Forest UCM Osc HookesLaw

Hooke's Law

Derivation

Equation of Motion from Cons of Energy

In the previous chapter Forest_UCM_Energy_Line1D, we saw how the equations of motion could from the requirement that Energy be conserved.

[math]E = T + U[/math]

[math] T = E - U[/math]

[math] \frac{1}{2} m v^2 = E- U[/math]

in 1-D

[math] \dot {x}^2 = \frac{2}{m} \left ( E-U(x) \right )[/math]

[math] \dot {x}^2= \frac{2}{m} \left ( E-U(x) \right )[/math]

[math] \dot {x}= \sqrt{\frac{2}{m} \left ( E-U(x) \right )}[/math]

[math] \frac{dx}{dt}= \sqrt{\frac{2}{m} \left ( E-U(x) \right )}[/math]

[math] \frac{dx}{ \sqrt{\frac{2}{m} \left ( E-U(x) \right )}}=dt[/math]

[math] \sqrt{\frac{m}{2}} \int \frac{dx}{ \sqrt{\left ( E-U(x) \right )}}=\int dt[/math]

Let consider the case where an object is oscillating about a point of stability [math](x_0)[/math]

A Taylor expansion of the Potential function U(x) about the equilibrium point [math](x_0)[/math] is

[math]U(x) = U(x_0) \; + \; \left . \frac{\partial U}{\partial x} \right |_{x=x_0} (x-x_0) \; + \; \frac{1}{2!}\left . \frac{\partial^2 U}{\partial x^2} \right |_{x=x_0} (x-x_0)^2 \; + \; \frac{1}{3!}\left . \frac{\partial^3 U}{\partial x^3} \right |_{x=x_0} (x-x_0)^3 \; + \dots [/math]

Further consider the case the the potential is symmetric about the equilibrium point [math](x_0)[/math]

at the equilibrium point

[math]\left . \frac{\partial U}{\partial x} \right |_{x=x_0} = 0 [/math]: Force = 0 at equilibrium

in order to have stable equilibrium

[math]\left . \frac{\partial^{2} U}{\partial x^{2}} \right |_{x=x_0} \gt 0 [/math]: stable equilibirium

also the odd (2n-1) terms must be zero in order to have stable equilibrium for a symmetric potential (the potential needs to be a max at the end points of the motion).

[math]\left . \frac{\partial^{2n-1} U}{\partial x^{2n-1}} \right |_{x=x_0} = 0 [/math]:

and the leading term is just a constant which can be dropped by redefining the zero point of the potential

[math]U(x_0) = 0[/math]

This leaves us with

[math]U(x) = \frac{1}{2!}\left . \frac{\partial^2 U}{\partial x^2} \right |_{x=x_0} (x-x_0)^2 \; + \; \frac{1}{4!}\left . \frac{\partial^4 U}{\partial x^4} \right |_{x=x_0} (x-x_0)^4 \; + \dots [/math]

if we make the following definitions

[math] \left . \frac{\partial^2 U}{\partial x^2} \right |_{x=x_0} (x-x_0)^2 \; = k [/math]

[math]U(x) = \frac{1}{6} \left . \frac{\partial^4 U}{\partial x^4} \right |_{x=x_0} (x-x_0)^4 \; = \epsilon [/math]

and that the equilibrium point is located at the orgin

[math]x_0 = 0[/math]

Then

[math]U(x) = \frac{1}{2}kx^2 \; + \; \frac{1}{4}\epsilon x^4 \; + \dots [/math]

Since we began this derivation with the assumption that energy was conserved then the force must be conservative such that

[math]: \vec F = - \vec \nabla U[/math]

or this 1-D force can be written as

[math]F = - \frac{\partial }{\partial x} U (x) = - kx - \epsilon x^3 - \dots[/math]

Interpretation (Hooke's law)

Returning back to the conservation of energy equation

[math] E = T + U = \frac{m}{2} \dot {x}^2 + \frac{1}{2}kx^2 \; + \; \frac{1}{4}\epsilon x^4 \; + \dots [/math]

Lets consider only the first term in the expansion of the potential U(x)

[math] E = \frac{m}{2} \dot {x}^2 + \frac{1}{2}kx^2 [/math]

[math] \frac{dE}{dt} = \frac{m}{2} 2 \dot {x} \ddot x + \frac{1}{2}k2 x \dot x = 0 [/math] energy is constant with time

[math] m\ddot x =-kx [/math] energy is constant with time

A Force exerted by a spring is proportional to the spring displacement from equilibrium and is directed towards restoring the equilibrium condition. (a linear restoring force).

In 1-D this force may be written as

[math]F = - kx[/math]

While the above was derived from the assumption of conservation of energy we can apply our two tests for conservative forces as a double check:

1.) The force only depends on position.

2.) The work done is independent of path ( [math]\vec \nabla \times \vec F = 0[/math] in 1-D and 3-D)

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]

Oscillator properties

[math] x = Ae^{i \omega t} + Be^{-i\omega t} [/math]

Using Euler's formula for complex variables

[math]e^{\pm ix} = \cos(x) \pm i \sin(x)[/math]

[math]\Rightarrow x = A\left ( \cos(\omega t) + i \sin(\omega t)\right ) + B\left ( \cos(\omega t) - i \sin(\omega t)\right )[/math]

[math]= (A + B) \cos (\omega t) + i(A-B) \sin(\omega t)[/math]

[math]= (A + B) \cos (\omega t) + i(A-B) \sin(\omega t)[/math]

[math]= A^{\prime}\cos (\omega t) + B^{\prime} \sin(\omega t)[/math]

let

[math]C = \sqrt{\left(A^{\prime}\right)^2+ \left ( B^{\prime}\right)^2}[/math]

[math]\cos\delta = \frac{A^{\prime}}{C}[/math]

[math]\sin\delta = \frac{B^{\prime}}{C}[/math]

then

[math]x = A^{\prime}\cos (\omega t) + B^{\prime} \sin(\omega t)[/math]

[math] = C \left [ \frac{A^{\prime}}{C}\cos (\omega t) + \frac{B^{\prime}}{C} \sin(\omega t) \right ][/math]

[math] = C \left [\cos\delta\cos (\omega t) + \sin\delta \sin(\omega t) \right ][/math]

[math] = C \cos (\omega t - \delta) [/math]

[math]\cos(A-B) = \cos(A)\cos(B)+\sin(B)\sin(A)[/math]

[math]x=A \cos(\omega t - \delta)[/math]

[math]v = \dot x = \omega A \sin(\omega t - \delta)[/math]

Kinetic (T) and potential (U) Energy

[math]U_{\mbox{max}}= \frac{1}{2} k x^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=U_{\mbox{max}}[/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=U_{\mbox{max}}[/math]

2-D Oscillators

Graph of the Energy

Forest_UCM_Osc#Hooke.27s_Law