Physics, please help.? Most detailed explanation gets 10 pts.?

Question

A LCR circuit consists of a coil with an inductance of L=0.3 H and a capacitor with a capacitance of Csub1= 5X10^-6 F.
a) What is the resonant frequency of that circuit?
b) How much is the change in period of oscillation if instead the capacitor Csub1 uses the other capacitor with a capacitance of Csub2= 5X10^-8 F?

Answer 1

a) Resonant frequency = 1/(2π√LC) = 130 Hz

Use same equation to find new resonant frequency

Then use period = 1/frequency to find change in period.

Answer 2

The resonant frequency is given by the formula:

f = squareroot(L*C)

f is in radians/second
L is in Henries
C is in Farads

Since capacitor C2 is 1/100th the value of C1, the new resonant frequency will be 1/10th the original resonant frequency.
.

Answer 3

To do an LRC series circuit, write out the equation for voltage around the loop.

L Q'' + R Q' + Q/C = voltage

Characteristic equation:
(L d^2 + Rd + 1/C) Q = 0

This is like a quadratic equation in d. Use the quadratic formula to get:

d = (-R +- sqrt (R^2 - 4L/C) )/2L

= -R/2L +- sqrt (R^2 / 4L^2 - 1/LC)

The resonant frequency is:
omega0 = sqrt (1/LC).

The damping coefficient (zeta) is R/2L.

Q = exp (- zeta t) cos ((omega0^2 - zeta^2)t + delta)

a)Calculate omega0 = sqrt (1/L C1)

b) The period is: T = 2 pi / omega, so calculate the period for C1 and C2 and take the difference.

difference = 2 pi sqrt (L) (sqrt (C1) - sqrt (C2))