Physics 116 ELECTROMAGNETISM AND OSCILLATORY MOTION Lecture 3 - PowerPoint PPT Presentation

Physics 116 ELECTROMAGNETISM AND OSCILLATORY MOTION Lecture 3 Energy in SHM Oct 3, 2011 R. J. Wilkes Email: ph116@u.washington.edu

Announcements - PHYS 116 Course home page: visit frequently for updated course info! http://faculty.washington.edu/wilkes/116/ - Question box: shy people can drop questions into the box at front of room, I’ll answer next time - Civil behavior guidelines: Turn off cell phones while in class • If you prefer to text or browse instead of listening, please sit in • BACK of room – your screen distracts others who are paying attention 3-Oct-2011 Physics 116 - Au11 2

Announcements - Office hours - JW: Monday and Friday, 2:30-3:00pm (B303) (note: changed from earlier posting on website) - Kyle Armour (TA): T 3:30-4:30 office (B442), W 2:30-3:30 in the study center, Th 11:30-12:30 office Study Center hours expanded: now 9:30-5:30 M-F • Q: How to succeed in 116…? 3-Oct-2011 Physics 116 - Au11 3

Mindful time on task 5 credits! you should spend about 1/3 of your time on 116 • Class has many components to help you: – Textbook readings ( read relevant sections BEFORE class) – Lecture with demonstrations – Review lecture slides (posted few days later) – Assigned/graded homework problems – Additional homework problems in textbook – Labs ( = hands-on demonstrations!) – TAs (Kyle’s office hrs, Study Center anytime) – Fellow students (form study groups) • Don’t just sleepwalk through homework! – No value unless you are thinking as you work – Don’t treat as busywork, to wade through as soon as possible – Most value if done after you have attended lecture on subject 3-Oct-2011 Physics 116 - Au11 4

5 “Clickers” are required iCue / H-ITT Clickers (TX-3100) • Required to enter answers in pop quizzes – Be sure to get radio (RF), not infrared (IR) – Other brands/models WILL NOT WORK • We’ll practice using them today, and begin using them for pop quizzes starting tomorrow – Bring your clicker to class every day from now on – Quizzes are designed to be easy I F you are paying attention • Questions will be about something we just discussed! 3-Oct-2011 Physics 116 - Au11 5

6 Clicker registration • Required to connect your name to clicker response data! – Don’t worry, all data saved with clicker’s unique hardware ID • Go to https:/ / catalyst.uw.edu/ webq/ survey/ wilkes/ 142773 (or just follow link on class home page) • Follow instructions carefully! – Clicker serial number: look on the outside of the case, of the newer models, or underneath the battery on the older models. It consists of 6 or 7 numbers. Do NOT include any letters in your response. – Personal screen name: optional, helps you spot your own response on the screen. (Note: no guarantee someone else won’t use the same! No way to enforce uniqueness) If none, your screen name = last 3 digits of clicker ID – CHECK FOR TYPOS CAREFULLY BEFORE YOU PRESS “submit” !! – May be a few days before classroom database is updated • If you have to buy a new clicker later in the term, go back and re-register 3-Oct-2011 Physics 116 - Au11 6

Lecture Schedule (up to exam 1) Today 3-Oct-2011 Physics 116 - Au11 7

Phase relationships and max values Last time: • Maximum values occur when sin/cos = 1, so ( ) = = ω max x A x t ( ) A cos t ( ) = ω = − ω ω max v A ( ) sin v t A t ( ) = − ω ω = ω 2 2 a t ( ) A cos t max a A • x, v and a have phase relationships determined by their trig functions: cos, -sin and -cos, all with the same value of ( ω t ) so at t = 0, x = +max, v = 0, and a = - max. After t=0, x and v are 90 deg out of phase (¼ cycle shift) x and a are 180 deg out of phase (opposite signs – ½ cycle shift) 1.5 1 0.5 x, v, a x=Acos(wt) 0 v= - Awsin(wt) a= - Aw^2cos(wt) -0.5 ( v and a have been -1 scaled to fit in this -1.5 graph ) 0 0.25 0.5 0.75 1 t / T = fraction of period 30-Sept-2011 Physics 116 - Au11 8

Examples / applications • Metal shelf in Space Shuttle vibrates during takeoff – its outer end moves with amplitude 0.25 mm and vibrates 110 times per second. (see prob. 13-27) • What is max acceleration and max speed of the shelf’s edge? – We are told A=0.25 mm = 0.00025 m, and f = 110 Hz ( ) – So ω = π = = = ฀ 2 f 6.28 radians 110 Hz 1/ sec 691radians/ sec * = ω = = ฀ max v A 0.00025 m 691radians/ sec 0.173m/ sec * ( ) = ω = 2 = 2 2 ฀ max a A 0.00025 m 691radians/ sec 120m/ sec * = = 2 g 9.8m/ sec , so max a 12 ! g – Notice how high frequency vibrations can produce dangerous accelerations, even if amplitude of motion is tiny! – Same A but at f=11 Hz (10 times lower f) would have max a that is 100 times smaller 3-Oct-2011 Physics 116 - Au11 9

Period for SHM • Now that we know a (t), we can find period T without calculus: ( ) = − x − mA ω 2 cos ω t F = ma → − kx = ma → ⎛ ⎞ k ( ) = − ( ) − A ω 2 cos ω t ⎟ A cos ω t ⎜ So ⎝ ⎠ m ⎛ ⎞ ω 2 = k k → ω = ⎜ ⎟ ⎝ ⎠ m m ⎛ ⎞ 1 ω = 2 π f = 2 π • We defined ⎜ ⎟ ⎝ ⎠ T ⎛ ⎞ 1 m T = 2 π → T = 2 π ⎜ ⎟ • So ⎝ ⎠ ω k • This tells us that T increases for larger m, or smaller k 3-Oct-2011 Physics 116 - Au11 10

Examples / applications • 0.46 kg mass on a spring has period T=0.77 s • What is k of spring? Frictionless! ( ) ( ) m m m = π → = π → = π 2 2 2 T 2 T 2 k 2 2 k k T ⎛ ⎞ ( ) 0 . 46 kg kg = = 2 ⎜ ⎟ 6 . 28 31 k ⎝ ⎠ 2 2 ( 0 . 77 s ) s − ⎛ ⎞ kg m = ⎜ ⎟ Note: N ⎝ ⎠ 2 s So units of result are N/m • If spring is hung vertically, how much will the spring stretch to its new rest position? = + = − F spring mg kx ( ) ⎛ ⎞ ⎛ ⎞ 2 0 . 46 9 . 8 / mg kg m s ⎜ ⎟ = − = − ⎜ ⎟ x ⎜ ⎟ ( ) ⎝ ⎠ ⎝ ⎠ k 31 N / m = 0 . 15 m 3-Oct-2011 Physics 116 - Au11 11

Energy considerations in SHM • The spring force is an example of a conservative force: – Total energy (= sum of kinetic and potential energy) will be constant – If we can ignore air resistance and friction within the spring… • Potential energy for a spring is ( ) ( ) 2 kx 2 = 1 U = 1 2 2 kA 2 cos ω t ( ) ( ) • Kinetic energy is, as usual, 2 mv 2 = 1 K = 1 2 2 m − A ω sin ω t ( ) ( ) K = 1 2 2 mA 2 ω 2 sin ω t • So total energy of spring-mass system is E = U + K ( ) + 1 ( ) 2 kA 2 cos 2 ω t 2 mA 2 ω 2 sin 2 ω t = 1 ( ) ( ) + m ω 2 sin 2 ω t ( ) 2 A 2 k cos 2 ω t = 1 3-Oct-2011 Physics 116 - Au11 12

Maximum U and K = maximum E ⎛ ⎞ ω 2 = k • Since ⎜ ⎟ ⎝ ⎠ m ( ) = 1 ( ) ( ) + m ω 2 sin 2 ω t ( ) ( ) + k sin 2 ω t ( ) 2 A 2 k cos 2 ω t 2 A 2 k cos 2 ω t E = U + K = 1 ( ) = 1 ( ) + sin 2 ω t ( ) 2 kA 2 cos 2 ω t E = U + K = 1 • So, the total energy is 2 kA 2 Notice: ½ k A 2 is the maximum potential or kinetic energy • – Energy swaps between K and U as the mass moves from max to minimum displacement K, U and E vs time over 1 cycle 1.2 1 0.8 Energy K=sin^2(wt) U=cos^2(wt) 0.6 E 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 time, in units of T 3-Oct-2011 Physics 116 - Au11 13

Examples / applications • 0.40 kg mass on a vertical spring with k=26 N/m • Stretched to 3.2 cm below its hanging equilibrium position and released • What is its speed as it goes through the equilibrium position? 1 = 2 U kA 2 1 ( )( ) = 2 26 N/m 0 . 032 m 2 ( ) = 0 . 013 N - m ( ) 1 = = 2 K 0 . 013 N - m mv 2 ( ) 2 K 0 . 026 N - m = = = v 0 . 25 m / s m 0 . 4 kg ( ) ⎡ ⎤ 2 N - m m m = = ⎢ ⎥ note : 2 ⎢ ⎥ kg s s ⎣ ⎦ 3-Oct-2011 Physics 116 - Au11 14 14

Pendulum motion as example of SHM • A pendulum oscillates about its rest position IF its max displacement is small, this motion is SHM • SHM occurs only when restoring force is F = -kx • For small angles (you decide what’s small!), θ ≈ θ → θ ≈ θ sin mg sin mg So F is proportional to θ (for larger θ , nonlinear so not exactly SHM) Initial height (max angle) of bob determines E U=mgh (h=initial height relative to rest position) E swaps between K and U as with spring-mass system: K=mgh when bob is at h=0 “Galileo’s pendulum” demo illustrates this… http://paws.kettering.edu/~drussell/Demos/Pendulum/Pendula.html If m of string ~ 0 this is a simple pendulum If we have to worry about m of bob’s support, it’s a physical pendulum 3-Oct-2011 Physics 116 - Au11 15

16 Let’s try a practice clicker quiz (no grade!) Four questions: 1 st three are a poll, 4 th is typical pop quiz for class DON’T CLI CK YET ! Just read the questions, turn on your clicker 1. What year are you? A. Freshman B. Sophomore C. Junior D. Senior, or beyond 3-Oct-2011 Physics 116 - Au11 16

Practice clicker quiz Q. 2 (no grade!) 2. Have you taken a calculus course? A. Yes (or taking calculus now) B. No

Practice clicker quiz Q. 3 (no grade!) 3. Why are you taking 116? A. Required for my major, I wouldn’t be here otherwise! B. Required for my major, but I would take physics anyway C. Not required, but wanted to take physics