Question

A damped oscillator loses 2.0% of its energy during each cycle. (a) How many cycles elapse before half of its original energy is dissipated? (Use the 2.0% information to get a relation between γ and T, then use that to find t1/2 in terms of T) (b) What is its Q factor? (c) If the natural frequency is 150 Hz, what is the width of the resonance curve (in rad/s) when a sinusoidal force drives the oscillator?

Answer 1

answer)a) after one cycle the energy is

E(1-0.02)

so let the number of cycles be n

so after n cycles we have

0.89Eⁿ=1/2*E

n=log0.5/log0.98=34

so the answer is 34 cycles

b) the formula for Q factor is

Q=2pi/energy lost

Q=2pi/0.02=314

so answer is 314

c) the width is given by

w=2pif/Q=2*pi*150/314= 3 rad/s

so the answer is 3 rad/s or 3.0 rad/s