Difference between revisions of "Forest UCM Energy Line1D"

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:: <math> = \int \pm \sqrt{\frac{m}{2\left (0-(-mgx) \right )}} dx  </math>
 
:: <math> = \int \pm \sqrt{\frac{m}{2\left (0-(-mgx) \right )}} dx  </math>
 
:: <math> = \int \pm \sqrt{\frac{1}{2gx}} dx  = \int \pm (2gx)^{-\frac{1}{2}}dx  </math>
 
:: <math> = \int \pm \sqrt{\frac{1}{2gx}} dx  = \int \pm (2gx)^{-\frac{1}{2}}dx  </math>
:: <math> =  \pm (2g)^{-\frac{1}{2}} \sqrt x =  \sqrt{\frac{2x}{g} } </math>
+
:: <math> =  \pm (2g)^{-\frac{1}{2}} 2\sqrt x =  \sqrt{\frac{2x}{g} } </math>
  
 
== spring example==
 
== spring example==

Revision as of 15:43, 26 September 2014

The equation of motion for a system restricted to 1-D is readily solved from conservation of energy when the force is conservative.

T+U(x)= cosntant E
T=EU(x)
12m˙x2=EU(x)
˙x=±2(EU(x))m
±m2(EU(x))dx=dt=tti=t

The ambiguity in the sign of the above relation, due to the square root operation, is easily resolved in one dimension by inspection and more difficult to resolve in 3-D.

The velocity can change direction (signs) during the motion. In such cases it is best to separte the inegral into a part for one direction of the velocity and a second integral for the case of a negative velocity.


Free fall

Consider a rock dropped at t=0 from a tower of height h.

The potential energy stored in the rock at any instant is given by

U(x)=mgx

Note
The potential is highest at x=0 and becomes negative as x increases

The initial total energy is

Etot=T+U=00=0
t=±m2(EU(x))dx
=±m2(0(mgx))dx
=±12gxdx=±(2gx)12dx
=±(2g)122x=2xg

spring example

Consider the problem of a mass attached to a spring in 1-D.

F=kx

The potential is given by

U(x)=F(x)dx=12kx2
t=±m2(EU(x))dx=dt
=m2xx0(EU(x))12dx
=m2xx0(E12kx2)12dx
=m2Exx0(1(xk2E)2)12dx

let

sinθ=xk2E and ω=km
cosθdθ=dxk2E

then

t=m2Exx0(1sin2θ)12dx
=m2Exx0dxcosθ
=m2Exx0cosθdθcosθ2Ek
=mkxx0dθ
=1ωθθ0dθ
θ=ωt+θ0
sinθ=sin(ωt+θ0)
2Emsinθ=2Emsin(ωt+θ0)
x=2Emsin(ωt+θ0)
x=Asin(ωt+θ0)
A=2Em = amplitude of oscillating motion
U(x)=12kx2=12kA2sin2(ωt+θ0)
E=T+U(x)=12kA2


Forest_UCM_Energy#Energy_for_Linear_1-D_systems