Chapter 15 Oscillations and Waves

1 Chapter 15. Oscillations and Waves Oscillations and Waves Simple Harmonic motion energy in SHM. Some Oscillating Systems Damped Oscillations Driven Oscillations Resonance MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 2. Simple Harmonic motion Simple harmonic motion (SHM). occurs when the restoring force (the force directed toward a stable equilibrium point) is proportional to the displacement from equilibrium. MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 3. Characteristics of SHM. Repetitive motion through a central equilibrium point.

2 Symmetry of maximum displacement. Period of each cycle is constant. Force causing the motion is directed toward the equilibrium point (minus sign). F directly proportional to the displacement from equilibrium. Acceleration = - 2 x Displacement MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 4. A Simple Harmonic Oscillator (SHO). Frictionless surface The restoring force is F = kx. MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 5. Two Springs with Different Amplitudes Frictionless surface MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 6.

3 SHO Period is Independent of the Amplitude MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 7. The Period and the Angular Frequency 2 . The period of oscillation is T= .. where is the angular frequency of k the Oscillations , k is the spring =. m constant and m is the mass of the block. MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 8. Simple Harmonic motion At the equilibrium point x = 0 so, a = 0 also. When the stretch is a maximum, a will be a maximum too. The velocity at the end points will be zero, and it is a maximum at the equilibrium point.

4 MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 9. Representing Simple Harmonic motion MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 10. Representing Simple Harmonic motion MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 11. Representing Simple Harmonic motion Position - xmax = A. Velocity - vmax = A. Acceleration - amax = 2A. MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 12. A simple harmonic oscillator can be described mathematically by: x ( t ) = Acos t dx v (t ) = = -A sin t dt dv a (t ) = = -A 2 cos t dt where A is the amplitude of the motion , the maximum Or by: displacement from equilibrium, A = vmax, and x ( t ) = Asin t A 2 = amax.

5 Dx v (t ) = = A cos t dt dv a (t ) = = -A 2 sin t dt MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 13. Linear motion - Circular Functions MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 14. Projection of Circular motion MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 15. Circular motion is the superposition of two linear SHO that are 900 out of phase with each other y = A sin( t ). x = A cos( t ). MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 16. Shifting Trig Functions sin The minus sign means that the x= A t.

6 Cos phase is shifted to the right. x= A { }. sin t 2 . cos T. - .. A plus sign indicated the phase is shifted to the left x = Asin t - 2 . x = A ( sin t cos 2 - sin 2 cos t ). x = A ( sin t (0) - (1)cos t ). x = -Acos t MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 17. Shifting Trig Functions . sin t - = 0 Shifted Trig Functions 2 sin( t). sin( t- ). t - = 0. 2 . t = 2 1 1 T. t= ; = 2 2 . T T. t= = Time 2 2 4. MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 18. energy MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 19.

7 Equation of motion & energy Assuming the table is frictionless: F x = - kx = ma x k Classic form for SHM a x ( t ) = - x ( t ) = - 2 x ( t ). m 1 2 1 2. Also, E = K ( t ) + U ( t ) = mv ( t ) + kx ( t ). 2 2. MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 20. Spring Potential energy MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 21. Spring Total energy MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 22. Approximating Simple Harmonic motion MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 23.

8 Approximating Simple Harmonic motion MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 24. Potential and Kinetic energy MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 25. The period of oscillation of an object in an ideal mass-spring system is sec and the amplitude is cm. What is the speed at the equilibrium point? At equilibrium x = 0: 1 2 1 2 1 2. E = K + U = mv + kx = mv 2 2 2. Since E = constant, at equilibrium (x = 0) the KE must be a maximum. Here v = vmax = A . MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 26.

9 Example continued: The amplitude A is given, but is not. 2 2 . = = = rads/sec T s and v = A = ( cm )( rads/sec ) = cm/sec MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 27. The diaphragm of a speaker has a mass of g and responds to a signal of kHz by moving back and forth with an amplitude of 10 4 m at that frequency. (a) What is the maximum force acting on the diaphragm? F = Fmax = ma max = m A = (. mA(2 f ) 2. = 4 )2. mAf 2 2. The value is Fmax=1400 N. MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 28.

10 Example continued: (b) What is the mechanical energy of the diaphragm? Since mechanical energy is conserved, E = Kmax = Umax. 1 2 The value of k is unknown so use Kmax. U max = kA. 2. 1 2 1 2 1 1. mvmax = m( A ) = mA2 (2 f ). 2 2. K max = mvmax K max =. 2 2 2 2. The value is Kmax= J. MFMcGraw-PHY 2425 Chap 15Ha- Oscillations -Revised 10/13/2012 29. Example: The displacement of an object in SHM is given by: y (t ) = ( cm )sin [( rads/sec ) t ]. What is the frequency of the Oscillations ? Comparing to y(t) = A sin t gives A = cm and = rads/sec.