

Description

The equation of spring motion is as follows: \[ m a = - kx - bv \] where \( k \) is the spring constant, \( b \) is the damping constant and \( x \) and \( v \) are the position and velocity of the oscillating tip. \(m\) is the mass of the spring. We plug in these quantities to obtain the acceleration \(a\). The simulation is done using the Euler's method, which dictates: \[ \Delta v = a \Delta t \implies v_{new} = v_{old} + \Delta v \] \[ \Delta x = v \Delta t \implies x_{new} = x_{old} + \Delta x \] The value of \(\Delta t\) determines the accuracy of the simulation. A precision of 2 (which implies \(\Delta t = 10^{-2}\)) is usually the correct balance between accuracy and speed, in this case.

1. Related: Planetary Motion, Projectile Motion, Double Pendulum.