Forest UCM Osc Damped - Revision history

Foretony: /* Solve for the Equation of Motion */

2021-10-14T23:12:01Z

Solve for the Equation of Motion

2014-10-06T12:36:29Z

Energy

2014-10-06T12:34:06Z

Energy

2014-10-06T12:27:36Z

Energy

2014-10-06T12:27:12Z

Energy

2014-10-06T12:22:26Z

Quality factor (Q)

2014-10-06T12:20:52Z

Physics Example; The LRC circuit

2014-10-06T12:15:03Z

Under damped Oscillator

2014-10-06T12:12:28Z

Newton's 2nd Law

2014-10-06T12:04:12Z

Physics Example; The LRC circuit

← Older revision		Revision as of 23:12, 14 October 2021
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−	~~You have change the~~ second order differential equation into two first order differential equations	+	The second order differential equation has changed into two first order differential equations

	:<math> \left ( \frac{d}{dt} + \beta + \sqrt{\beta^2 -\omega^2_0} \right ) x = 0 </math>		:<math> \left ( \frac{d}{dt} + \beta + \sqrt{\beta^2 -\omega^2_0} \right ) x = 0 </math>

@@ Line 158: / Line 158: @@
 :<math>W_f = \int (-2\beta \dot x) dx = -2 \beta \int \dot x \frac{dx}{dt} dt = - 2 \beta \int v^2 dt</math>
 for the case of an underdamped oscillator
 [[Forest_UCM_Osc#Damped_Oscillations]]

← Older revision		Revision as of 12:34, 6 October 2014
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	:<math>E(t) = E(0) + W_f</math>		:<math>E(t) = E(0) + W_f</math>
		+
		+
		+	The dissipative force may be represented based on the general form of the differential equation as:
		+
		+	:<math>F=-2\beta \dot x</math>
		+
		+	The work done by this force is given as
		+	:<math>W_f = \int (-2\beta \dot x) dx = -2 \beta \int \dot x \frac{dx}{dt} dt = - 2 \beta \int v^2 dt</math>
		+
		+	The rate of energy loss is then given by
		+	:<math> \frac{dE}{dt} = \frac{W_f}{dt} = -2\beta v^2</math>
		+
		+	for the case of an underdamped oscillator
		+


	[[Forest_UCM_Osc#Damped_Oscillations]]		[[Forest_UCM_Osc#Damped_Oscillations]]

@@ Line 146: / Line 146: @@
 ==Energy ==
-A damped oscillator is a system that is loosing energy as a function of time ( energy is leaving the system usually in the form of heat).  The system starts out with a total energy at time <math>t=0</math> of <math>E(0)</math>.  A frictional force exists causing the system to do work (W_f) and thereby lose energy.
+A damped oscillator is a system that is loosing energy as a function of time ( energy is leaving the system usually in the form of heat).  The system starts out with a total energy at time <math>t=0</math> of <math>E(0)</math>.  A frictional force exists causing the system to do work <math>(W_f)</math> and thereby lose energy.
-:E(t) = E(0) + W_f
+:<math>E(t) = E(0) + W_f</math>
 [[Forest_UCM_Osc#Damped_Oscillations]]

← Older revision		Revision as of 12:27, 6 October 2014
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	==Energy ==		==Energy ==
		+
		+	A damped oscillator is a system that is loosing energy as a function of time ( energy is leaving the system usually in the form of heat). The system starts out with a total energy at time <math>t=0</math> of <math>E(0)</math>. A frictional force exists causing the system to do work (W_f) and thereby lose energy.
		+
		+	:E(t) = E(0) + W_f


	[[Forest_UCM_Osc#Damped_Oscillations]]		[[Forest_UCM_Osc#Damped_Oscillations]]

← Older revision		Revision as of 12:20, 6 October 2014
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	: <math>2\beta = \frac{R}{L}</math> and <math> \omega_0^2 = \frac{1}{LC}</math>		: <math>2\beta = \frac{R}{L}</math> and <math> \omega_0^2 = \frac{1}{LC}</math>

		+	==Energy ==

		+	==Quality factor (Q) ==
	[[Forest_UCM_Osc#Damped_Oscillations]]		[[Forest_UCM_Osc#Damped_Oscillations]]

← Older revision		Revision as of 12:15, 6 October 2014
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−	There are two terms above, the first term is an exponential decay and the ~~secons~~ is the usual harmonic oscilator.	+	There are two terms above, the first term is an exponential decay and the second is the usual harmonic oscilator.

	They combine to produce oscillations whoase amplitudes decay with time.		They combine to produce oscillations whoase amplitudes decay with time.

@@ Line 1: / Line 1: @@
 =1-D Damped Oscillations=
 ==Newton's 2nd Law==
-As in the case of air resistance, assume there is frictional force proportional to the velocity of the oscillation body.
+As in the case of air resistance, it was assume that there was a frictional force proportional to the velocity of the oscillation body.  The same assumption is made for a block oscillating via a spring with friction between the block and the table top.

@@ Line 121: / Line 121: @@
 An electrical circuit composed of an inductor <math>(L)</math> , a resister <math>(R)</math>, and a capacitor <math>(C)</math> is known as an LRC circuit.
-The resistor in the circuit serves to simply resists the movement of charge through the circuit ( analogous to friction) such that the voltage drop accross the resistor is given by
+The resistor in the circuit serves to simply resists the movement of charge through the circuit ( analogous to friction; causing energy to leave the system in teh form of heat) such that the voltage drop accross the resistor is given by
 :<math>V= R \dot q</math>