Lecture
Coverage (PHY303, Fall ‘03)
- Sep. 3: Ch
0 : course objective: systematic approach to non-relativistic
mechanics; CHPT. 1: defintion of vector and basic operations, scalar
product and projection interpretation, vector product and geometric
interpretation, determinant, triple products;
- Sep. 8: Ch
1: coordinate system transformations (example: rotation), derivatives of
cartesian vectors and scalar/vector products, velocity and acceleration in
cartesian and plane polar coordinates;
- Sep. 10: Ch
1: angular coordinate systems in 3-D: position and velocity in cylindrical
coordinates, spherical coordinates: position vector and derivation of velocity
via an intermediate coordinate transformation to cartesian basis vectors;
Ch 2: Newton's first law of
motion; Newton's 2.law of motion, mass as measure of inertia,
linear momentum, Newton's 3.law, action and re-action principle, momentum
conservation, general objective of 1-body Newtonian dynamics, 2.order
differential equation+ 2 boundary conditions,solution of the equation of
motion for a constant force;
- Sep 15: Ch
2: position-dependent forces,
equation of motion, kinetic and potential energy, work function,
conservation of total (mechanical) energy, solution of the equation of
motion for velocity and position, turning points and forbidden regions of
the motion in x; example: gravitational
force without/with height dependence, velocity dependent forces, linear
and quadratic drag constants, approximate values in air, linear drag
without external force: equation of motion, velocity v(t) and position
x(t), maximal range.
- Sep
17: Ch 2: equation
of motion and its solution for quadratic drag force without external
force, linear drag force with constant gravitational force, terminal velocity and
characteristic time, time-behavior of initial and terminal velocity
components, quadratic drag + constant force; Ch 3: approximation of typical potentials inducing
oscillations with quadratic term, simple harmonic motion (SHM): equation
of motion and its solution; amplitude and phase, generality of solution,
velocity;
- Sep
22: Ch 3: effect of constant external
force on SHM, simple pendulum, energy equation for SHM: potential and
total energy, velocity as function of displacement, turning points;
Damped harmonic motion (DHM) with linear drag
force, equation of motion, general solution, structure of the solutions,
overdamping, critical damping (extra solution), underdamping: complex
displacement; requirement of reality for
physical displacement, underdamped solution, energy equation, energy loss
rate, definition and derivation of quality factor;
- Sep 24: Ch3: Forced Harmonic Motion: equation of motion with
oscillating external driving force, ansatz for steady-state solution,
derivation of frequency-dependence for phase shift and amplitude,
resonance phenomenon;
- Sep 29: Ch3: Discussion of frequency-dependence of FHM:
maximal amplitude, resonance frequency, loss of resonance behavior,
resonance width and quality factor in the weak-damping limit, zero- and
high-frequency limits of the solution and physical interpretation, LCR
circuit as electrical-mechanical analog; Fourier series.
- Oct 1: Ch4: general equation of motion in 3-D (Newton's
2.law), superposition principle, work function as line-integral in 3-D,
example, closed loop integral for work function, condition on the force
field for its vanishing (path-independence of work function), Stokes'
Theorem in 3-D; definition of 3-D differential
('Del') operator and curl of a vector field, definition of conservative
force, existence of potential function for conservative force, gradient of
potential, path-independence and total differential of work function,
total mechanical energy conservation in 3-D, separable forces and ensuing
equation of motion for projectile motion;
- Oct 8 :
Ch4: Harmonic oscillator in 2- and 3-D,
vectorial form of Hooke's law, example: 3 orthogonal springs, general
solution for the 2-D isotropic SHM, quadratic equation for trajectory,
discriminant, ellipse solution, inclination angle, equivalence of 3-D
isotropic solution, nonisotropic oscillator, energy equation; Charged particle in electromagnetic field, Lorentz
force.
- Oct 13:
Ch4: Constrained motion of a particle,
force of constraint (e.g. from curved surface), example: particle rolling
down a sphere; Ch 5: Newton's law in accelerated coordinate systems,
example: linear acceleration, angular accelerating (rotating) systems,
velocity transformation; transformation
of acceleration from inertial to rotating system, discussion of
centripetal, Coriolis and transverse acceleration from the inertial
observer's point of view.
- Oct 15: MIDTERM I
- Oct 20:
Ch5: Uniform rotation around the z-axis, dynamics in a
rotating coordinate system, fictitious forces (Coriolis, transverse and
centrifugal). Effects of
Earth rotation: static case, deformation of Earth, plumb line, geocentric
latitude, dynamic case: projectile motion on the Earth surface, Coriolis
force to lowest order in omega, equations of motion to that order, their
decoupling and general solution;
- Oct 22:
Focault pendulum, components of tension force, small
angle approximation, equations of motion in x'- and y'-components, ansatz
for time-dependent coordinate rotation, decoupling of the differential
equations, elliptic motion with superimposed rotation, precession period;
CHPT. 6: Newton's law of universal gravitation,
interchangeability of source and test mass, central force, spherical mass
distribution: integration of total force on test mass outside the sphere,
general independence of spherical mass distributions on their radius:
total mass as concentrated in the center as a specific feature of the
1/r^2 force law;
- Oct 27: Kepler's 3 laws of planetary motion: law of equal
areas from angular momentum conservation, law of ellipsis from from 2-D
equations of motion in spherical coordinates, differential equation of the
orbit and its solution, conic sections (ellipse, parabola and hyperbola),
ellipse discussion;
- Oct 29: Kepler's 'Harmonic Law' from Newton's 2. law and
gravitational law, Keplerian motion, prediction of Neptune, dark matter
problem, definition of gravitational field strength and potential,
- Nov 3: energy equation for the orbit and rederivation of the
orbit equation, relation between total energy and orbit type; comet
motion, astronomical units, eccentricity and orbital period, centrifugal
and effective radial potential, 1-D radial equation of motion, turning
points, minimal energy, bound and open orbit solutions;
- Nov 5: Rutherford scattering: repulsive Coulomb potential,
orbit equation, impact parameter and scattering angle and their relation,
differential cross section, derivation of Rutherford scattering formula, remark
on deviations from the Rutherford behavior; CHPT. 7: definition of center
of mass position, momentum and velocity for an ensemble of particles;
Equation of motion for center of mass motion in the presence of external
and internal forces, vanishing of the internal part for central forces,
total angular momentum and its decomposition in internal (spin) and
external (orbital) contributions
- Nov 10:, total kinetic energy as a sum of relative and cm
motion; interacting 2-body system:
reduced mass, equation of relative motion, examples for bound state
problem: corrections to Kepler's 3.law for planetary motion, binary star
system: mass determination from revolution velocities and period; Two-body
collisions, short-range forces, conservation laws (momentum and energy)
and energy-loss/-gain term, 1-D (central) collisions and coefficient of
restitution, 2-D scattering: kinematics and cm-coordinates, general goal
of scattering experiments, relation between incoming lab- and center of
mass-velocity,
- Nov
12: MIDTERM II
- Nov 17:, relation between lab- and cm-scattering angle,
relation between in- and outgoing cm-velocity; 2-D
elastic collision limit (Q=0), heavy target limit and Rutherford
scattering, equal mass case: lab scatt.-angle is half the cm-angle, inelastic
collision: relation between energy loss and restitution coefficient,
motion with variable mass, derivation of Newton's 2.law including mass
loss term, example: rocket motion in free space;
- Nov 19 CHPT 8: Laminar
objects – Center of mass, moment of inertia. Parallel axis theorem,
perpendicular axis theorem, torque as time derivative of angular momentum.
Object rolling down incline without slipping. Energy considerations.
Conditions for rolling, critical angle.
- Nov 24:
Marble rolling in hemispherical dish – treatment in a plane. CHPT
10: The Lagrangian. Hamilton’s principle. Variations. Illustration using
Fermat’s principle of least time (reflection and refraction) example of free fall: definition of variations in
position and velocity, Lagrange function, derivation of Newton's 2. law
for free fall from Hamilton's variational principle using partial
integration; parametrization of a variation of the true solution for the
free-fall case, verification that the true solution is indeed a minimum of
the action, definition of generalized coordinates, examples: simple
pendulum , masses connected with string on an incline; general procedure
of determining the number of generalized coordinates using equations of
(holonomic) constraints
- Nov 26:
Problem solving session.
- Dec 1: Setting up the Lagrangian; Lagrange’s
equations. Applications: projectile motion, central forces, Attwood’s
machine, double Attwood’s machine.
- Dec 3: Object sliding off incline that slides
on horizontal support.
Generalized momenta.
Ignorable coordinates. Pendulum suspended from moveable support.
- Dec 8: The
Hamiltonian. Hamilton’s equations. Two masses connected with a spring. CHPT 11: Oscillating systems. Stability
and instability.
- Dec 10: Single degree of freedom. Frequency of
small oscillations. Two degrees of freedom. Normal modes. General solution
of symmetric coupled oscillators with equal mass.
- Dec 17:
FINAL EXAM 11:00 – 1:30