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Lecture 1
• Oscillations (chapter 14)
• Fluids (15)
• Waves (20, 21) (Optics in PHYS 270)
• Thermodynamics (16-19)
• Electrostatics (26-30)
• Electric Currents (31-32)
Outline of PHYS 260: 6 topics
(applications of Newton’s laws from PHYS 161: increasing complexity)
(beyond Newton’s laws: continue with magnetism in PHYS 270)
Not much connection between topics: survey course
Outline for today
• Kinematics of simple harmonic oscillations (mathematical description of motion): relation to uniform circular motion
• Dynamics: use conservation of energy and Newton’s laws to relate kinematics to physical parameters (mass...)
Chapter 14 (Oscillations)
Review: uniform circular motion (4.5);restoring forces; elastic potential energy (10.4, 10.5);
conservation of energy; energy diagrams (10.7)
Features of Oscillations
• back-and-forth motion about equilibrium position
• period (T): time for 1 cycle
• frequency (f = 1/ T): number of cycles per second
• units: 1 hertz (Hz) = 1 cycle/second = 1/ s
• 1 k Hz = 1000 Hz T = 1 / ( 1000 / s ) = 1 ms
Special case: Simple Harmonic Motion
• amplitude (A): max. displacement from equilibrium position (x=0)
• velocity (v) = dx/dt
• v=0 at x = +A, -A
• v = at x = 0vmax
3 Questions
• related to A?
• T (or f) related to physical parameters (mass, spring constant)
• derive motion from Newton’s laws
vmax
1. Mathematical description• focus on spring-mass (but general)
• empirical data (theory in next lecture):
• from graph:
• using calculus:
• stretch spring more mass moves faster
! (angular frequency) = 2"f = 2"/T(in radions/second, not cycles/second)
vx(t) = !vmax sin!
2!tT
"
x(t) = A cos!
2!t
T
"
= A cos (2!ft)= A cos "t
vx(t) =dx
dt
= !!A sin!
2"t
T
"
Example• An object undergoing SHM has a maximum displacement of 4.7 m
at t = 0 s. If the angular frequency of oscillation is 1.6 rad/s, what is the object’s (a) displacement and (b) speed when t = 3.5 s?
Relation to Circular motion (I)
• general initial condition:
• SHM: projection of uniform circular motion onto 1 dimension
• x = A cos ! = A cos "t(! = "t: uniform circular motion with ! = 0 at t = 0)
x0 != A at t = 0
Relation to Circular motion (II)
• In general:
• phase of oscillation (or angle of circular motion)
• phase constant (sets initial condition: starting point on circle):!0
!(t)
x0 = A cos !0;v0 x = !"A sin!0
!(t = 0) ! !0 "= 0 #x(t) = A cos !(t) = A cos ("t + !0);vx(t) = $"A sin ("t + !0) = vmax sin ("t + !0)
vs. cos !0 in x0
2. Use conservation of energy to...• ...relate
• assume no friction (energy conserved)
• potential energy:
• mechanical energy:
• at turning point:
• at equilibrium:(independent
of A)
(period of oscillation is half)
E = 12mv2
max
E = U = 12kA2
E = K + U = 12mv2
x + 12kx2
U = 12k (!x)2 (!x = x! xc)
set to 0A, ! to m, k (spring constant)
3. Newton’s laws
• no friction/gravity
• acceleration not constant (2nd order differential equation):
• unique solution guess:
• verify:
• satisfied if (same as energy...)
• assumption justified by Newton’s + Hooke’s laws (theory agrees with experiment!)
ax = dvxdt = d2x
dt2 = ! kmx
(Fnet)x = (Fspring)x
= !k!x = !kx
= max
ax =dvx
dt= !!2A cos !t
= !!2x(t)
dxdt = !!A sin ... " dx2
dt2 = !!2A cos ...
x(t) = A cos (!t + "0)
x(t) = A cos (!t + "0)
!2 = k/m
unspecified constants