Difference between revisions of "Forest UCM NLM"

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:<math>\vec{r} = r \hat{r}</math>
 
:<math>\vec{r} = r \hat{r}</math>
  
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:<math>\vec{r} \ne r \hat{r} +\phi \hat{\phi} </math>: [[\phi]] does not have the hits of length
  
  

Revision as of 21:13, 17 June 2014


Newton's Laws of Motion

Limits of Classical Mechanic

Classical Mechanics is the formulations of physics developed by Newton (1642-1727), Lagrange(1736-1813), and Hamilton(1805-1865).

It may be used to describe the motion of objects which are not moving at high speeds (0.1[math] c[/math]) nor are microscopically small ( [math]10^{-9} m[/math]).

The laws are formulated in terms of space, time, mass, and force:

Space and Time

Space

Cartesian, Spherical, and Cylindrical coordinate systems are commonly used to describe three-dimensional space.

Cartesian

TF UCM CartCoordSys.png


Vector Notation convention:

Position:

[math]\vec{r} = x \hat{i} + y \hat{j} + z \hat{k} = (x,y,z) = \sum_1^3 r_i \hat{e}_i[/math]

Velocity:

[math]\vec{v}[/math] = [math]\frac{d \vec{r}}{dt}[/math] = [math]\frac{d x}{dt}\hat{i} + x\frac{d \hat{i}}{dt} + cdots[/math]


cartesian unit vectors do not change with time (unit vectors for other coordinate system types do)


[math]\frac{d \hat{i}}{dt} =0 =\frac{d \hat{j}}{dt} =\frac{d \hat{k}}{dt}[/math]
[math]\vec{v}[/math] = [math]\frac{d \vec{r}}{dt}[/math] = [math]\frac{d x}{dt}\hat{i} + \frac{d y}{dt}\hat{i} + \frac{d z}{dt}\hat{i} [/math]

Polar

TF UCM PolarCoordSys.png Vector Notation convention:

Position:

[math]\vec{r} = r \hat{r}[/math]
[math]\vec{r} \ne r \hat{r} +\phi \hat{\phi} [/math]: \phi does not have the hits of length


Velocity:

[math]\vec{v}[/math] = [math]\frac{d \vec{r}}{dt}[/math] = [math]\frac{d x}{dt}\hat{i} + x\frac{d \hat{i}}{dt} + cdots[/math]


cartesian unit vectors do not change with time (unit vectors for other coordinate system types do)


[math]\frac{d \hat{i}}{dt} =0 =\frac{d \hat{j}}{dt} =\frac{d \hat{k}}{dt}[/math]
[math]\vec{v}[/math] = [math]\frac{d \vec{r}}{dt}[/math] = [math]\frac{d x}{dt}\hat{i} + \frac{d y}{dt}\hat{i} + \frac{d z}{dt}\hat{i} [/math]


The unit vectors are changing in time. You could express the position vector in terms of cartesian unit vector in order to avoid this

[math]\vec{r} = r \cos(\phi) \hat{i} + r \sin(\phi)\hat{j}[/math]

Spherical

TF UCM SphericalCoordSys.png

Cylindrical

TF UCM CylCoordSys.png

Vectors

Scaler ( Dot ) product

Vector ( Cross ) product

Forest_Ugrad_ClassicalMechanics