Physics 116 ELECTROMAGNETISM AND OSCILLATORY MOTION Lecture 2 SHM - - PowerPoint PPT Presentation

• R. J. Wilkes

Email: ph116@u.washington.edu

Physics 116

ELECTROMAGNETISM AND OSCILLATORY MOTION Lecture 2

SHM and circular motion

Sept 30, 2011

First: I’m not Wilkes!

Your lecturer today: Victor Polinger Substituting for R. J. Wilkes (He will return on Monday)

In case you were not here yesterday:

• PHYS 116 Course home page:

Visit frequently for updated course info! http://faculty.washington.edu/wilkes/116/

• Physics Dept’s general-information page for

100 courses: http://www.phys.washington.edu/1xx/

• Webassign page (for homework and grades):

http://webassign.net/washington/login.html

Announcements

(we’ll have a slide like this one most days) In case you were not here yesterday:

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All HW Assignments and Grades on WebAssign

• Homework solutions are submitted online.

Homework solutions are submitted online.

• No work is accepted past due date.

No work is accepted past due date.

• Click on assignment

Click on assignment name name to begin working to begin working.

• Grades

Grades for homework are stored in lecture section. for homework are stored in lecture section.

NOTE: Lab grades stored separately, under their NOTE: Lab grades stored separately, under their own sections

• wn sections.

.

Other Class Resources

• Your Fellow Students

– You’re encouraged to study and discuss class together!

• Physics Study Center (A-wing Mezzanine, just under
• ur lecture room)

– Meeting place / work space for students – Get guidance and help from TAs outside 116 office hours – Look over other textbooks on the same subjects

• Physics Library (6th floor of C-wing)

– Best view from a comfy chair on campus!

• Class discussion board

– https://catalyst.uw.edu/gopost/board/wilkes/23253/

• Your TA

– Kyle Armour (take a bow, Kyle) – He will post office hours next week – Email questions to him via class email, ph116@uw.edu

In case you were not here yesterday:

“Clickers” are required

iCue / H-ITT Clickers (TX-3100)

• Required to enter answers in quizzes

– Be sure to get radio (RF), not infrared (IR)

• We will not use them this week
• We’ll practice using them next week, and begin using

them for pop quizzes thereafter

– Bring your clicker to class every day from Oct 3 onward

• Why must we torture you with pop quizzes?

– Motivation to attend class (physically and mentally)! – Quizzes are designed to be easy I F you are paying attention

• Questions will be about something we just discussed!

In case you were not here yesterday:

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Today

Lecture Schedule

(up to exam 1)

• Period T = length of one full cycle (time between peaks)
• Frequency f = number of cycles per unit time (units = 1 / time)
• Amplitude A = maximum displacement (length)
• Graph of oscillating object’s position vs time for T= 1 sec, f = 1 Hz, A = 2 m
• 2
• 1

1 2 0.25 0.5 0.75 1 1.25 1.5 1.75 2 Distance, meters

T = 1 sec

A = 2 m f = 1 / T Unit of frequency = 1 cycle/sec = 1 hertz (Hz)

time, seconds

Oscillation terminology

summary from yesterday:

position along the cycle is the phase of the wave: ( t / T ) = (φ / 2π) so φ = 2π ( t / T ) , or φ = 360° ( t / T )

9

• 1. Pulse rate is 70 beats / minute
• What is f in Hz ?

This is just conversion of units: 1 min = 60 sec so f in Hz (cycles/sec) = (cycles/min)*(min/sec) f = { 70 (1/min) }* { 1/60 (min/sec) } = 70/60 (1/sec=Hz) = 1.17 Hz

• What is period T in sec ?

connection between f and T : f = 1 / T So in this case T = 1 / f = 1 / 1.17 Hz (cycles/sec) = 0.85 sec

• 2. Mass on a spring moves with amplitude A and period T
• How long does it take to move a net distance A ?

Definition of A = distance displaced in ¼ of a cycle (1 cycle takes t = T) so it reaches x = A at time t = T/4

Examples / applications

t=T A T/4 A/2 T/12

Note: it does NOT reach A/2 in T/8! Speed is not constant: x = A sin(2p { t / T }) x=A/2 when sin(2p { t / T }) = ½ here t / T = fraction of a full cycle at time t, 2p { t / T } = fraction of 2p = phase of sine curve at t sin-1( ½ ) = 0.52 radians = 0.083*2p So x=A/2 when { t / T }=0.083, or t = 0.083T = T / 12 sin-1 (x) means “angle whose sine is x” (arcsine, or inverse sine)

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• 1.5
• 1
• 0.5

0.5 1 1.5 0.2 0.4 0.6 0.8 1

t / T = fraction of period displacement

1 2 3 4 5 6

phase, 0 - 2π

• Restoring force (spring-mass example): F = - k x
• For Simple Harmonic Motion (SHM), k = constant

– force is proportional to displacement

• Acceleration a = F/m = -(k/m)*x
• Now comes magic (i.e. calculus, beyond our scope): as you know,

– a = rate of change of velocity v (“derivative” of v) – v = rate of change of position x (“derivative” of x) – In the language of calculus, – This is a “differential equation” for x vs time …whose solution (magic!) is Where A=displacement at t=0 And Graph’s lower axis shows time/T Upper axis shows phase

Position vs time in simple harmonic motion

x m k dt x d dt dx dt d m kx ma F ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = → ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − → =

2 2

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = T t A t x π 2 cos ) ( ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ T t π 2 k m T π 2 =

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• Circular motion is periodic
• Coordinate of a point on a circle is given by sin/cos functions
• So x or y coordinate of object moving in a circle has exactly the

same mathematical description as object in SHM

– Plot the x coordinate of a pin on a turntable – and we… – Get the x coordinate of a mass on a spring with same period and A=R – Here’s a web applet that illustrates this: http://www.ngsir.netfirms.com/englishhtm/SpringSHM.htm

SHM’s connection to circular motion

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• We can use the mathematical equivalence of circular motion

and SHM to find useful formulas about SHM

– The angular position of a peg on a turntable is just – here θ is the total angle swept out, and ω is the angular velocity

• ω (omega) is the rate of change of angle with time

(this assumes we accumulate the total number of radians the peg sweeps out, we do not reset to 0 radians whenever it passes its starting point) – We found that the x coordinate of the peg has the same form as in SHM: – Notice:

– So if we define we can write – ω = “angular frequency”

• number of radians per second of rotation, for peg on turntable
• Nothing actually rotating in SHM, but we use ω = 2 π f anyway

Angular velocity in SHM and circular motion

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = T t A t x π 2 cos ) (

t t ω θ = ) (

f T π π 2 1 2 = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛

2 f ω π =

( )

( ) cos x t A t ω =

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• Recall from PHYS 114:

– For uniform (constant ω ) circular motion

• peg’s speed is v = R ω
• x component of v points in – x direction for 0 < θ < π radians
• From triangle shown, we can see vx= - (v sin (θ) )
• so x component of velocity is vx= -Rω sin (ωt)
• So for SHM and circular motion we find:

Velocity in SHM and circular motion

( ) ( )

( ) cos 2 cos ( ) sin t x t A A t T v t A t π ω ω ω ⎛ ⎞ ⎛ ⎞ = = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ = −

t t ω θ = ) (

f T π π 2 1 2 = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛

2 f ω π =

θ

x y

R

θ v vy vx

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• Again recall from PHYS 114:

– For uniform (constant ω ) circular motion

• Centripetal acceleration has magnitude a = v2 / R =(R ω)2/R = R ω2
• Centripetal acceleration points toward center
• x component of a points in – x direction, for 0 < θ < π/2 radians
• From triangle shown, we see x component ax= - a cos (θ)
• so x component of acceleration is ax(t)= R ω2 cos (ωt)
• So for both SHM and circular motion we find:

Acceleration in SHM and circular motion

( ) ( ) ( )

2

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

t t ω θ = ) (

f T π π 2 1 2 = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛

2 f ω π =

θ

x

R

a ay ax

θ

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• 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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