Lecture 1 Outline of PHYS 260 6 topics Oscillations chapter 14 Fluids 15 Waves 20 21 Optics in PHYS 270 Thermodynamics 16 19 applications of Newton s laws from PHYS 161 increasing complexity Electrostatics 26 30 Electric Currents 31 32 beyond Newton s laws continue with magnetism in PHYS 270 Not much connection between topics survey course Outline for today Chapter 14 Oscillations 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 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 Sinusoidal amplitude A max displacement from equilibrium position x 0 velocity v dx dt v 0 at x A A v vmax at x 0 3 Questions vmax related to A T or f related to physical parameters mass spring constant derive motion from Newton s laws 1 Mathematical description focus on spring mass but general empirical data theory in next lecture 2 t x t A cos T A cos 2 f t A cos t angular frequency 2 f 2 T in radions second not cycles second from graph vx t vmax sin using calculus vx t dx dt A sin stretch spring more 2 t T 2 t T mass moves faster 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 x A at t 0 SHM projection of uniform circular motion onto 1 dimension x A cos A cos t t uniform circular motion with 0 at t 0 Relation to Circular motion II In general t 0 0 0 x t A cos t A cos t 0 vx t A sin t 0 vmax sin t 0 t phase of oscillation or angle of circular motion 0 phase constant sets initial condition starting point on circle x0 A cos 0 v0 x A sin 0 vs cos 0 in x0 2 Use conservation of energy to relate A to m k spring constant set to 0 assume no friction energy conserved potential energy U 12 k x 2 x x xc 1 1 2 2 E K U mv kx mechanical constant x 2 2 at turning point E U 12 kA2 K 0 2 U 0 at equilibrium E 12 mvmax independent of A period of oscillation is half 3 Newton s laws dv ax x dt 2 A cos t 2 x t no friction gravity Fnet x Fspring x k x kx max acceleration not constant 2nd order differential equation unique solution ax verify dx dt dvx dt d2 x dt2 k mx unspecified constants guess x t A cos t 0 A sin dx2 dt2 2 A cos satisfied if k m same as energy 2 assumption x t A cos t 0 justified by Newton s Hooke s laws theory agrees with experiment
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