Month: August 2016

135 Natural Frequency

When an oscillatory system is displaced from its equilibrium position, and then released, it goes into oscillation. Basically, if an oscillating mass is subjected to only the restoring force (i.e. no damping force and no external driving force), the resulting oscillation is called a free oscillation.

It turns out that for a SHM, a=-ω2x. This means that the a/x ratio determines the frequency of oscillation.

With this in mind, we can understand why for a spring-mass system, a stiffer spring (increases the restoring force per unit displacement) and a smaller mass (increases the acceleration for the same restoring force) results in a higher frequency of oscillation. In fact, it can be derived quite easily that for a spring-mass system ω2=k/m.

A pendulum is another famous SHM. The derivation is more complicated but it can be shown that for a pendulum ω2=g/L. (Notice the amplitude has no effect on the period of oscillation?)

134 Restoring Force

An oscillation is any periodic repetitive to-and-fro motion.

Just like circular motion requires a centripetal force, an oscillation requires a restoring force. The restoring force serves to accelerate the oscillating mass towards the equilibrium position. But upon arriving at the equilibrium position, inertia causes the mass to overshoot the equiblirium position. This repeats again and again, hence oscillations.

A simple harmonic motion is a special class of oscillation whose displacement variation with time can be described by one single sinusoidal function (hence simple harmonic). It can also be shown that for an oscillation to be simple harmonic, the restoring force must be proportional to displacement.

The spring-mass system and the pendulum are probably the two most famous SHMs. For the spring-mass system, it is clear that at the equilibrium position, weight and tension forces are balanced. But at any other displacement x, the mass experiences a net force of Fnet=kx. The restoring force is clearly proportional to displacement, resulting in a simple harmonic oscillation.

133 Pendulum Wave

In this simulation, 17 balls are programmed to oscillate at angular frequencies of ω, 17ω/16, 18ω/16, 19ω/16,… 2ω. Notice that the angular frequencies are evenly spaced out, which implies the periods are not.

Since angular frequency is the rate of change of phase angle, the leftmost ball lags behind the most in terms of phase. For example, after one period of the leftmost ball, the leftmost ball would have traversed only one cycle (2π), while the rightmost would have already traversed two cycles (4π), and the other balls evenly spaced out in between. This makes the balls line up along one cycle of cosine function.

1T

After two periods, the positions of the balls would be spaced out evenly along two cosine functions. (Because the rightmost ball would have completed 2 cycles more than the leftmost).

2T

3T4T

5T6T7T8T9T10T11T12T13T14T15T16T

132 Human Wave

We can apply the concept of phase to any repetitive periodic motion. One complete cycle corresponds to 360° or 2π rad. Half-a-cycle corresponds to 180° or π rad. One and a quarter-cycle coreesponds to 450° or 5π/4 rad. So on and forth.

See if you can recognize the phase differences among the following oscillations.

In-phase (phase difference of 0, 2π, 4 π, …)

Complete Out-of-Phase (phase difference of π, 3π, 5 π, …)

Quarter-cycle Phase Difference (Each oscillation leads the one on the right by π/2 rad)

Each oscillation leads the one on the right by 1/8 of a cycle (45° or π/4 rad). The profiles of two progressive waves moving from left to right are now clearly visible. There is the transverse wave in the bobbing heads, and the longitudinal wave in the gyrating hips. 😛

In an actual progressive wave, there is a continuous increasing phase lag in the direction of wave propagation. Each wave element lags the preceding wave element by a bit.

131 Fortune Telling Sticks

The NO stick has a much lower natural frequency than the YES sticks.

When the base is moved to-and-fro subtlely but periodically, both sticks experience a periodic driving force.

When the base is shaken at the natural frequency of the NO stick, energy is transfered efficiently to the NO stick. The NO stick goes into resonance and attain a large amplitude of oscillation. The YES stick, being very quick, easily moves in line with the base and attains negligible amplitude of oscilation.

When the base is shaken at the natural frequency of the YES stick, the YES stick goes into resonance and attain a large amplitude of oscillation. The NO stick is too slow to react to the rapidly changing driving force, and remains motionless.