Oscillations: Review (Chapter 1) Oscillations: otions that are periodic in tie (i.e. repetitive) o Swinging object (pendulu) o Vibrating object (spring, guitar string, etc.) o Part of ediu (i.e. string, water) as wave passes by Oscillation requires o Restoring force (pushes syste back toward equilibriu) o Inertia (keeps syste going past equilibriu)
Oscillator eaple: Mass on a Spring: o Restoring force fro spring. Varies with etension Hooke s Law: F k (linear restoring force) If ass displaced fro equilibriu then released will oscillate If force is linear and restoring (and thus conservative) o Motion is siple haronic otion (SHM) o Syste is a siple haronic oscillator (SHO)
Acceleration in SHM: not constant depends on Newton s nd law says F a k o For SHM, iplies k a so that a k At a. positive displaceent fro equil. A have a A and v 0 At equilibriu 0 have a 0 and v va k At a. negative displaceent fro equil. A have a A and v 0
How do we find an epression for (t) in SHM? First: find an equation for which (t) is the solution. Reeber: d dv d v and a But for ass on spring: a k So for ass on spring: d k This is a differential equation. More general for: d So identify k for ass/spring Function (t) that satisfies equation: o Oscillates like particle doing SHM ore general than spring case o What is?
How do we solve d to get (t)? To solve differential equation guess solution then test! d cos d sin Notice that: cos d and sin d So try: t Acos t o Iportant: arguent t ust be in radians o Constants A,, and ust be found fro details of otion Test t A t cos : d v A sin t d a A cos t d cos satisfies So: t A t
Iportant! d ust be satisfied by any SHM (i.e. t A t For a ass on a spring (one eaple of SHM) k o a o cos ) k (Natural angular frequency for spring/ass syste) For other eaples of SHM, depends on specific for of restoring force and inertia is natural angular frequency of SHO. Units are radians/second (really s 1 )
Paraeters that describe SHM: t Acos t t is the phase in radians. Period (T): o Motion at t T is the sae as otion at t o Means that phase increases by in tie T o So t T t o Result: T Frequency: o (angular frequency) is nuber of radians by which phase increases per second o f 1 T (frequency) is nuber of ties per second that otion repeats units are cycles/second or Hertz (Hz) o so f
Phase constant o Depends on where otion is in cycle at t=0. i.e. Aplitude (A) o t oscillates between +A and A 0 Acos
Relations between Position, Velocity, and Acceleration in SHM t Acos t A d v (slope of vs t), t A sin t v Aω d a (slope of v t A cos t vs t), a Aω
How do we get paraeters,, and A fro description of specific otion? depends on properties of oscillator Angular frequency o i.e. for spring/ass syste, k / Get Aplitude and phase constant fro and v at specific tie (i.e. t =0) o E.: ass, spring k. Start clock (set t=0) when ass at arbitrary i 0 Acos and v0 vi A sin So i Solve for A (use trigonoetry identity): sin i vi cos 1 A A vi gives: A i Solve for phase constant: sin vi tan cos i
Check for two siple cases: o If v 0, otion starts fro rest at aiu or iniu displaceent i A i i and tan 0 so that 0
o If 0, otion starts fro equilibriu with aiu speed i v i A and tan 1 So v i cost
EXAMPLES (to finish yourself if not copleted on board): Chapter 1, Proble 14: A piston oscillates in an engine with position 5.00 ccost / 6. At t 0 s, find (a) position, (b) velocity, and (c) acceleration. (d) Find the period and Aplitude of the otion.
Chapter 1, Proble 9: A particle s otion along the ais is SHM. It starts fro equilibriu at t 0 s and oves to the right. Aplitude is.00 c and frequency is 1.50 Hz. (a) Find an epression for t. (b) Find the aiu speed and earliest tie t 0 at which particle has this speed. (c) Find the aiu acceleration and earliest tie t 0 at which particle has this acceleration.
Energy and SHM Mechanical energy (E=K+U) conserved if no non conservative forces o For SHM, alternates between kinetic energy and potential energy Eaple: for ass on spring k so that k / o Acos t and v A sin t 1 1 o Kinetic Energy: K v ka sin t 1 1 o Potential Energy: U k ka cos t 1 o Mechanical Energy: E K U ka sin t cos t 1 o So total echanical energy only depends on k and A: E ka o Fro E K U, also get: 1 1 1 ka k v 1 ka k Gives ass speed as function of : v A A