T m. Fapplied. Thur Oct 29. ω = 2πf f = (ω/2π) T = 1/f. k m. ω =

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Claude Richards
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1 Thur Oct 9 Assignent 10 Mass-Spring Kineatics (x, v, a, t) Dynaics (F,, a) Tie dependence Energy Pendulu Daping and Resonances x Acos( ωt) = v = Aω sin( ωt) a = Aω cos( ωt) ω = spring k f spring = 1 k T π spring = π k Fapplied = kx ω = πf f = (ω/π) T = 1/f

2 f, T, ω, A: Frequency f nuber of cycles per second (1/s) or Hz Period T tie for one coplete cycle (s) T = 1/f Angular frequency also angular speed πf (rad/s) Aplitude axiu distance fro equilibriu () f = # cycles tie T = # tie cycles

3 A ass is hung fro a spring and set into oscillation. It oscillates with a given frequency f 1. Now a second identical spring is also attached to the ass (sae k, sae length). How does the new frequency copare to the old? 1)The new frequency is double the old )The new frequency is sqrt() ties the old 3)The new frequency is the sae as the old 4)The new frequency is 1/sqrt() ties the old 5)The new frequency is half the old Double force with two springs. As if single spring of double strength. (k eff = k) Stiffer spring, greater frequency. f 1 π k 1 k 1 k f = π π = = 1 = ( ) 1 f

4 Which of the following will increase the period of a ass spring syste? 1. Increasing the ass. Increasing the stiffness of the spring 3. Increasing the aplitude 4. (1) and () 5. (1) and (3) 6. () and (3) 7. All three T spring = π k

5 Mass-Spring Syste - Horizontal Siulation Horizontal ass and spring Mass and Spring PRS 1 Mass and Spring PRS Full Siulation Will place links on LON-CAPA

6 Mass and Spring Syste Force greatest at extrees where spring stretched or copressed the ost (zero at equilibriu) Note: Aplitude is the greatest displaceent of the object fro equilibriu, (ie. the Extrees ) F=a or a = F/, so where the force is biggest, the acceleration is biggest. Object speeds up as it heads towards equilibriu (F and a in direction of travel) As it goes through equilibriu, F and a change direction, so starts to slow down. Max v at equilibriu, zero v at extrees.

7 The graph represents displaceent of a ass in a ass-spring syste. At what point(s) is the agnitude of the force at a axiu? (1) A () C (3) E (4) G (5) I (6) A and I (7) A, E, and I (8) C and G (9) A,C,G, and I The agnitude of the force is greatest at the extrees. axiu acceleration at what point(s)? A, E, and I iniu force & acceleration at what point(s)? C and G iniu (zero) velocity at what point(s)? A, E, and I axiu velocity at what point(s)? C and G

8 Describe Teporal (Tie) Behavior Copare to Circular otion Frequency (1/s) or Hz Period (s) = T = 1/f x(t),v(t), a(t) θ = ωt x = Acos( ωt) Coparison of Circular Motion and Haronic Motion loncapa.phy.ohiou.edu/res/ohiou/physlets/sh/sh_circle_1.htl

9 Position as a function of tie x = Acos( ωt) A = Aplitude ω = πf USE RADIANS ode in Calculator ω is angular speed or angular frequency

10 Velocity as function of tie Acceleration as function of tie v = ωa sin( ωt) a = ω A cos( ωt)

11 Spring-Mass Review forulas! Sect 10. ω = spring k spring Note: T, f, ω are independent of Aplitude! f = 1 k T π spring = π ω (lowercase greek oega) is angular frequency or angular speed k x = Acos( ωt) xax = A v = Aω sin( ωt) sin and cos range (-1 to 1) vax = Aω a = Aω cos( ωt) aax = Aω Suggest aking list of equations while reading for reference in class

acquisition DataStudio Part of lab the physics Part

12 Lab Next Week Investigate Hooke s Law and Siple Haronic Motion Will be using coputer-based acquisition DataStudio Part of lab the physics Part of the lab learning advantages and disadvantages of coputers in lab

13 Position, Velocity, and Acceleration vs. Tie B D E C x id = ( )/ = 0.535

14 The graph represents velocity of a ass in a ass-spring syste. At what point(s) is the agnitude of the force at a axiu? (1) A () C (3) E (4) G (5) I (6) A and I (7) A, E, and I (8) C and G (9) A, C, G, I The agnitude of the force is greatest at the extrees. The velocity is 0 at the extrees.

15 Frequency and Period Which of the following are true about a ass spring syste? (1) Increasing the ass increases the period () A stiffer spring increases the frequency (3) Decreasing the ass increases the frequency (4) All of the above f spring = 1 k T π spring = π k

16 A ass attached to a spring is sliding back and forth on a horizontal frictionless table. The kinetic energy is a axiu at: 1. When the spring is copressed the ost. When the spring is stretched the ost 3. At equilibriu 4. At both extrees KE = ½ v Largest KE at greatest v

17 Energy Can store energy in spring PEspring =! Sect kx Work done on spring = Δ PE Total Mech Energy Constant if know Energy at any one point in tie, know energy at all tie oves between KE and PE Can use to find v as function of x Siulation Horizontal ass and spring sh_ass_spring_1.htl

18 Total Mechanical Energy Total Mechanical Energy: Officially 1 1 E = v + Iω + gh + 1 kx If horizontal ass and spring can siplify E = v + Not rotating, no change in height 1 1 kx

19 Velocity vs Position for SHM PE + KE = PE( x) + KE( x) INITIAL INITIAL Choose initial at extree: x = A 1 The final result is: 1 ka + 0 = kx + 1 v k v A x ( ) Max when x=0 = ±

20 The graph represents velocity of a ass in a assspring syste. At what point(s) is the PE due to the spring at a axiu? (1) A () C (3) E (4) G (5) I (6) A and I (7) A, E, and I (8) C and G (9) A,C,G, and I PE greatest when spring stretched or copressed the ost. The spring is stretched or copressed ost at the extrees. The velocity is 0 at the extrees.

21 Soe coents: 1 k Careful with this calculation. Soe people accidentally change it into either on paper or in their calculator π ( 1/ ) π k / Total echanical Energy = KE + PE. If want E TOT then pick a given tie and find both KE and PE. Hint: it s easier if one of these is zero. What is sign of KE if v is positive? negative? What is sign of elastic PE if x is positive? negative? Lots of equations Write down in one place Try grouping Make sure you understand liits or conditions

22 Pendulu Siple pendulu: ost of ass concentrated at one point Physical pendulu: ass spread around Periodic otion Describe v, KE, PE, a What does period depend on? What is the 'restoring force'?

23 Pendulu Forces F NET = - g sinθ Restoring? Linear? SHM? If θ sall ( < 15º or 0.6rad), sin(θ(rad)) θ(rad) θ (rad) = s/l s = arc length dist fro equilibriu F -g θ F -(g/l) s For sall angles, pendulu undergoes SHM F -k s ; k=(g/l)

24 Pendulu- Mass/Spring Coparison Mass-Spring Pendulu "stiffness" k (g/l) inertia ang freq ω k g L = g L f = (1/π)ω T = 1/f 1 π π k k Clock "counts ticks" - Hands ove fixed aount each cycle 1 π π L g g L Both quantities are Mass Independent & Aplitude Independent!

25 Two identical pendulu clocks are set in otion, one on the earth and one on the oon. The acceleration due to gravity on the Moon is 1.67 /s (1/6 that of Earth). 1)The clock on the Moon runs faster than the clock on Earth )The clock on the Moon runs the sae speed as the clock on Earth 3)The clock on the Moon runs slower than the clock on Earth g saller. Longer period and lower frequency. Fewer ticks in sae aount of tie. Ticks related to frequency.

26 In order to get the clock on the Moon to give proper earth tie you would: 1. Shorten the pendulu. Keep the pendulu the sae length 3. Lengthen the pendulu Want the sae period or frequency. Cutting length by 1/6 and g by 1/6 gives sae period. f = 1 π g L = 1 π g L

Resonance Lots of systes have natural or

atos, usical instruents, car on road, car

27 Resonance Lots of systes have natural or resonant frequencies If drive at that frequency, energy builds up Swing, rope, tuning forks, airplane wings, buildings, atos, usical instruents, car on road, car itself Taipei ton Steel Pendulu daper Reduce Oscillations

Resonance Lots of systes have a natural frequency at which

Tacoa Narrows Bridge Collapse 1940 Fro University of

28 Resonance Lots of systes have a natural frequency at which they oscillate. If driven at that frequency, will build up energy. Can be good usical instruents or bad bridges, skyscrapers Tacoa Narrows Bridge Collapse 1940 Fro University of Washington Library Special online exhibit

29 A car rests on springs and has a certain resonant frequency. If a driver picks up passengers, how will the resonant frequency be affected? 1. The resonant frequency will increase with passengers in the car.. The resonant frequency will stay the sae. 3. The resonant frequency will decrease with passengers in the car. Greater ass, ore sluggish, lower frequency.

30 Daped Motion Energy pulled away fro oscillation friction, air resistance,... Aplitude decreases over tie. (FLASH)! Sect 10.5 Daped Daped even ore Big-tie Daped Underdaped Underdaped Overdaped Passes through equilibriu at least once Does not pass through equilibriu Critically daped least aount of daping where syste does not oscillate. (Return to equilibriu as quickly as possible

WileyPLUS Assignment 3. Next Week

WileyPLUS Assignment 3. Next Week WileyPLUS Assignent 3 Chapters 6 & 7 Due Wednesday, Noveber 11 at 11 p Next Wee No labs of tutorials Reebrance Day holiday on Wednesday (no classes) 24 Displaceent, x Mass on a spring ωt = 2π x = A cos