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Simple Harmonic Motion

Simple Harmonic Motion. Simple Harmonic Motion. Vibrations Vocal cords when singing/speaking String/rubber band Simple Harmonic Motion Restoring force proportional to displacement Springs F = -kx. 11. x. +A. t. -A. Question II.

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Simple Harmonic Motion

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  1. Simple Harmonic Motion

  2. Simple Harmonic Motion • Vibrations • Vocal cords when singing/speaking • String/rubber band • Simple Harmonic Motion • Restoring force proportional to displacement • Springs F = -kx 11

  3. x +A t -A Question II A mass on a spring oscillates back & forth with simple harmonic motion of amplitude A. A plot of displacement (x) versus time (t) is shown below. At what points during its oscillation is the magnitude of the acceleration of the block biggest? 1. When x = +A or -A (i.e. maximum displacement) 2. When x = 0 (i.e. zero displacement) 3. The acceleration of the mass is constant 17

  4. Potential Energy in Spring • Force of spring is Conservative • F = -k x • W = -1/2 k x2 • Work done only depends on initial and final position • Define Potential Energy PEspring = ½ k x2 Force work x 20

  5. m x x=0 PES x 0 ***Energy in SHM*** • A mass is attached to a spring and set to motion. The maximum displacement is x=A • Energy = PE + KE = constant! = ½ k x2 + ½ m v2 • At maximum displacement x=A, v = 0 Energy = ½ k A2 + 0 • At zero displacement x = 0 Energy = 0 + ½ mvm2 Since Total Energy is same ½ k A2 = ½ m vm2 vm = sqrt(k/m) A 25

  6. x +A t -A Question 3 A mass on a spring oscillates back & forth with simple harmonic motion of amplitude A. A plot of displacement (x) versus time (t) is shown below. At what points during its oscillation is the speed of the block biggest? 1. When x = +A or -A (i.e. maximum displacement) 2. When x = 0 (i.e. zero displacement) 3. The speed of the mass is constant 29

  7. Question 4 • A spring oscillates back and forth on a frictionless horizontal surface. A camera takes pictures of the position every 1/10th of a second. Which plot best shows the positions of the mass. 1 2 3 EndPoint Equilibrium EndPoint EndPoint Equilibrium EndPoint EndPoint Equilibrium EndPoint 38

  8. Springs and Simple Harmonic Motion X=0 X=A; v=0; a=-amax X=0; v=-vmax; a=0 X=-A; v=0; a=amax X=0; v=vmax; a=0 X=A; v=0; a=-amax X=-A X=A 32

  9. x = R cos q =R cos (wt) sinceq = wt x 1 1 R 2 2 3 3  0 y 4 6 -R 4 6 5 5 What does moving in a circlehave to do with moving back & forth in a straight line ?? x 8 8 q R 7 7 34

  10. SHM and Circles

  11. Simple Harmonic Motion: x(t) = [A]cos(t) v(t) = -[A]sin(t) a(t) = -[A2]cos(t) x(t) = [A]sin(t) v(t) = [A]cos(t) a(t) = -[A2]sin(t) OR Period = T (seconds per cycle) Frequency = f = 1/T (cycles per second) Angular frequency =  = 2f = 2/T For spring: 2 = k/m xmax = A vmax = A amax = A2 36

  12. Example A 3 kg mass is attached to a spring (k=24 N/m). It is stretched 5 cm. At time t=0 it is released and oscillates. Which equation describes the position as a function of time x(t) = A) 5 sin(wt) B) 5 cos(wt) C) 24 sin(wt) D) 24 cos(wt) E) -24 cos(wt) 39

  13. Example A 3 kg mass is attached to a spring (k=24 N/m). It is stretched 5 cm. At time t=0 it is released and oscillates. What is the total energy of the block spring system? 43

  14. Example A 3 kg mass is attached to a spring (k=24 N/m). It is stretched 5 cm. At time t=0 it is released and oscillates. What is the maximum speed of the block? 46

  15. Example A 3 kg mass is attached to a spring (k=24 N/m). It is stretched 5 cm. At time t=0 it is released and oscillates. How long does it take for the block to return to x=+5cm? 49

  16. Pendulum Motion • For small angles • T = mg • Tx = -mg (x/L) Note: F proportional to x! • S Fx = m ax -mg (x/L) = m ax ax = -(g/L) x • Recall for SHO a = -w2 x • w = sqrt(g/L) • T = 2 p sqrt(L/g) • Period does not depend on A, or m! L T m x mg 37

  17. Example: Clock • If we want to make a grandfather clock so that the pendulum makes one complete cycle each sec, how long should the pendulum be?

  18. Question 1 Suppose a grandfather clock (a simple pendulum) runs slow. In order to make it run on time you should: 1. Make the pendulum shorter 2. Make the pendulum longer 38

  19. Summary • Simple Harmonic Motion • Occurs when have linear restoring force F= -kx • x(t) = [A] cos(wt) • v(t) = -[Aw] sin(wt) • a(t) = -[Aw2] cos(wt) • Springs • F = -kx • U = ½ k x2 • w = sqrt(k/m) • Pendulum (Small oscillations) • w = sqrt(L/g) 50

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