Difference between revisions of "Forest UCM Energy Line1D"
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:: <math>= \sqrt{\frac{m}{k}} \int_{x_0}^x d\theta</math> | :: <math>= \sqrt{\frac{m}{k}} \int_{x_0}^x d\theta</math> | ||
:: <math>= \frac{1}{\omega} \int_{\theta_0}^{\theta} d \theta</math> | :: <math>= \frac{1}{\omega} \int_{\theta_0}^{\theta} d \theta</math> | ||
+ | |||
+ | :<math> \theta = \omega t + \theta_0 </math> | ||
+ | :<math> \sin \theta = \sin {\omega t + \that+0}</math> | ||
+ | :<math>\sqrt{\frac{2E}{m}} \sin \theta = \sqrt{\frac{2E}{m}} \sin {\omega t + \that+0}</math> | ||
+ | :<math>x = \sqrt{\frac{2E}{m}} \sin {\omega t + \that+0}</math> | ||
+ | |||
+ | :\Rightarrow x = \sqrt{ | ||
[[Forest_UCM_Energy#Energy_for_Linear_1-D_systems]] | [[Forest_UCM_Energy#Energy_for_Linear_1-D_systems]] |
Revision as of 12:40, 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.
- cosntant
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.
spring example
Consider the problem of a mass attached to a spring in 1-D.
The potential is given by
let
- and
then
- \Rightarrow x = \sqrt{