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Lecture 39

Lecture 39. Room Frequency BA. Clicker Question. What is the sign of cos(225 o )? Sign of sin(225 o )? Don’t use a calculator! A) cos(225 o ) = (+) , sin(225 o ) = (–) B) cos(225 o ) = (–) , sin(225 o ) = (+) C) cos(225 o ) = (+) , sin(225 o ) = (+)

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Lecture 39

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  1. Lecture 39

  2. Room Frequency BA Clicker Question What is the sign of cos(225o)? Sign of sin(225o)? Don’t use a calculator! A) cos(225o) = (+) , sin(225o) = (–) B) cos(225o) = (–) , sin(225o) = (+) C) cos(225o) = (+) , sin(225o) = (+) D) cos(225o) = (–) , sin(225o) = (–) E) None of these. One of them is zero

  3. Announcements • CAPA assignment #14 is due on Friday at 10 pm. • This week in Section: Lab #6 (with prelab) • Read Chapter 12 on Sound

  4. Horizontal Spring and Mass Oscillation x = 0, v = ±vmax x = A, v = 0 x = -A, v = 0 Amplitude A = E At turning points x = ±A, v = 0, ET = only PE= At equilibrium point x = 0, v =±vmax, ET = only KE=

  5. vm A θ x –A +A 0 Harmonic Time Dependence of SHM SHM is mathematically the same as one component of circular motion at constant speed vm, with ω is constant and θ = ωt. For horizontal mass m oscillating with spring, spring constant k x = A cosωt at t = 0 at t = 0 x = A sin ωt

  6. Room Frequency BA Clicker Question 270o = (3/2)π. What is cos[(3/2) π] and what is sin[(3/2) π] ? A) cos[(3/2) π] = 1, sin[(3/2) π] = 0 B) cos[(3/2) π] = 0, sin[(3/2) π] = 1 C) cos[(3/2) π] = 1, sin[(3/2) π] = 1 D) cos[(3/2) π] = 0, sin[(3/2) π] = 0 E) None of these cos[(3/2) π] = 0, sin[(3/2) π] = -1

  7. Simple Harmonic Oscillator with x = 0 at t = 0

  8. Room Frequency BA Clicker Question The position of a mass on a spring as a function of time is shown below. When the mass is at point P on the graph A) The velocity v > 0 B) v < 0 C) v = 0 v is the slope at P

  9. Room Frequency BA Clicker Question The position of a mass on a spring as a function of time is shown below. When the mass is at point P on the graph A) The acceleration a > 0 B) a < 0 C) a = 0 As the mass approaches its extreme position, it is slowing down (velocity positive but decreasing) so the acceleration must be negative

  10. Vertical Spring and Mass Oscillation y Spring equilibrium without gravity y=0 Spring Force Spring equilibrium with gravity y = yE Gravity New spring equilibrium length where –mg - kyE = 0 yE = -mg/k Oscillation frequency is NOT changed! ω2 = k/m

  11. Vertical Spring and Mass Oscillation y Spring equilibrium without gravity y=0 Spring equilibrium with gravity y = yE Energy Still Conserved! Now ET has gravity PE term: With a little algebra you can rewrite this as

  12. Simple Pendulum Oscillation T = mg cosθ θ L Fnet = -mg sin θ m θ mg s =θL There is a net force back towards the vertical equilibrium position! This gives oscillation, but is it SHM? No! For pure SHM we would need Fnet = -mg θ BUT! For small θ, sin θ ≈ θ = s/L, so we get matan = Fnet ≈ -mgs/L

  13. Simple Pendulum Oscillation for Small Angles We found matan ≈ -mgs/L for small θ Cancelling m gives atan ≈ -(g/L)s For horizontal spring we had ax = -(k/m)x Use SHM formulas with g/L in place of k/m !!! • Frequency independent of amplitude θ0 • Frequency independent of mass m

  14. Room Frequency BA Clicker Question Will a given pendulum have a shorter or longer or equal period on the moon compared to the period on earth? A) Equal periods B) Shorter on Moon C) Longer on Moon “g” is smaller on the moon so T is longer

  15. The Physical Pendulum Any object suspended from any point in the object except the center-of-mass will swing back and forth! This is called a physical pendulum, as opposed to a simple pendulum. “L” is now distance from pivot to Center of Mass Now, changing distribution of mass will change period, frequency, if the Center of Mass is changed

  16. Waves are Everywhere!!! Whenever you have a bunch of stuff or many things which can interact with each other, you can get waves.

  17. Wave Simulation A great simulation to learn about one-dimensional waves can be found athttp://phet.colorado.edu/en/simulation/wave-on-a-string

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