Forest UCM NLM

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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
or in more compact form
˙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


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