Difference between revisions of "Forest UCM NLM"

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The unit vectors (<math>\hat{r}</math> and  <math>\hat{\phi}</math> ) are changing in time.  You could express the position vector in terms of cartesian unit vector in order to avoid this
+
The unit vectors (<math>\hat{r}</math> and  <math>\hat{\phi}</math> ) are changing in time.  You could express the position vector in terms of the cartesian unit vectors in order to avoid this
  
 
:<math>\vec{r} = r \cos(\phi) \hat{i} + r \sin(\phi)\hat{j}</math>
 
:<math>\vec{r} = r \cos(\phi) \hat{i} + r \sin(\phi)\hat{j}</math>

Revision as of 19:03, 18 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 (0.1c) nor are microscopically small ( 109m).

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:

r=xˆi+yˆj+zˆk=(x,y,z)=31riˆei

Velocity:

v = drdt = dxdtˆi+xdˆidt+cdots


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


dˆidt=0=dˆjdt=dˆkdt
v = drdt = dxdtˆi+dydtˆi+dzdtˆi

Polar

TF UCM PolarCoordSys.png Vector Notation convention:

Position:

Because ˆr points in a unique direction we can write the position vector as

r=rˆr
rrˆr+ϕˆϕ: ϕ does not have the units of length


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

r=rcos(ϕ)ˆi+rsin(ϕ)ˆj

The dependence of position with \phi can be seen if you look at how the position changes with.

Velocity:

v = drdt = dxdtˆi+xdˆidt+cdots


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


dˆidt=0=dˆjdt=dˆkdt
v = drdt = dxdtˆi+dydtˆi+dzdtˆi

Spherical

TF UCM SphericalCoordSys.png

Cylindrical

TF UCM CylCoordSys.png

Vectors

Scaler ( Dot ) product

Vector ( Cross ) product

Forest_Ugrad_ClassicalMechanics