The “half-amplitude problems” can be solved by just “look-look-see-see”!

v-x is elliptical and a-x is straight!

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a-x graph animated

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v-x graph animated

– x-t, v-t and a-t relationships, they are all sinusoidal!

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x-t graph

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v-t graph

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a-t graph

The spring-mass system is a SHM because its motion can be described by one single sinusoidal function.

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Some of you may be curious why SHM is given such an exotic name. Actually, it comes from mathematicians’ knowledge that any periodic function can be described as a summation of many sinusoidal functions of different frequencies, or harmonics.

For example, the periodical rectangular function (drawn in red) can be constructed by summing five harmonics (drawn in blue). If you want a more perfect rectangular function, more sinusoidal functions of higher frequencies, or higher harmonics are required. If you’re interested, Discrete Fourier Transform is how one can solve for the harmonics of any periodic function.

Video

Barton’s Pendulum <xmdemo>

Driver-Driven Phase Relationship <xmdemo>

The Tacoma Narrows Bridge <xmphysics>

The Tuning Fork Resonance <xmdemo>

The Coupled Pendulum <xmdemo>

The Coupled Inverted Pendulum <xmdemo>

 

Applets

Damped Oscillation: {ngsir applet}

Resonance: {ngsir applet}

Resonance: {walter fendt applet}

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 progressed by only one cycle (2π), while the rightmost would have already progressed by two cycles (4π), and the other balls evenly spaced out in between. This makes the balls line up along one cycle of cosine function.

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).

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🍏: <ngsir applet>

🍏:{walter fendt applet}