Physics 2D Spring 1999 Lecture 2: Classical Physics

Classical Physics Review

Newtonian Mechanics

Newton's Laws of Motion - mass, momentum and energy

Simple harmonic oscillator

Inertial Reference Frames — Principle of Relativity

Rotational motion - torques and angular momentum

Electric and Magnetic fields

Electric field and potential

Magnetic fields - forces on charges and currents

Electromagnetic Radiation

Light as a wave - finite speed of light

Wave properties (e.g. sound or water waves):

- Basics: speed, amplitude, phase

- linear superposition -> interference

Doppler Effect

Classical Relativity

Galilean Transformation

Invariance of Newton's Laws

Newtonian Mechanics
Newton's Laws of Motion:

Galileo's Principle of Inertia - An object will remain at rest or in uniform motion in a straight line unless acted on by an external force.

The rate of change of momentum of a body is proportional to the force applied:

For a constant mass we have the familiar forms:

For every action there is an equal and opposite reaction

-> Conservation of momentum (equal and opposite forces acting over the same time).

Important Application: Simple Harmonic Oscillator

Force opposite and prop. to displacement, e.g. a pendulum or a spring:

General Solutions:

where the Angular frequency (radian/s).

Frequency (Hz) = 1/Period (s).

Energy:

Kinetic energy: work done to accelerate/decelerate a body from rest:

= force applied x distance:

Principle of Conservation of Energy: in a closed system, energy is neither created nor destroyed - it can only change form:

Particle: Total Energy = Kinetic Energy + Potential Energy: E = K + U

Principle of Relativity:

Define inertial reference frame: coordinate system in which Newton's Laws of Motion are valid.

Principle: All the laws of Physics are the same for all inertial reference frames.

Examples of Inertial Reference Frames:

vehicle "coasting" at constant velocity (speed and direction)
freely falling or orbiting space-ship

Counter-examples (non-inertial frames):

accelerating reference frames
this includes rotating (spinning) reference frames
the earth is not an inertial frame, but most of the time we can ignore its slow rotation (exceptions: weather systems, Foucault's pendulum).

Rotation:

Moment or Torque: Force x perpendicular distance from Pivot (center of rotation) :

Angular Momentum = "moment of momentum":

L is a vector defining the plane (and sense) of rotation.

Newton's 3^rd law guarantees conservation of angular momentum (as long as there are no external torques).

Circular (orbital) motion: constant speed but changing velocity

Centripetal acceleration (towards center).

Electric & Magnetic Fields
Electric fields:

Electrostatic force between point charges: (Newton)

Charge q is measured in Coulomb ( C). Charge of electron e =-1.6x10^-19 C

Electric field = force per unit charge: (Newton)

e.g. field around a point charge (Newton/Coulomb)

Potential Energy = work required to move a charge into an electric field from infinity, e.g. for two point charges (Joule)

Electrical Potential = potential energy per unit charge, e.g. around a point charge (Volt = Joule/Coulomb).

So "Work done" = charge x potential difference = qV

-> new unit of energy: 1electron-Volt (1 eV) = work done when a charge of 1 electron crosses a potential difference of 1V. 1eV = 1.6x10^-19J.

Magnetic Fields:

Force on a charged particle in a magnetic field:

-> Perpendicular to velocity and field direction - causes charge to rotate around field lines. Moving charges = a current!

Currents also generate a magnetic field, e.g. at the center of a "current loop":

Electromagnetic Radiation (Krane 3.1)

Solution of Maxwell's Equations - electric and magnetic fields propagate together as an electromagnetic wave (demonstrated by Hertz, 1887).

Direction of propagation: E x B - i.e. perp. to electric/magnetic fields

Speed of light in vacuo: =2.9979x10⁸ m/s

Measured by Romer (moons of Jupiter), and Fizeau (cogs on wheel)

General Properties of Waves

Waves transport energy and momentum (later!)

Speed (m/s)= Frequency (Hz) x Wavelength (m):

Instantaneous displacement ("snapshot") given by Amplitude and Phase

Diffraction/Refraction — i.e. "bending around corners"

Superposition -> interference (later!)

Polarization (transverse waves only, later!)

Doppler Effect:

(i) Observer moving (v_O): changes perceived frequency or wave velocity

(ii) Source moving (v_S): changes actual wavelength

-> different results depending on who does the moving (try it!).

i.e. The wave-carrying medium defines the reference frame!

Classical (Galilean) Relativity (Krane 2.1)

Transforming between Inertial Frames:

Q. How do observers in different inertial frames relate measurements?

A. Principle of Relativity states that all physical laws are the same

(N.B. not just Newton's Laws — can use the same thermometers & barometers too!)

BUT: coordinates of events different - we need a coordinate transformation.

Two observers at rest in their frame of reference, but moving relative to

each other with velocity u. Synchronize clocks and origins at t=0, and

make the x-axis the direction of motion.

Galilean Transformation:

Consequences:

Velocities:

Accelerations:

-> Newton's Laws hold for both observers (as long as u is constant).

e.g. O' tosses an apple straight up:

O observes a parabolic trajectory:

…but both agree that O' catches the apple.

See also this week's Homework (on the WWW, solutions next week!)