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 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 x +A t -A 17 Potential Energy in Spring Force of spring is Conservative F = -k x W = -1/2 k x2 Force work x Work done only depends on initial and final position Define Potential Energy PEspring = ½ k x2 20 ***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 PE 2 Energy = ½ k A + 0 S At zero displacement x = 0 Energy = 0 + ½ mvm2 Since Total Energy is same 0 2 2 ½ k A = ½ m vm m vm = sqrt(k/m) A x=0 x x 25 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 x +A t -A 29 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 EndPoint Equilibrium EndPoint EndPoint Equilibrium EndPoint EndPoint Equilibrium EndPoint 2 3 38 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 What does moving in a circle have to do with moving back & forth in a straight line ?? x = R cos = R cos (wt) since = w t x x 1 1 2 R 3 R 8 2 8 7 3 y 7 4 6 5 0 -R 2 4 3 2 6 5 34 SHM and Circles Simple Harmonic Motion: x(t) = [A]cos(wt) v(t) = -[Aw]sin(wt) a(t) = -[Aw2]cos(wt) x(t) = [A]sin(wt) OR v(t) = [Aw]cos(wt) a(t) = -[Aw2]sin(wt) xmax = A Period = T (seconds per cycle) vmax = Aw Frequency = f = 1/T (cycles per second) amax = Aw2 Angular frequency = w = 2f = 2/T For spring: w2 = k/m 36 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 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 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 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 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 sqrt(L/g) Period does not depend on A, or m! L T x m mg 37 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? 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 g w L T 2 L 2 w g 38 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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