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Damped Oscillators

Damping and amplitude decreases

         

The above is a schematic diagram of a damped oscillator, where the block has a mass of \(m=8.0\text{ kg},\) spring constant \(k=130\text{ N/m},\) and damping constant \(b=70\text{ g/s}.\) Approximately, how long does it take for the mechanical energy to drop to one-half its initial value?

The above is a schematic diagram of a damped oscillator, where the block has a mass of \(1.7\text{ kg}\) and the spring constant is \(9\text{ N/m}.\) The damping force can be expressed as \(-b(dx/dt),\) where \(b=210\text{ g/s}.\) If the block is pulled down \(12.0\text{ cm}\) and released, approximately, what is the time required for the amplitude of the resulting oscillations to fall to one-third of its initial value?

The above is a schematic diagram of a damped oscillator. If the damped oscillator has mass \(m=240\text{ g},\) spring constant \(k=85\text{ N/m},\) and damping constant \(b=70\text{ g/s},\) what is the approximate ratio of the amplitude of the damped oscillations to the initial amplitude at the end of \(10\) cycles?

The above is a schematic diagram of a damped oscillator, where the block has a mass of \(m=250\text{ g}\) and the spring constant is \(k=95 \text{ N/m}.\) It takes \(4.0\text{ s}\) for the amplitude of the damped oscillations to drop to half its initial value. If we replace the block with mass \(4m\) and the spring with spring constant \(3k,\) how long will it take for the amplitude of the damped oscillations to drop to half its initial value?

The above is a schematic diagram of a damped oscillator, where the block has a mass of \(m\) and spring constant \(k.\) If the ratio of the amplitude of the damped oscillations at the end of \(30\) cycles to the initial amplitude is \(0.57,\) what will be the ratio of the amplitude of the damped oscillations at the end of \(60\) cycles to the initial amplitude?

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