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Wednesday, December 2, 1998

Lecture 36. Chapter 13: Simple Harmonic Motion velocity vs position circular motion periods of springs. Wednesday, December 2, 1998. Physics 111. Free Food!. Other Notes. There will be two more meetings of Lab: - Thursday, December 3 (Lab 10).

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Wednesday, December 2, 1998

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  1. Lecture 36 Chapter 13: Simple Harmonic Motion velocity vs position circular motion periods of springs Wednesday, December 2, 1998 Physics 111

  2. Free Food! Other Notes There will be two more meetings of Lab: - Thursday, December 3 (Lab 10) - Thursday, December 10 (Lab Final=Party) MLK Center Pizza & Beverages provided Orders taken in lab tomorrow 5:00 - 7:00 pm

  3. h Now, sketch a plot of the height of the block above the floor as a function of time. What kind of mathematical functions (with which you’re familiar) result in such a pattern?

  4. Equilibrium position Amplitude Amplitude period

  5. This type of oscillatory behavior is known as Simple Harmonic Motion An object in simple harmonic motion displays an acceleration that is proportional to the displacement and in the opposite direction.

  6. h Now, sketch a plot of the velocity of the block as a function of time as it goes through its oscillating motion. How is this plot related to that for the height?

  7. Height vs Time Velocity vs Time The wave for the height is 1/4 period behind the wave for the velocity!

  8. h Using your sketch of velocity vs time, try to sketch the acceleration of the block as a function of time. How is this plot related to the velocity plot?

  9. Velocity vs Time Acceleration vs Time The wave for the velocity is 1/4 period behind the wave for the velocity!

  10. Concept Quiz! Springs: x, v, and a

  11. q0 q1 Okay, so we’ve seen that objects in simple harmonic motion can be described by trigonometric functions (sines and cosines). We know that these trigonometric functions complete one cycle (peak-to-peak) over what angular displacement? That is to say, what is q1-q0? 2p

  12. 3 s t = 0 s t = 3 s So what if we observe our spring to oscillate with a period of 3 seconds. How would we write our function for the height of our block versus time? Take a stab at it!

  13. 3 s t = 0 s t = 3 s What is the Period of this wave? 3 seconds What is the Frequency of this wave? 1/3 s

  14. Where have we seen this quantity before? 2pf Angular Frequency!!! Circular motion! w = 2pf Now…dust off the cobwebs…

  15. What does a spring have to do with circular motion? Now…you’ve got to be thinking to yourself... Well…I’m gonna tell ya!

  16. What if I were now to shine a spotlight on this system from the right side and look at the motion of the shadow on a wall to the left? Here’s my tennis ball on a string again. We’re looking down on the plane of motion.

  17. L I G H T! What will a graph of the height of the shadow on the wall look like? Just like the position vs time for the mass on a spring!

  18. Height of Shadow vs Time

  19. So, the trigonometric functions describe BOTH circular motion (as we saw in Chapter 7) and the motion of a mass on a spring! WOW! On what quantities did the period of our tennis ball moving in a circle at the end of a string depend? its speed and the radius of the circle.

  20. ! r If we look at the motion of the shadow, what will be the amplitude of the oscillation? So we might surmise that the period for our mass-spring system will also be related to the amplitude of the motion! Where A represents the amplitude of the oscillation.

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