Physics 123A - Lecture 11 Oscillatory Motion An oscillator is an object or system of objects that undergoes periodic oscillatory motion or behavior. Example: a mass and spring system. Characteristics: 1.Oscillatory motion is about some (energy=0) equilibrium position; 2.Oscillatory motion is periodic, with a definite period or cycle time. Simple Harmonic Motion September 30, 2009

Physics 123A - Lecture 12 Example: Frequency and Period of a Radio Station What is the oscillation period for the broadcast of a 100 MHz FM radio station? September 30, 2009

Physics 123A - Lecture 13 Simple Harmonic Motion (SHM) Prototype Experiment Consider Fig. (a). An air-track glider attached to a spring. The glider is pulled a distance A from its rest position and released. Fig. (b) shows a graph of the motion of the glider, as measured each 1/20 of a second. The graphs on the right show the position and velocity of the glider from the same measurements. We see that A =0.17 m and T =1.60 s. Therefore the oscillation frequency of the system is f = Hz September 30, 2009

Physics 123A - Lecture 14 Kinematics of SHM September 30, 2009

Physics 123A - Lecture 15 Example: A System in SHM An air-track glider is attached to a spring, pulled 20 cm to the right, and released at t=0. It makes 15 oscillations in 10 s. (a) What is the period T of oscillation? (b) What is the object’s maximum speed? (c) What is the position and velocity at t=0.8 s? September 30, 2009

Physics 123A - Lecture 16 Example: Finding the Time A mass on a spring oscillating in simple harmonic motion (SHM) starts at x=A at t=0 and has period T. At what time, as a fraction of T, does the object first pass through x = ½A ? 60 0 =  /3 2 1 √3√3 September 30, 2009

Physics 123A - Lecture 17 SHM and Circular Motion Uniform circular motion projected into one dimension is simple harmonic motion (SHM). Consider a particle rotating ccw, with the angle  increasing linearly with time: September 30, 2009

Physics 123A - Lecture 18 The Phase Constant But what if  is not zero at t =0? A phase constant  0 means that the rotation starts at a different point on the circle, implying different initial conditions. September 30, 2009

Physics 123A - Lecture 19   Phases and Oscillations Here are three examples of differing initial conditions:   , implying x 0 = A/2 and moving to the left (v<0) ;   , implying x 0 = A/2 and moving to the right (v>0) ;    implying x 0 =  A and momentarily at rest ( v=0 ).  September 30, 2009

Physics 123A - Lecture 110 Example: Using Initial Conditions A mass on a spring oscillates with a period of 0.80 s and an amplitude of 10 cm. At t=0, it is 5.0 cm to the left of equilibrium and moving to the left. What is the position and direction of motion at t=2.0 s? September 30, 2009

Physics 123A - Lecture 111 The Energy of SHM September 30, 2009

Physics 123A - Lecture 112 Velocity and Amplitude of SHM September 30, 2009

Physics 123A - Lecture 113 Example: Using Conservation of Energy A 500 g block on a spring is pulled a distance of 20 cm and released. The subsequent oscillation is measured to have a period of 0.80 s. At what position(s) is the block’s speed 1.0 m/s? September 30, 2009

Physics 123A - Lecture 114 The Dynamics of SHM The acceleration is proportional to the negative of the displacement at any time. September 30, 2009

Physics 123A - Lecture 115 The Equation of Motion in SHM This is the equation of motion for the system. It is a homogeneous linear 2 nd order differential equation. Its solutions are sinusoids. September 30, 2009

Physics 123A - Lecture 116 Solving the Equation of Motion Q: How do you solve this differential equation? A: One method: guess a solution and see if it works. Our guess: the 2 nd derivative of a cosine is a cosine, so... It works! September 30, 2009

Physics 123A - Lecture 117 Example: Analyzing an Oscillator At t=0, a 500 g block oscillating on a spring is observed to be moving to the right at x=15 cm. At t=0.30 s, it reaches its maximum displacement of 25 cm. (a)Draw a graph of the motion for one cycle. (b)At what time in the first cycle is x=20 cm? September 30, 2009

End of Lecture 1 Before the next lecture, read T&M, Chapters 14.3 through 14.5 Lecture Homework #1 will be posted soon on the WebAssign system and will due at 11:59 PM on Wednesday, October 7, (next week.)