A mass M is suspended from a spring and oscillates with a period of 0.940 s. Each complete oscillation results in an amplitude reduction of a factor of 0.96 due to a small velocity dependent frictional effect. Calculate the time it takes for the total energy of the oscillator to decrease to 0.50 of its initial value.

Question

Alexia Mercado

Physics

11 March, 05:22

A mass M is suspended from a spring and oscillates with a period of 0.940 s. Each complete oscillation results in an amplitude reduction of a factor of 0.96 due to a small velocity dependent frictional effect. Calculate the time it takes for the total energy of the oscillator to decrease to 0.50 of its initial value.

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Answers (1)

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Willow Rodriguez · Answer 1

The time, t = 7.99seconds

Explanation:

Total energy of SHM system is:

E = ½kA²

0.50E = 0.50 (½kAo²) = ½kA²

A = Ao/√2 = 0.707Ao

The amplitude after each oscillation is given by:

A (n) = Ao (0.96) ^n

Solve for n by setting this equal to half-energy amplitude:

Ao (0.96) ^n = Ao/√2

ln (Ao (0.96) ^n) = ln (Ao/√2)

ln (Ao) + n*ln (0.96) = ln (Ao) - ln (√2)

n = - ln (√2) / ln (0.96)

n = 8.5

Find time from n & period T:

Period, T = 0.940seconds

t = 8.5 * 0.940

t = nT = 7.99s