Difference between revisions of "Forest UCM NLM"

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:: or in more compact form
 
:: or in more compact form
  
:<math>\vec{\dot{v}} =  \vec{\dot{r}}\hat{r} + r \dot{\phi} \hat{\phi}</math>
+
:<math>\vec{v}=\vec{\dot{r}} =  \dot{r} \hat{r} + r \dot{\phi} \hat{\phi}</math>
  
 
cartesian unit vectors do not change with time (unit vectors for other coordinate system types do)
 
cartesian unit vectors do not change with time (unit vectors for other coordinate system types do)

Revision as of 20:45, 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+


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 given by ˆr=r|r| 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 ϕ can be seen if you look at how the position changes with time.

Consider the motion of a particle in a circle. At time t1 the particle is at r(t1) and at time t2 the particle is at r(t2)


TF UCM PolarVectDiff.png


If we take the limit t2t1 ( or Δt0) then we can write the velocity of this particle traveling in a circle as

ˆr(t2)ˆr(t1)Δˆr=Δϕˆϕ
or
dˆrt=dϕdtˆϕ

Thus for circular motion at a constraint radius we get the familiar expression

v=limΔt0r(t2)r(t1)Δt=limΔt0r(ˆr(t2)ˆr(t1))Δt=rΔϕΔtˆϕ=rωˆϕ
v=rdϕdtˆϕ


If the particle is not constrained to circular motion ( i.e.: r can change with time) then the velocity vector in polar coordinates is


Velocity:

v = drdtˆr+rdϕdtˆϕ
or in more compact form
v=˙r=˙rˆr+r˙ϕˆϕ

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