Simple Harmonic Motion 1

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 +A t -A x 17

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

***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 x PES m x x=0 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 +A t -A x 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 2 3 EndPoint Equilibrium EndPoint EndPoint Equilibrium EndPoint EndPoint Equilibrium EndPoint 38

Springs and Simple Harmonic Motion X=0 X=A X=-A 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 32

What does moving in a circle have to do with moving back & forth in a straight line ?? x = R cos q = R cos (wt) since q = w t y x -R R  1 2 3 4 5 6 x 8 8 q R 7 7 34

SHM and Circles

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 xmax = A vmax = A amax = A2 Period = T (seconds per cycle) Frequency = f = 1/T (cycles per second) Angular frequency =  = 2f = 2/T For spring: 2 = 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) Example: Given m,k and A and x(0), choose x(t), v(t), a(t), Calculate v_max and a_max, period, energy, K, U 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? Example: Given m,k and A and x(0), choose x(t), v(t), a(t), Calculate v_max and a_max, period, energy, K, U 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? Example: Given m,k and A and x(0), choose x(t), v(t), a(t), Calculate v_max and a_max, period, energy, K, U 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? Example: Given m,k and A and x(0), choose x(t), v(t), a(t), Calculate v_max and a_max, period, energy, K, U 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 p sqrt(L/g) Period does not depend on A, or m! L T m x 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 38

Summary Simple Harmonic Motion Springs Pendulum (Small oscillations) 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