PHYS12 Physics 1 FUNDAMENTALS Module 3 OSCILLATIONS & WAVES Text Physics by Hecht Chapter 1 OSCILLATIONS Sections: 1.5 1.6 Exaples: 1.6 1.7 1.8 1.9 CHECKLIST Haronic otion, periodic otion, siple haronic otion (SHM), aplitude A (), cycle, period T (s), frequency f (Hz), angular frequency ω (rad.s -1 ) T = 1 / f ω = 2 π f = 2 π / T (Eq. 1.11, 1.12) displaceent x (), velocity v x (.s -1 ) and acceleration a x (.s -1 ) in SHM displaceent aplitude A (), velocity aplitude Aω (.s -1 ), acceleration aplitude Aω 2 (.s -2 ) phase φ (rad), initial phase ε (rad), in-phase, out-of-phase SHM: x vs t, v vs t and a vs t graphs SHM: acceleration proportional to displaceent x = A cos(ω t + ε) (Eq. 1.13) v x = - Aω sin(ω t + ε) (Eq. 1.16) a x = - Aω cos(ω t + ε) = - ω 2 x (Eq. 1.18) Elastic restoring force F e (N) SHM: conservation of energy, total energy E (J), inetic energy KE (J) and potential energy PE e (J) E = KE + PE KE = ½ v 2 PE = ½ x 2 Oscillating ass-spring syste Newton s Second Law r r F = a natural frequency ω o (rad.s -1 ) ωo = 1 f o = T = 2π 2π (Eq. 1.19 1.21) a3/p1/waves/waves156.doc 9:19 AM 29/3/5 1
NOTES Vibratory or oscillatory otion otion that repeats itself is referred to as haronic otion. aplitude A ω = dθ/dt ω = 2π / T A T = 1 / f f = 1 / T ω = 2 π f ω=2 π / T θ One cycle: period T (s) Cycles in 1 s: frequency f (Hz) angular frequency ω (rad.s -1 ) Siple Haronic Motion (SHM) Vibratory otion at a constant single frequency and with constant aplitude. Describing the otion Define coordinate syste, origin (equilibriu point) and observer (OBS) Origin equilibriu position displaceent x () velocity v (.s -1 ) + x ax acceleration a (.s -2 ) Force F e (N) a3/p1/waves/waves156.doc 9:19 AM 29/3/5 2
Displaceent x(t) = x ax cos(ω t + ε) x(t) = x ax cos(2 π f t + ε) x(t) = x ax cos(2 π t / T + ε) Velocity v(t) = dx(t)/dt = - ω x ax sin(ω t + ε) = - v ax sin(ω t + ε) slope of the x(t) vs t graph v(t) Acceleration a(t) = dv(t)/dt = - ω 2 x ax cos(ω t + ε) = - a ax cos(ω t + ε) slope of the v(t) vs t graph a(t) a(t) = - ω 2 x(t) Aplitude displaceent aplitude x ax A (ax value of displaceent fro origin) velocity aplitude v ax = ω x ax (ax value of velocity) acceleration aplitude a ax = ω 2 x ax (ax value of acceleration) Phase (rad) all angles ust be easured in radians (rad) calculator in rad ode. phase angle (ω t + ε) initial phase angle (t = ) ε 1 SHM position x -1 1 2 3 4 5 6 7 8 9 1 5 velocity v acceleration a -5 1 2 3 4 5 6 7 8 9 1 1-1 1 2 3 4 5 6 7 8 9 1 tie t a3/p1/waves/waves156.doc 9:19 AM 29/3/5 3
Exaple: A SHM oscillating syste of a bloc and spring is shown below in several different positions. For each situation, indicate if the position x, velocity v, acceleration a and force F are + or or. position velocity acceleration force oving right + x ax oving left + x ax oving left + x ax oving right Stopped for an instant Stopped for an instant oving right a3/p1/waves/waves156.doc 9:19 AM 29/3/5 4
Oscillating Bloc Spring Syste Consider a bloc on a frictionless surface that is attached to a spring. When the bloc is disturbed fro its equilibriu position, it will oscillate naturally without being driven by soe external source of energy. Ignoring any frictional effects or daping, the bloc attached to spring will vibrate bac-and-forth with a single frequency. The restoring force acting on the bloc always is directed bac to the equilibriu position. When the bloc reaches the equilibriu position it will be oving with its axiu velocity and overshoots. The echanical energy of the bloc-spring syste is conserved. total energy E = PE + KE = constant This type of otion is referred to as siple haronic otion (SHM). Define coordinate syste, origin (equilibriu point) and observer (OBS) equilibriu F e = x = + copressed F e (t 1 ) x(t 1 ) stretched Restoring force x(t 2 ) F e (t)= - x(t) F e (t 2 ) Newton s Second Law and Hooe's Law applied to the vibrating bloc-spring syste (assuption x is not too large) r r Σ F = a r r r F = x = a For SHM r 2 r a = ω x e r r a = x Therefore, the natural angular frequency ω o the specific frequency at which the physical syste will oscillate all by itself once set in otion is a3/p1/waves/waves156.doc 9:19 AM 29/3/5 5
natural angular frequency (rad.s -1 ) ω = o natural (linear) frequency (Hz) f o 1 = 2π period (s) T = 2π Energy Considerations Total energy of bloc and spring syste E (J) Kinetic energy of bloc KE (J) Potential energy of bloc-spring syste PE (J) E = K(x, t) + U(x, t) E = constant (assuing no loss of echanical energy) KE = ½ v 2 PE = ½ x 2 E = ½ v 2 + ½ x 2 E = ½ v ax 2 = ½ x ax 2 E = ½ x ax 2 = ½ x 2 + ½ v 2 2 2 2 2 x x ax ω ax 2 ax 2 xax xax vx ( ) = ω x x = x 1 = v 1 a3/p1/waves/waves156.doc 9:19 AM 29/3/5 6
Exaple: Coplete the table below: t x V a KE PE A - ω 2 A ½ A 2 -A +A -A +A T/4 -A +A T/2 -A +A 3T/4 -A +A T a3/p1/waves/waves156.doc 9:19 AM 29/3/5 7
Proble A spring is hanging fro a support without any object attached to it and its length is 5. An object of ass 25 g is attached to the end of the spring. The length of the spring is now 85. (a) What is the spring constant? The spring is pulled down 12 and then released fro rest. (b) Describe the otion on the object attached to the end of the spring. (c) What is the displaceent aplitude? (d) What are the natural frequency of oscillation and period of otion? Another object of ass 25 g is attached to the end of the spring. (e) Assuing the spring is in its new equilibriu position, what is the length of the spring? (f) If the object is set vibrating, what is the ratio of the periods of oscillation for the two situations? Solution Setup equilibriu 1 2 L = 5 =.5 g = 9.8.s -2 1 = 25 g =.25 g 2 =.5 g L 1 = 85 =.85 L 2 =? x 1ax = 12 =.12 x 2ax = A 2 =? f 1 =? Hz T 1 =? s T 2 / T 1 =? 1 Hooe s Law & SHM: F = x fo = 2π a3/p1/waves/waves156.doc 9:19 AM 29/3/5 8
Action (a) spring constant Newton s Laws: g = s = g / s s 1 = L 1 L = (.85.5) =.35 = 1 g / s 1 = (.25)(9.8)/(.35) N. -1 = 7. N. -1 (c) aplitude: spring pulled down 12 x 1ax = A 1 =.12 (d) frequency and period (does not depend upon aplitude) 1 1 7 f1 2π 2π.25 Hz.84 Hz T 1 = 1 / f 1 = 1.2 s (e) (f) Newton s Laws: g = s s = g / s 2 = 2 g / = (.5)(9.8) / (7.) =.7 L 2 = s 2 + L =.7 +.5 = 1.2 1 1 = 2π T 2 2 = 2π T T2 2.5 = = = 2 = 1.4 T.25 1 1 a3/p1/waves/waves156.doc 9:19 AM 29/3/5 9