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Chapter 13

Chapter 13. Periodic Motion. Describing oscillations. The spring drives the glider back and forth on the air-track and you can observe the changes in the free-body diagram as the motion proceeds from –A to A and back. Refer to Example 13.1. Simple harmonic motion.

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Chapter 13

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  1. Chapter 13 Periodic Motion

  2. Describing oscillations • The spring drives the glider back and forth on the air-track and you can observe the changes in the free-body diagram as the motion proceeds from –A to A and back. • Refer to Example 13.1.

  3. Simple harmonic motion • An ideal spring responds to stretch and compression linearly, obeying Hooke’s Law. • For a real spring, • Hookes’ Law is a good • approximation.

  4. Simple harmonic motion viewed as a projection • If you illuminate uniform circular motion (say by shining a flashlight on a candle placed on a rotating lazy-Susan spice rack), the shadow projection that will be cast will be undergoing simple harmonic motion.

  5. X versus t for SHO then simple variations on a theme

  6. Q13.1 An object on the end of a spring is oscillating in simple harmonic motion. If the amplitude of oscillation is doubled, how does this affect the oscillation period T and the object’s maximum speed vmax? A. T and vmax both double. B. T remains the same and vmax doubles. C. T and vmax both remain the same. D. T doubles and vmax remains the same. E. T remains the same and vmax increases by a factor of .

  7. A13.1 An object on the end of a spring is oscillating in simple harmonic motion. If the amplitude of oscillation is doubled, how does this affect the oscillation period T and the object’s maximum speed vmax? A. T and vmax both double. B. T remains the same and vmax doubles. C. T and vmax both remain the same. D. T doubles and vmax remains the same. E. T remains the same and vmax increases by a factor of .

  8. SHM phase, position, velocity, and acceleration • SHM can occur with various phase angles. • For a given phase we can examine position, velocity, and acceleration.

  9. Q13.2 This is an x-t graph for an object in simple harmonic motion. At which of the following times does the object have the most negativevelocityvx? A. t = T/4 B. t = T/2 C. t = 3T/4 D. t = T

  10. A13.2 This is an x-t graph for an object in simple harmonic motion. At which of the following times does the object have the most negativevelocityvx? A. t = T/4 B. t = T/2 C. t = 3T/4 D. t = T

  11. Watch variables change for a glider example • As the glider undergoes SHM, you can track changes in velocity and acceleration as the position changes between the classical turning points. • Refer to Problem-Solving Strategy 13.1 and Example 13.3.

  12. Q13.3 This is an x-t graph for an object in simple harmonic motion. At which of the following times does the object have the most negativeaccelerationax? A. t = T/4 B. t = T/2 C. t = 3T/4 D. t = T

  13. A13.3 This is an x-t graph for an object in simple harmonic motion. At which of the following times does the object have the most negativeaccelerationax? A. t = T/4 B. t = T/2 C. t = 3T/4 D. t = T

  14. Q13.5 This is an ax-t graph for an object in simple harmonic motion. At which of the following times does the object have the most negativevelocityvx? A. t = 0.10 s B. t = 0.15 s C. t = 0.20 s D. t = 0.25 s

  15. A13.5 This is an ax-t graph for an object in simple harmonic motion. At which of the following times does the object have the most negativevelocityvx? A. t = 0.10 s B. t = 0.15 s C. t = 0.20 s D. t = 0.25 s

  16. Energy in SHM • Energy is conserved during SHM and the forms (potential and kinetic) interconvert as the position of the object in motion changes.

  17. Energy in SHM II • Figure 13.15 shows the interconversion of kinetic and potential energy with an energy versus position graphic. • Refer to Problem-Solving Strategy 13.2. • Follow Example 13.4.

  18. Q13.7 This is an x-t graph for an object connected to a spring and moving in simple harmonic motion. At which of the following times is the kinetic energy of the object the greatest? A. t = T/8 B. t = T/4 C. t = 3T/8 D. t = T/2 E. more than one of the above

  19. A13.7 This is an x-t graph for an object connected to a spring and moving in simple harmonic motion. At which of the following times is the kinetic energy of the object the greatest? A. t = T/8 B. t = T/4 C. t = 3T/8 D. t = T/2 E. more than one of the above

  20. Q13.8 To double the total energy of a mass-spring system oscillating in simple harmonic motion, the amplitude must increase by a factor of A. 4. B. C. 2. D. E.

  21. A13.8 To double the total energy of a mass-spring system oscillating in simple harmonic motion, the amplitude must increase by a factor of A. 4. B. C. 2. D. E.

  22. Find velocity • 1) What is the velocity as a function of the position v(x) for a SHO glider with mass m and spring constant k? • Use conservation of energy • 2) What is the maximum velocity of the glider?

  23. Vibrations of molecules • Two atoms separated by their internuclear distance r can be pondered as two balls on a spring. The potential energy of such a model is constructed many different ways. The Leonard–Jones potential shown as Equation 13.25 is sketched in Figure 13.20 below. The atoms on a molecule vibrate as shown in Example 13.7.

  24. Old car • The shock absorbers in my 1989 Mazda with mass 1000 kg are completely worn out (true). When a 980-N person climbs slowly into the car, the car sinks 2.8 cm. When the car with the person aboard hits a bump, the car starts oscillating in SHM. Find the period and frequency of oscillation. • How big of a bump (amplitude of oscillation) before you fly up out of your seat?

  25. Damped oscillations • A person may not wish for the object they study to remain in SHM. Consider shock absorbers and your automobile. Without damping the oscillation, hitting a pothole would set your car into SHM on the springs that support it.

  26. Damped oscillations II

  27. Forced (driven) oscillations and resonance • A force applied “in synch” with a motion already in progress will resonate and add energy to the oscillation (refer to Figure 13.28). • A singer can shatter a glass with a pure tone in tune with the natural “ring” of a thin wine glass.

  28. Forced (driven) oscillations and resonance II • The Tacoma Narrows Bridge suffered spectacular structural • failure after absorbing too much resonant energy (refer to Figure • 13.29).

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