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Chapter 4: Q38E (page 1015)

An L-R-Cseries circuit has L= 0.600 H and C= 3.00 mF. (a) Calculate the angular frequency of oscillation for the circuit when R= 0. (b) What value of Rgives critical damping? (c) What is the oscillation frequency 蝇' when Rhas half of the value that produces critical damping?

Short Answer

Expert verified
  1. Themagnitude of the angular frequency when R = 0 is 745 rad/s.
  2. The value of R that gives critical damping is894惟
  3. The magnitude of the angular frequency when R has half the value producing critical damping is 645 rad/s.

Step by step solution

01

(a) Determination of the magnitude of the angular frequency when R = 0 .

In a L-C-R circuit, the expression for the angular frequency is,

蝇'=1LC-R24L2 ... (i)

In absence of any resistance, R = 0,

蝇=1LC

This is the resonant frequency. Substitute the values in the above equation,

蝇=10.600H3.00=745rad/s

Thus the magnitude of the angular frequency when R = 0 is745 rad/s.

02

(b) Determination of the height the value of R that gives critical damping.

The condition of critical damping is,

1LC=R2L2

Solve for R,

R=LC2=0.600H3.00渭贵2=894惟

Thus the resistance that gives critical damping is 894惟.

03

(c) Determination of the magnitude of the angular frequency when R has half the value producing critical damping.

Half the value of R as in part (b) is,

R=12LC2R=LC

Substitute the value of R in equation (i),

蝇'=1LC-R24L2=1LC-L/C24L2=1LC-14LC=341LC=341LC=34蝇=34745rad/s=645rad/s

Thus with damping the frequency is less value of 645 rad/s.

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