Classical Mechanics

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 kg , m=8.0\text{ kg}, 弹簧常数 k = 130 N/m , k=130\text{ N/m}, and damping constant b = 70 g/s . 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 kg 1.7\text{ kg} and the spring constant is 9 N/m . 9\text{ N/m}. The damping force can be expressed as b ( d x / d t ) , -b(dx/dt), where b = 210 g/s . b=210\text{ g/s}. If the block is pulled down 12.0 cm 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 g , m=240\text{ g}, 弹簧常数 k = 85 N/m , k=85\text{ N/m}, and damping constant b = 70 g/s , 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 10 cycles?

The above is a schematic diagram of a damped oscillator, where the block has a mass of m = 250 g m=250\text{ g} and the spring constant is k = 95 N/m . k=95 \text{ N/m}. 这需要 4.0 s 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 4 m 4m and the spring with spring constant 3 k , 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 m and spring constant k . k. If the ratio of the amplitude of the damped oscillations at the end of 30 30 cycles to the initial amplitude is 0.57 , 0.57, what will be the ratio of the amplitude of the damped oscillations at the end of 60 60 cycles to the initial amplitude?

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