Forest UCM Osc SHM - Revision history

Foretony: /* Equation of motion */

2014-10-05T12:43:00Z

Equation of motion

Foretony: /* Kinetic (T) and potential (U) Energy */

2014-10-04T14:30:02Z

Kinetic (T) and potential (U) Energy

Foretony: /* Equation of motion */

2014-10-04T14:28:46Z

Equation of motion

Foretony: /* Equation of motion */

2014-10-04T14:27:23Z

Equation of motion

Foretony at 14:27, 4 October 2014

2014-10-04T14:27:01Z

Foretony: Created page with " Forest_UCM_Osc#Simple_Harmonic_Motion_.28SHM.29"

2014-10-04T14:23:13Z

Created page with " Forest_UCM_Osc#Simple_Harmonic_Motion_.28SHM.29"

New page

[[Forest_UCM_Osc#Simple_Harmonic_Motion_.28SHM.29]]

← Older revision		Revision as of 12:43, 5 October 2014
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−	then the analogous equation becomes	+	then the analogous (known as the auxilary ) equation becomes

	:<math> \left ( mO^2 + aO + k \right ) x = 0 </math>		:<math> \left ( mO^2 + aO + k \right ) x = 0 </math>

@@ Line 68: / Line 68: @@
 :<math> x =  Ae^{\gamma t}  +  Be^{\beta t}   </math>
 ::<math>  =  Ae^{i \omega t}  +  Be^{-i\omega t}   </math>
 ==Kinetic (T) and potential (U) Energy ==

← Older revision		Revision as of 14:28, 4 October 2014
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	::<math> = Ae^{i \omega t} + Be^{-i\omega t} </math>		::<math> = Ae^{i \omega t} + Be^{-i\omega t} </math>

		+	==Kinetic (T) and potential (U) Energy ==
		+
		+	:<math>U= \frac{1}{2} k x^2 =\frac{1}{2} m \omega^2 A^2 \cos^2(\omega t - \delta)</math>
		+	:<math>T= \frac{1}{2} m v^2 =\frac{1}{2} m \omega^2 A^2 \sin^2(\omega t - \delta)</math>
		+
		+	:<math>E = T + U= \frac{1}{2} m \omega^2 A^2 \sin^2(\omega t - \delta) +\frac{1}{2} m \omega^2 A^2 \cos^2(\omega t - \delta)</math>
		+	:::<math>= \frac{1}{2} m \omega^2 A^2 =</math>constant
		+
		+
		+	:<math>U_{\mbox{max}}= \frac{1}{2} k x^2_{\mbox{max}} =\frac{1}{2} m \omega^2 A^2 \cos^2(\omega t - \delta)_{\mbox{max}}= \frac{1}{2} m \omega^2 A^2=E</math>
		+	:<math>T_{\mbox{max}}= \frac{1}{2} m v^2_{\mbox{max}} =\frac{1}{2} m \omega^2 A^2 \sin^2(\omega t - \delta)_{\mbox{max}}= \frac{1}{2} m \omega^2 A^2=E</math>

	[[Forest_UCM_Osc#Simple_Harmonic_Motion_.28SHM.29]]		[[Forest_UCM_Osc#Simple_Harmonic_Motion_.28SHM.29]]

← Older revision		Revision as of 14:27, 4 October 2014
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		+
		+	=Simple Harmonic Motion=

	==Equation of motion==		==Equation of motion==

← Older revision		Revision as of 14:27, 4 October 2014
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		+
		+
		+	==Equation of motion==
		+
		+	In solving the differential equation
		+
		+	:<math> m\ddot x =-kx </math>energy is constant with time
		+
		+
		+	and observe that the above differential equation is a special case of the more general differential equation
		+
		+	:<math> m\ddot x + c \dot x =-kx </math>energy is constant with time
		+
		+	one could rewrite the above as
		+
		+	:<math> \left ( \frac{d}{dt}\frac{d}{dt} +\frac{d}{dt} + k \right ) x = 0 </math>
		+
		+	One could cast the above differential equation into an analogous quadratic equation if you
		+
		+	let
		+
		+	:<math>O = \frac{d}{dt}</math>
		+
		+
		+	then the analogous equation becomes
		+
		+	:<math> \left ( mO^2 + aO + k \right ) x = 0 </math>
		+
		+	where m, a, and k are constants
		+
		+	factoring this quadratic you would have
		+
		+	:<math> \left ( O - \gamma \right )\left ( O - \beta \right ) x = 0 </math>
		+
		+	where a non-trivial solution would exist if one of the terms in the parentheses were zero
		+
		+	this basically reduces our 2nd order differential equation down to two first order differential equations
		+
		+	:<math> \left ( \frac{d}{dt} - \gamma \right ) \left ( \frac{d}{dt} + \beta \right ) x = 0 </math>
		+
		+	one of the solutions would be
		+
		+	:<math> \frac{d}{dt} x - \gamma x= 0 </math>
		+	:<math> \frac{dx}{x} = \gamma dt </math>
		+	:<math> x = Ae^{\gamma t} </math>
		+
		+
		+	:<math> x = Ae^{\gamma t} + Be^{\beta t} </math>
		+
		+
		+	For the special case where there isn't a first derivative term (a=0)
		+
		+	You simply have
		+
		+	:<math> \left ( m\frac{d}{dt}\frac{d}{dt} + k \right ) x = 0 </math>
		+
		+	or
		+	:<math> \left ( mO^2 + k \right ) x = 0 </math>
		+
		+	:<math> O= \pm \sqrt{-\frac{k}{m}} = \pm i \sqrt{-\frac{k}{m}}</math>
		+	:<math> \gamma \equiv +i\sqrt{\frac{k}{m}} = i \omega</math>
		+	:<math> \beta = -i\sqrt{\frac{k}{m}} = -i \omega</math>
		+
		+	then you have
		+
		+	:<math> x = Ae^{\gamma t} + Be^{\beta t} </math>
		+	::<math> = Ae^{i \omega t} + Be^{-i\omega t} </math>


	[[Forest_UCM_Osc#Simple_Harmonic_Motion_.28SHM.29]]		[[Forest_UCM_Osc#Simple_Harmonic_Motion_.28SHM.29]]