Physics 201, Lecture 27

Transcription

1 Physics 201, Lecture 27 Today s Topics n Oscillations (chapter 15) n Simple Harmonic Oscillation (SHM) n Mathematical Representations n Properties of SHM n Examples: Block-Spring Oscillator (More examples next lecture)

2 q When and where About Final Thursday Dec 21st 2:45-4:45 pm Room Allocation: See my on Dec 4th. q Format Closed book 1+3 8x11 formula sheets allowed, must be self prepared. (Absolutely no photocopying/printing of sample problems, examples, class lectures, HW etc.) 30 multiple choice questions. (count as 200 points) Bring a calculator (but no computer). Only basic calculation functionality can be used. q Special needs/ conflicts: All special requests should be made by 6pm Thursday December 7 th (today!). (except for medical emergency) All alternative test sessions are in our lab room, only for approved requests.

3 Chapters Covered q The final exam is cumulative. q ~50 % will be on old chapters (Ch 1-12) q ~50 % will be on new chapters (Ch 13,14,15) Chapter 13: Gravitation Chapter 14: Fluid Mechanics Chapter 15: Oscillation Motion. Special Review Lecture: Thursday December 14 th 9:55-10:45am : Chapters since midterm 3 (Note the special date which is after the last class day) Super Friday: December 15 th. 10am-5pm in the lab room.

4 College of Engineering Policies On Final Exam Rescheduling q Regulation 9 Student responsibility for scheduling: " Each student is responsible for arranging a course list that will permit satisfactory progress towards degree requirements and a class schedule that (a) avoids class and final exam scheduling conflicts, (b) avoids an excessively demanding final exam schedule, and (c) verifies registration in chosen classes. q Regulation 25 Final exam rescheduling: " A student may be permitted to take an examination at other than the regularly scheduled time only with permission of the instructor. Permission will be granted only for illness or other unusual and substantiated cause beyond the student's control. (See also Regulation 9). ( )

5 Motion of Spring-Mass System q (see demo) After the mass is (initially) pulled away from equilibrium and released (from rest) at displacement A, the mass block will move under restoring force F s in a back and forth fashion (oscillation). The motion is periodic. The motion is between A and A. At any point, acceleration a = F s /m = - k/m x The motion continues indefinitely Ø This type of motion is called Simple Harmonic Motion (SHM) q Key quantities of SHM Amplitude A Period T Frequency f = 1/T (unit 1/s: Hertz Hz) Angular frequency ω=2πf

6 Quizzes/Exercises q A particle m moving in x direction has its position described by expression x(t)=acos(ωt+φ) (A>0): What is its maximum displacement in +x direction? x + max = +A What is its maximum in x direction? x max = A So the particle is moving between A and +A è A is called the amplitude.

7 Quizzes/Exercises (2) q A particle m moving in x direction has its position described by expression x(t)=acos(ωt+φ) (A>0): Compare the displacement x(t) and x(t+2π/ω) x(t) = Acos(ωt + φ) x(t + 2π /ω) = Acos(ωt + 2π + φ) = x(t) Ø so the motion is periodic in time è we can define T = 2π/ω as the period For a periodic motion x(t+t) = x(t) A related quantity f = 1/T is called the frequency. (unit Hertz = 1/s) ω itself is called the angular frequency. ω = 2π/T = 2πf

8 Quizzes/Exercises q A particle m moving in x direction has its position described by expression x(t)=acos(ωt+φ) (A>0): What is its position at t=0? x(0) = A cos(φ) or x(0)/a=cos(φ) Ø The initial position is defined by φ. φ is called the phase angle (or simply phase) of the motion.

9 Harmonic Oscillation q Motion described by expression x(t)=acos(ωt+φ) is called harmonic oscillation A: amplitude ω: angular frequency Quick Quiz: Is x(t)=asin(ωt+φ) a harmonic oscillaton? x(t)=acos(ωt+φ) T= 1/f = 2π/ω φ: phase constant x(0)=acos(φ)

10 Quizzes/Exercises(3) q For a harmonic motion x(t)=acos(ωt+φ) : what is its velocity as function of t? v(t) = dx dt = Aω sin(ωt + φ) = Aω cos(ωt + φ + π /2) what is its acceleration as function of t? a(t) = dv dt = Aω 2 sin(ωt + φ + π /2) = Aω 2 cos(ωt + φ)

11 Oscillator Equation q Expression x=acos(ωt+φ) is a solution to harmonic oscillator equation: 2 d x = ω 2 x dt 2 q Many physical systems follow this equation: mechanic oscillators, sound, electromagnetic fields, quantum fields

12 Example 1: Spring-Block Oscillator q A block attached on an ideal spring forms a simple harmonic oscillator Why? Ø Hooke s Law: F = -kx Newton s 2 nd Law: ma=md 2 x/dt 2 = F = -kx è md 2 x/dt 2 = -kx è d 2 x/dt 2 = -(k/m)x Compare to Harmonic equation: d 2 2 x 2 dt = ω x à ω = sqrt(k/m) A and φ set by initial condition

13 Quizzes/Exercises(4) q For a harmonic motion x(t)=acos(ωt+φ) : what are x, v, a at t=0? (initial conditions) Ø x t=0 =Acos(φ) ; Ø v t=0 =- Aωsin(φ); Ø a t=0 =-Aω 2 cos(φ) what are maximum x,v,a? (extremes) Ø x max =A ; Ø v max =Aω; Ø a max =Aω 2

14 Quizzes/Exercises(3 again) q For a harmonic motion x(t)=acos(ωt+φ) : what is its velocity as function of t? v(t) = dx dt = Aω sin(ωt + φ) = Aω cos(ωt + φ + π /2) what is its acceleration as function of t? a(t) = dv dt = Aω 2 sin(ωt +φ + π / 2) = Aω 2 cos(ωt +φ) what are x,v,a at t=0? Ø x t=0 =Acos(φ) ; v t=0 =- Aωsin(φ); a t=0 =-Aω 2 cos(φ) what are maximum x,v,a? Ø x max =A ; v max =Aω; a max =Aω 2

15 Exercises (continue) q For a particle of mass m in harmonic motion x(t)=acos(ωt+φ) : what is the force on it as function of t? Ø F(t)=ma(t)= - Amω 2 cos(ωt+φ) i.e. F = - mω 2 x = - kx; k=mω 2 or ω 2 =k/m what is the kinetic energy as function of t? Ø K(t) = ½ mv 2 = ½ ma 2 ω 2 sin 2 (ωt+φ) Is the force on it a conservative force? Ø Yes. (recall: spring force is conservative) If yes, what is the potential energy associated? Ø U(t) = ½ kx 2 = ½ ma 2 ω 2 cos 2 (ωt+φ) what is total mechanic energy as function of t? Ø E(t) =K(t)+U(t) = ½ ma 2 ω 2 Is total mechanic energy a constant? Ø Yes, E(t) = ½ ma 2 ω 2 independent of t.

16 Energy Consideration for SHM q SHM: x(t) = A cos( ωt + φ ) E Ø Kinetic Energy (K): K(t) = ½ mv 2 = ½ ma 2 ω 2 sin 2 (ωt+φ) Ø Potential Energy (U): U(t) = ½ kx 2 = ½ ma 2 ω 2 cos 2 (ωt+φ) Ø Total Energy (E): E(t) =K(t)+U(t) = ½ ma 2 ω 2 =Constant

17 Harmonic Oscillation: Summary Of Properties q Harmonic Oscillation Eq: d 2 2 x 2 dt = ω x q Solution (Harmonic oscillation) : x=acos(ωt+φ) q Amplitude A: set by initial condition q Phase φ: set by initial condition q ω (or f = ω/2π or T= 1/f = 2π/ω ): determined by intrinsic and geometric features Intrinsic frequency. q Total mechanic Energy: E= ½ ma 2 ω 2

18 SHM: Summary of Formulas q Simple harmonic motion equation: x(t)=acos(ωt) : x = Acos(ωt) ; x max =A ; v = - Aωsin(ωt); v max =Aω; a = -Aω 2 cos(ωt)= -ω 2 x(t); a max =Aω 2 q Frequency and Periods: f = ω/2π T =1/f =2π/ω; q Energy: KE(t) = ½ mv 2 = ½ ma 2 ω 2 sin 2 (ωt) PE(t) = ½ kx 2 = ½ ma 2 ω 2 cos 2 (ωt) Total E(t) = KE+PE= ½ ma 2 ω 2 = constant q Spring-Mass and Pendulums: Spring-mass: ω = k / m Simple Pendulum: ω = g / L Physical Pendulum: ω = mgl / I

19 Practical Technique: Recognize Phase Constant q An oscillation is described by x=acos(ωt+φ). Find out φ for each of the following figures: x t Answer φ= 0 x t φ= π/2 x t φ= π