In a damped oscillating circuit the energy is dissipated In the resistor. The Q-factor Is a measure of the persistence of the oscillator against the dissipative loss. (a) Prove that for a ligt1y danced circuit the energy, U, in the circuit decreases according to the following equation. d u d t = − 2 β u , w h e r e β = R 2 L (b) Using the definition of the Q-factor as energy divided by the loss over the next cycle, prove that Q-factor of a lightly damped oscillator as defined in this problem is Q = U b e g i n △ U o n e c y c l e = 1 R L C c (Hint: For (b), to obtain Q, divide E at the beginning of one cycle by the change E over the next cycle.)
In a damped oscillating circuit the energy is dissipated In the resistor. The Q-factor Is a measure of the persistence of the oscillator against the dissipative loss. (a) Prove that for a ligt1y danced circuit the energy, U, in the circuit decreases according to the following equation. d u d t = − 2 β u , w h e r e β = R 2 L (b) Using the definition of the Q-factor as energy divided by the loss over the next cycle, prove that Q-factor of a lightly damped oscillator as defined in this problem is Q = U b e g i n △ U o n e c y c l e = 1 R L C c (Hint: For (b), to obtain Q, divide E at the beginning of one cycle by the change E over the next cycle.)
In a damped oscillating circuit the energy is dissipated In the resistor. The Q-factor Is a measure of the persistence of the oscillator against the dissipative loss. (a) Prove that for a ligt1y danced circuit the energy, U, in the circuit decreases according to the following equation.
d
u
d
t
=
−
2
β
u
,
w
h
e
r
e
β
=
R
2
L
(b) Using the definition of the Q-factor as energy divided by the loss over the next cycle, prove that Q-factor of a lightly damped oscillator as defined in this problem is
Q
=
U
b
e
g
i
n
△
U
o
n
e
c
y
c
l
e
=
1
R
L
C
c
(Hint: For (b), to obtain Q, divide E at the beginning of one
which of the following oscillations in an rlc series circuit is characterized nu a decreasing amplitude that is also oscillating?
a. overdamped
b. simple harmonic
c. critically damped
d. underdamped
An L-R-C series circuit has L = 0.800 H and C = 9.00 μF. Calculate the angular frequency of oscillation for the circuit when R = 0. What value of R gives critical damping? What is the oscillation frequency ω′ when R has half of the value that produces critical damping?
The charge q on a capacitor in a simple a-c circuit varies with time according to the equation q = 3 sin(120πt + π/4). Find the amplitude, period, and frequency of this oscillation. By definition, the current flowing in the circuit at time t is I = dq/dt. Show that I is also a sinusoidal function of t, and find its amplitude, period, and frequency.
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