No need to do problem 1. I simply put it so problem number 2 can be done. Thanks

Question

No need to do problem 1. I simply put it so problem number 2 can be done. Thanks for any help in advance.

1.) A simple harmonic oscillator consists of a 150g mass attached to a spring whose force constant is 10^4dy/cm. The mass displaced by 2.5 cm and released from                    from rest. Calculate a) the natural frequency Vo and the period To. b) the total energy and the maximum speed.                 

2.) allow the motion in the preceding problem to take the place in a resisting medium, after oscillating for 8s, the maximum amplitude decreases to half the                    initial value. Calculate a) the damping parameter(calling it B), b) the frequency V1 and compare it with the undamped frequency c) and the decrement of the                    motion.

3) critically damped oscillator is x(t)=(A+Bt)e^(??t)  . Show that x(t)=Bte^(??t) can  satisfty the equation of motion for the critically damped oscillator.

Explanation / Answer

a. Xm = Ao e^-Bt, here t= 8s and Xm = = A0/2

thus Ao/2 = Ao e^-Bt

solving 0.5 = e^-Bt, taking natural log , Ln(0.5) = -B(8)

B = 86.6 *10^-3 secs^-1

b. V1:w1 = sqrt(wo^2-B^2)  

w0 = 8.16 rad/s from previous bit,

w1 = sqrt(66.586 - 0.0074)

w1 = 8.159 rad/sec

frequency v1 = 8.159/2*3.14 = 1.229 sec^-1

C. from ths result we conclude that angular frequency doesnopt change much whenb the dampng is small

compared with the natural angular frequency but this is not the case when the damping cofficient is comparable with natural frequency

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