Periodic Motion Chapter 11: Sections 1-3, 5
Oscillation • A complete fluctuation of the value of a quantity above and below some median value. Physics chapter 11
Simple Harmonic Motion • Equilibrium position - Central position where forces are balanced. • Amplitude (A) - the maximum displacement from equilibrium.(m) • Period (T) - Time for one complete oscillation.(s) • Frequency (f)- Number of oscillations that occur in a given unit of time. (cyc/s or Hz) • Angular frequency (w) - the frequency given in rad/s. Physics chapter 11
Simple Harmonic Motion • The displacement vector and the vector for the restoring force are always in opposite directions. • Hooke's Law • F = kx • The force a spring exerts is proportional to the distance that the spring is stretched or compressed. • k = F⁄x (N/m) Physics chapter 11
Simple Harmonic Motion mass & spring Physics chapter 11
Example • When a ball is suspended from a spring, the spring stretches by 70 mm. • If the ball oscillates up and down, what is its period? • What is its frequency? Physics chapter 11
Example • Write an equation for the force constant: Physics chapter 11
Example • We don't know the mass of the ball. • Write the equation for the period of oscillation: Physics chapter 11
Example 2 • A piston undergoes simple harmonic motion in a vertical direction with an amplitude of 6 cm. • A coin is placed on top of the piston. • What is the lowest frequency at which the coin will be left behind by the piston on its downstroke? Physics chapter 11
Example 2 Physics chapter 11
Example 2 • The coin will leave the piston when the downward acceleration of the latter exceeds the acceleration of gravity g. • The maximum downward acceleration of piston occurs at the highest point of its motion. Physics chapter 11
Example 2 = 2.0 Hz Physics chapter 11
Simple Pendulum • All of the mass may be considered to be located at the end or bob. Physics chapter 11
Example • A certain simple pendulum has a period on the earth of 0.500 s. What is its period on the surface of the moon, where g = 1.67 m/s2? • 1.21s Physics chapter 11
Energy in SHM Physics chapter 11
Simple Harmonic Motion Physics chapter 11
Motion Equations • x = 0.5cos2πt (in standard units) • Amplitude = 0.5 m • Angular Frequency = 2π rad/s • f = /2π • T = 1/f Physics chapter 11
Motion Equations • x = Acost • v = –Asint • a = –2Acost • vmax = A= 2πfA where x = 0. • a = –2x • amax = 2A at x = ±A Physics chapter 11
x vs. t x = Acoswt Physics chapter 11
x and v vs. t v = –wAsinwt vmax = wA= 2πfA where x = 0. Physics chapter 11
x, v, and a vs. t • a = –w2Acoswt • a = w2x • amax = w2A at x = ±A Physics chapter 11
More equations Physics chapter 11
Example • As shown on the next slide, a rotating wheel drives a piston by means of a long connecting rod pivoted at both ends. • The wheel's radius is 20 cm and it is turning at 4.0 rev/s. • A) Write an equation for the displacement of the piston as a function of time. • B) Find the maximum speed and acceleration of the piston. Physics chapter 11
Example x = ? vmax = ? amax = ? Physics chapter 11
Example • x = Acoswt • A = 20 cm = 0.2 m • w = 4 rev/s x 2p rad/1 rev = 8p rad/s • x = 20cos8pt cm • x = 0.2cos8pt m Physics chapter 11
Example • v = –Asint • v = – 1.6psin8pt m/s • vmax = A= 2πfA • vmax = 1.6p m/s Physics chapter 11
Example • a = –2Acost • a = – 12.8p2cos8pt m/s2 • amax = 2A • amax = 12.8p2 m/s2 Physics chapter 11
