Oscillation/Periodic Motion

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Goals for Chapter 13 • To examine when natural oscillation occurs Chapter 13 • To quantify a special type of oscillation called simple harmonic motion • To examine kinetic and potential energy in simple Oscillation/Periodic harmonic motion Motion • To understand the concept of driven oscillations and resonance

PowerPoint® Lectures for University Physics, Twelfth Edition – Hugh D. Young and Roger A. Freedman

Lectures by James Pazun Modified by P. Lam 6_14_2011

Introduction - when does natural oscillation occur? Another example of natural oscillation • Natural oscillation occurs whenever • A mass-spring system an object is displaced from its stable exhibits natural equilibrium position. oscillation when the • Example: A pendulum hanging mass is displaced from straight down is in a stable equilibrium position. its equilibrium position and then let go or give it • If you pull the pendulum to the side push. and then let go or give it a push, the pendulum will oscillate at its natural frequency, • Natural frequency g " = rad/s (will derive later) k ! " = rad/s m if there is no damping forces. Friction, air - resistance, and other • The restoring force is damping forces will cause the pendulum ! provided by the to oscillate at a slightly different frequency stretched (or and to stop eventually. compressed) spring. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

1 Oscillation and periodic motion A special type of oscillation - Simple harmonic motion • The text equates periodic motion with oscillation. • An ideal spring responds to stretch and compression Simple Harmonic motion occurs • Technically, a damped oscillation is not a periodic function. linearly, obeying Hooke’s when there is no damping force and The motion is periodic only when there is no damping force. In Law. real situations, all natural oscillations are damped by friction, the restoring force is given by: air-resistance, etc. F = - kx (Hook's Law)

For real springs, Hook's Law is a good approximation only when the displacement (x) is "small".

Undamped oscillation

Damped oscillation

Mathematics of simple harmonic motion (SHM) Solution to the simple harmonic motion (SHM)

F ma d 2 x = F = ma ! "kx = m 2 dt 2 d x "kx = m 2 (assumed no damping force and restoring force obeys Hook's Law) dt Solution: x(t) = A cos!t + B sin !t This equation is called a "differential equation" (D.E.). Substitute: That is an equation which relates the function, x(t), to "k(A cos!!t + B sin !t) = " m! 2 (A cos!t + B sin !t) 2 k dx # A & B can be any number but ! must be m its derivatives (in this case, 2 ). dt ! is called the natural (angular) frequency of mass-spring system. A differential equation is like a puzzle. Try to think of a function, x(t), which can Example : m = 0.1kg and k = 40 N/m 40 rad " # = = 20 (angular frequency) satisfy the relationship stated by the differential equation. 0.1 s # 20 cycles 1 2$ In this case, what function whose 2nd derivative frequency f = = % 3.2 ; period T = = = 0.32s 2$ 2$ s f # is proportional to the function with a negative sign?

2 Solution to the simple harmonic motion (SHM) -cont Solution to the simple harmonic motion (SHM) -cont What does x(t)=A cos!t + B sin!t represent? What are the physical meanings of A & B? d 2 x Most general solution to m = !kx is: Look at the situation at t=0 dt 2 x(t=0)=A ! A is the initial displacement. v x(t) =A cos t+ B sin t=x cos t+ o sin t dx " " o " " v(t)= = "A! sin!t + B! cos!t " dt 2 2 v vo o v(t = 0) = B! ! B = = xo + 2 cos("t + #) (another way of writing the solution) ! "

vo For example : x(t) = Acos!t represents the situation where tan# = ! "xo the spring is stretched (A>0) or compressed (A<0) at t=0 and then let go without giving the mass any initial velocity. The mass spring system will oscillate at the natural freqency no matter how large or small A is (the initial displacement) as long as F = - kx is valid.

Sample problems for SHM Simple Pendulum

! = I" 2 2 d # !L(mgsin#) = (mL ) dt 2 g d 2# ! sin# = L dt 2 For "small" #, sin# " # Given : Spring constant k = 800 N/m, m = 0.5kg, g d 2# # ! # = initial displacment = +0.02m, initial velocity = 0 L dt 2 Find : frequency of oscillation, period of oscillation, and x(t). k d 2 x compare with SHM: ! x = m dt 2 # pendulum exhibits a SHM when # is small (e.g. less than 1 radian) ! # natural angular frequency for a simple pendulum is g $= (Note : does not depend on m) L # #(t) = Acos$t + Bsin$t

3 Energy in SHM Conservation of energy in SHM

• Energy is

conserved vo Given : x(t) = xo cos!t + sin!t for SHM during SHM and ! 1 1 the potential and Calculate the kinetic energy (K= mv2 ) and potential energy (U= kx2 ) 2 2 kinetic energy and show that K+U does not change with time. interconvert as Try it yourselves first before looking at the answer below. the position of dx v(t) = = !x ! sin!t + v cos!t the object in dt o o 1 1 motion changes. K(t) = m[v(t)]2 = m"(x !)2 sin2 !t ! 2x !v sin!t cos!t + (v )2 cos2 !t$ 2 2 # o o o o %

1 2 1 " 2 2 vo vo 2 2 $ U(t) = k[x(t)] = k&(xo ) cos !t + 2xo sin!t cos!t + ( ) sin !t' 2 2 # ! ! % Note : k = m! 2 1 1 ( K(t)+U(t) = mv2 + kx2 = K +U = constant 2 o 2 o o o

Conservation of energy in SHM – graphical view Energy Expression for SHM • Figure 13.15 shows the interconversion of kinetic and potential energy with an energy versus position graphic.

Given : Spring constant k = 800 N/m, m = 0.5kg, initial displacment = +0.02m, initial velocity = 0 Find : The maximum velocity of the mass. Find : The kinetic energy, potential energy and total energy at t = 2 s.

4 Damped oscillations Forced (driven) oscillations and resonance

• Consider shock • A force applied “in synch” with a motion already in progress absorbers on your m!x!= !kx ! bx!; b = damping coefficient will resonate and add energy to the oscillation (refer to Figure automobile. Without 13.28). damping, hitting a pothole would set your • Resonance occurs when driven frequency=natural frequency car into SHM on the springs that support it. The system has the largest amplitude of oscillation at resonance, see Figure to the left. Damping decreases the amplitude.

Forced (driven) oscillations and resonance II Mathematics of driven oscillator

• The Tacoma Narrows Bridge suffered spectacular structural d 2 x k m + kx = 0 (natural oscillation, natural freq. = " = ) failure after absorbing too much resonant energy (refer to Figure dt 2 o m 13.29). d 2 x m + kx = F cos"t (driven ocillation, driven freq. = " ) dt 2 o (assumed no damping force)

Solution : Let x(t) = xo cos"t (system is forced to oscillate at driven freq.) Substittue : 2 m(#" )xo cos"t + kxo cos"t = Fo cos"t

Fo Fo /m Fo /m $ xo = 2 = 2 = 2 2 k # m" k /m #" "o #"

$ % amplitude when " = "o

with damping force, amplitude is largest when " = "o, but not infinite.