Download Introduction to Electromagnetism

Intro to Classical Mechanics
[email protected], 3.Oct.2002
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Study of motion
Space, time, mass
Newton’s laws
Vectors, derivatives
Coordinate systems
Force and momentum
Energies
Four realms of physics
Classical Mechanics
Quantum Mechanics
(big and slow:
everyday experience)
(small: particles, waves)
Special relativity
Quantum field theory
(fast: light, fast particles)
(small and fast: quarks)
Mechanics = study of motion of objects
in absolute space and time
Time and space are NOT absolute, but their
interrelatedness shows up only at very high speeds, where
moving objects contract and
moving clocks run slow.
Virtually all everyday (macroscopic, v<c) motions can be
described very well with classical mechanics,
even though Earth is not an inertial reference frame (its
spin and orbital motions are forms of acceleration).
Space and time are defined via speed of light.
• c ~ 3 x 108 m/s
• meter = distance light travels
in 1/(3 x 108) second
• second is fit to match:
period T = 1/frequency = 1/f
E = hf = 2mB (hyperfine
splitting in Cesium)
second ~ 9 x 1010 TCs
Vectors and derivatives
Practice differentiation vectors: #1.6 (p.36)
A = i a t + j b t2 + k g t3
Polar coordinates
r=
r
q
y = r sin q
x = r cos q
v = dr/dt =
a = dv/dt =
der/dt =
deq /dt =
Cylindrical and spherical coordinates
Practice #1.22 (p.36)
Ant’s motion on the surface of a ball of radius b is given by
r=b, f = w t, q = p/2 [1 + 1/4 cos (4 w t)]. Find the velocity.
Newton’s Laws
I. If F = 0, then v = constant
II. S F = dp/dt = m a
III. F12 = -F21
Momentum p = m v
a = F/m = dv/dt
v =  a dt = dx/dt
x =  v dt
Practice #2.1, 2.2
Given a force F, find the resultant velocity v.
For time-dependent forces, use a(t) = F(t)/m, v(t) =  a(t) dt.
For space-dependent forces, use F(x) = ma = m dv/dt where dv/dt
= dv/dx * dx/dt = v dv/dx and show that v dv = 1/m F dx.
2.1(a) F(t) = F0 + c t
2.2(a) F(x) = F0 + k x
Energies
F = m dv/dt = m v (dv/dx). Trick: d(v2)/dx =
Show that F = mv (
) = m/2 d(v2)/dx
Define F = dT/dx where T = Kinetic energy. Then
change in kinetic energy =  F dx = work done.
Define F = -dV/dx where V = Potential energy.
Total mechanical energy E = T + V
is conserved in the absence of friction or other dissipative forces.
Practice with energies
To solve for the motion x(t), integrate v = dx/dt where
T = 1/2 m v2 = E - V
Note: x is real only if V < E
 turning points where V=E.
#2.3: Find V = -  F dx for forces in 2.1 and 2.2.
Solve for v and find locations (x) of turning points.