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

• R. J. Wilkes

Email: ph116@u.washington.edu

Physics 116

ELECTROMAGNETISM AND OSCILLATORY MOTION Lecture 3

Energy in SHM

Oct 3, 2011

• 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

Announcements

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• 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…?

Announcements

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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

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“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!

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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,

• r 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

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Today

Lecture Schedule

(up to exam 1)

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8

• Maximum values occur when sin/cos = 1, so
• 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)

Phase relationships and max values

2

max max max x A v A a A ω ω = = = ( ) ( ) ( )

2

( ) cos ( ) sin ( ) cos x t A t v t A t a t A t ω ω ω ω ω = = − = −

• 1.5
• 1
• 0.5

0.5 1 1.5 0.25 0.5 0.75 1

t / T = fraction of period x, v, a

x=Acos(wt) v= - Awsin(wt) a= - Aw^2cos(wt)

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Last time: ( v and a have been scaled to fit in this graph )

9

• 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 – 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

Examples / applications

( )

2 6.28 radians 110 1/ sec 691radians/ sec f Hz ω π = = = = ฀

( )

2 2 2 2

max 0.00025 691radians/ sec 0.173m/ sec max 0.00025 691radians/ sec 120m/ sec 9.8m/ sec , max 12 ! v A m a A m g so a g ω ω = = = = = = = = ฀ ฀

* * *

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• Now that we know a(t), we can find period T without calculus:
• We defined
• So
• This tells us that T increases for larger m, or smaller k

Period for SHM

F = ma → − kx = ma → − mAω 2 cos ωt

( )= −x

So − Aω 2 cos ωt

( )= −

k m ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Acos ωt

( )

ω 2 = k m ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ → ω = k m

ω = 2π f = 2π 1 T ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ T = 2π 1 ω ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ → T = 2π m k

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11

• 0.46 kg mass on a spring has period T=0.77 s
• What is k of spring?

Note: So units of result are N/m

• If spring is hung vertically, how much will the spring stretch to

its new rest position?

Examples / applications

( ) ( ) ( )

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = = = → = → =

2 2 2 2 2 2 2 2

31 ) 77 . ( 46 . 28 . 6 2 2 2 s m kg N s kg s kg k T m k k m T k m T π π π

( )

( )

m m N s m kg k mg x kx mg Fspring 15 . / 31 / 8 . 9 46 .

2

= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = − = + =

3-Oct-2011 Physics 116 - Au11 Frictionless!

12

• 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
• Kinetic energy is, as usual,
• So total energy of spring-mass system is

Energy considerations in SHM

U = 1 2 kx2 = 1 2 kA2 cos ωt

( )

( )

2

K = 1 2 mv2 = 1 2 m − Aω sin ωt

( )

( )

2

K = 1 2 mA2ω 2 sin ωt

( )

( )

2

E = U + K = 1 2 kA2 cos2 ωt

( )+ 1

2 mA2ω 2 sin2 ωt

( )

= 1 2 A2 k cos2 ωt

( )+ mω 2 sin2 ωt ( )

( )

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K, U and E vs time over 1 cycle

0.2 0.4 0.6 0.8 1 1.2

0.2 0.4 0.6 0.8 1

time, in units of T

Energy

K=sin^2(wt) U=cos^2(wt) E

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• Since
• So, the total energy is
• Notice: ½ k A2 is the maximum potential or kinetic energy

– Energy swaps between K and U as the mass moves from max to minimum displacement

Maximum U and K = maximum E

ω 2 = k m ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ E = U + K = 1 2 A2 k cos2 ωt

( )+ mω 2 sin2 ωt ( )

( )= 1

2 A2 k cos2 ωt

( )+ k sin2 ωt ( )

( )

E = U + K = 1 2 kA2 cos2 ωt

( )+ sin2 ωt ( )

( )= 1

2 kA2

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• 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?

Examples / applications

( )( ) ( ) ( ) ( ) ( )

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = = = = = = = = = = s s kg note s m kg m K v mv K m kA U m m m

• N

: / 25 . 4 . m

• N

026 . 2 2 1 m

• N

013 . m

• N

013 . 032 . N/m 26 2 1 2 1

2 2 2 2 2

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• 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!),

So F is proportional to θ (for larger θ, nonlinear so not exactly SHM)

Pendulum motion as example of SHM

http://paws.kettering.edu/~drussell/Demos/Pendulum/Pendula.html

θ θ θ θ mg mg ≈ → ≈ sin sin

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…

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

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Let’s try a practice clicker quiz (no grade!)

Four questions: 1st three are a poll, 4th 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

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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
• therwise!
• B. Required for my major, but I would take physics

anyway

• C. Not required, but wanted to take physics

Practice clicker quiz Q. 4 (no grade!)

• 4. “Simple Harmonic Motion” refers to
• A. Motion in a circle
• B. Any kind of back-and-forth motion
• C. Motion due to a special kind of restoring

force: F = - kx

• D. A variety of folk dance

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