Simple Harmonic Motion This type of motion is the most pervasive motion in the universe. All atoms oscillate under harmonic motion. We can model this motion with a linear restoring force.

Examples of Vibrational Motion Mass on a spring Simple Pendulum We will model these systems with a linear restoring force using Hooke’s Law for the mass/spring system. Simple – linear force law Harmonic – restoring force Motion – moving F = - kx

Newton’s Laws F = ma -kx = ma 0 = ma + kx 0 = a + (k/m) x 0 = a +  o 2 x Where  o = √k/m Called the natural frequency.

Period of Motion The natural frequency is related to the linear frequency by f =  o /2π For our purposes, we will most often be measuring the period, T = 1/f = 2π√m/k. This is the time it takes for one repetition.

Equations of Motion Based upon the equation of motion derived from Newton’s Second Law we can define the following relationships: Position, x(t) = A sin(  o t +  ) Velocity, v(t) = A  o cos(  o t +  ) Acceleration, a(t) = -A  o 2 sin(  o t +  ) These functions describe the kinematics of any simple harmonically oscillating system.

Energy of SHM Systems The equations for x(t) and v(t) can be put into our energy expression: E = ½ mv 2 + ½ kx 2 E = ½ m(A  cos(  t+  )) 2 + ½ k(Asin(  t+  )) 2. E = ½ kA 2. The total energy is a constant.

Limits of Vibration Motion Maximum value of x is A. Occurs at extremes of oscillation Maximum value of v is A  o. Occurs at zero point. Maximum value of a is –A  o 2. Occurs at extremes of oscillation.

Example of SHM A 0.1 kg mass on a spring of spring constant 10 N/m is seen to oscillate 10 times in 20 seconds with an amplitude of 0.1 m. What is the natural frequency of the system, the period, and position when t = 1 second? Solution: m  2 = k, T =1/f = 2  / .  2 = k/m = (100 N/m)/0.1 kg = 100/s 2 So,  = 10 rad/s, therefore T = 2  /  rad/s Or T = seconds.

The Simple Pendulum The simple pendulum is a SHM system. Mass, m, attached to a string of length, L. Natural frequency, w = √g/L Period of motion, T = 2π√L/g The angular position of the bob is given by  (t) =  0 sin  t

Pendulum Example A brass ball is used to make a simple pendulum with a period of 5.5 seconds. How long is the cable that connects the ball to the support? Solution: T = 2π√L/g L = T 2 g / 4π 2 L = (5.5 s) 2 (9.8 m/s 2 )/(4(3.14) 2 ) L =7.51 m

Simple Harmonic Motion & Waves The oscillation of a medium that supports a wave is the same as the motion of a mass on a spring. The same words are used to describe waves, such as frequency, period and amplitude. Let’s look more closely at wave motion in the next section.