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Example 14.10 For the damped oscillator the mass m of the block is 200 g, k = 90 N m-1 and the damping constant b is 40 g s-1. Calculate (a) the period of oscillation, (b) time taken for its amplitude of vibrations to drop to half of its initial value and (c) the time taken for its mechanical energy to drop to half its initial value.

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Solution

Dear Student,
m=0.2 kg , K=90 N/m, b=0.04 kg/sω=Km-b2m2=900.2-0.040.42=21.21 rad /sT=2π21.21=0.3 sec.follow the eqn A=A0e-bt2mA=A02A02=A0e-bt2m-bt2m=-ln2t=ln2*2mb=0.693*2*0.20.04=6.93 secinitial energy of the system E=KA022let at A the system energy becomes halfKA22=KA024A=A02 time taken to reach tis value.A02=A0e-bt2m-bt2m=-ln2t=ln2*2mb=ln2*2*0.20.04=3.5 sec.regards

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