We’ve found that Simple Harmonic Oscillators have a frequency that depends on th

Question

We’ve found that Simple Harmonic Oscillators have a frequency that depends on their configurations. This is their natural frequency. Many unexpected things can be harmonic oscillators; including bridges, bedframes, coins and tabletops. When excited they always “ring” the same way regardless of how they’re struck.

If a harmonic oscillator is forced to oscillate at an arbitrary driving frequency, the amplitude changes. Driving an oscillator at the “wrong frequency” means that sometimes you’re pushing with the motion (to add energy by doing positive work on it) and sometimes you’re pushing against the motion (taking energy out by doing negative work). If the driving frequency is timed just right so that it’s always pushing in the same direction as the motion, it always adds energy to the oscillator. This happens at the natural frequency and is called resonance.

The lab coordinator was showing off once (at a time embarrassingly much later than childhood) by bouncing on a bed. Without warning, the oscillations suddenly became very large, the coordinator was violently ejected, and the steel bedframe buckled and deformed. Offer an explanation of how this event happened—then offer some insight on the bent frame. Was it behaving linearly according to Hooke’s Law when it buckled? Did it remain an ideal spring at that time?

Explanation / Answer

When you were jumping up and down on the bed, you were essentially adding energy to the bedframe. When u jumped up you gained potential energy. This energy depends on the height, so the higher u jumped and the more the enrgy you gained. But when you hit the frame, your kinetic energy was high, so inturn you were applying a lot of force n the bed frame. This is an action, the reaction of the bedframe is to provide a restoring force, which is given by hooke''s law as F = -kx, where k is the spring constant. This restoring force is depended on the spring's material and the length of compression.

As you added energy by jumping higher, resonance occured where the bed itself pushed u higher. At this point the force applied is pretty high, but the restoring force is still defined by the spring constant, which is same and the the compression. Thus the applied force is greater the restoring force, so the bed buckled.

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