Intro to Classical Mechanics 3.Oct.2002 Study of motion Space, time, mass Newton’s laws Vectors, derivatives Coordinate systems Force and momentum Energies

Four realms of physics

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 10 8 m/s meter = distance light travels in 1/(3 x 10 8 ) second second is fit to match: period T = 1/frequency = 1/f E = hf = 2  B (hyperfine splitting in Cesium) second ~ 9 x T Cs

Practice differentiation vectors: #1.6 (p.36) A = i  t + j  t 2 + k  t 3 Vectors and derivatives

Polar coordinates x = r cos  y = r sin  r  r = de r /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. F 12 = -F 21 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) = F 0 + c t 2.2(a) F(x) = F 0 + k x

Energies F = m dv/dt = m v (dv/dx). Trick: d(v 2 )/dx = Show that F = mv ( ) = m/2 d(v 2 )/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 v 2 = 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.