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For a damped oscillator with a mass of 300 g, a spring constant 150 N/m and a damping coefficient of 87 g/s, what is the ratio of the amplitude of the damped oscillations to the initial amplitude at the end of 17 cycles?

Respuesta :

Answer:

[tex]\frac{A}{A_o} = 0.500[/tex]

Explanation:

As we know that the Amplitude of Damped oscillation is given by the formula

[tex]A = A_o e^{-bt/2m}[/tex]

here we know that

[tex]b = 87 g/s[/tex]

m = 300 g

k = 150 N/m

so the time period is given as

[tex]T = 2\pi\sqrt{\frac{m}{k}}[/tex]

now we have

[tex]T = 2\pi\sqrt{\frac{0.300}{150}}[/tex]

[tex]T = 0.28 s[/tex]

now total time taken in 17 cycles

[tex]t = 17 T[/tex]

[tex]t = 17(0.28) = 4.78 s[/tex]

now plug in all values

[tex]A = A_oe^{-87(4.78)/2(300)}[/tex]

[tex]A = A_o(0.500)[/tex]

so we have

[tex]\frac{A}{A_o} = 0.500[/tex]

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