Intro to Classical Mechanics Zita@evergreen, 3.Oct.2002

Intro to Classical MechanicsZita@evergreen.edu, 3.Oct.2002

• Study of motion• Space, time, mass• Newton’s laws• Vectors, derivatives• Coordinate systems• Force and momentum• Energies

Four realms of physics

Classical Mechanics(big and slow:

everyday experience)

Quantum Mechanics(small: particles, waves)

Special relativity(fast: light, fast particles)

Quantum field theory(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 = 2B (hyperfine splitting in Cesium)

second ~ 9 x 1010 TCs

Practice differentiation vectors: #1.6 (p.36)

A = i t + j t2 + k t3

Vectors and derivatives

Polar coordinates

x = r cos

y = r sin r

r =

der/dt =

de/dt =

v = dr/dt =

a = dv/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, = t, = /2 [1 + 1/4 cos (4 t)]. Find the velocity.

Newton’s Laws

I. If F = 0, then v = constant

II. 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.