# Mh1: Simple Harmonic Motion. Chapter 15. Motion of a Spring-Mass System. Periodic Motion. Oscillatory Motion

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1 Mh1: Siple Haronic Motion Chapter 15 Siple block and spring Oscillatory Motion Exaple: the tides, a swing Professor Michael Burton School of Physics, UNSW Periodic Motion! Periodic otion is otion of an object that regularly repeats! The object returns to a given position after a fixed tie interval! Siple Haronic Motion! When the force acting on the object is proportional to the position of the object relative to soe equilibriu position, and is directed towards it! F = - k. x Motion of a Spring-Mass Syste! Active Figure 15.0! Block of ass is attached to a spring, and free to ove on a frictionless horizontal surface! When the spring is neither stretched nor copressed, the block is at the equilibriu position! x = 0

2 Hooke s Law! Active Figure 15.01! Hooke s Law for the stretched spring F s = - kx! F s is the restoring force! It is always directed toward the equilibriu position! Thus it is always opposite the displaceent fro equilibriu! k is the force (spring) constant! x is the displaceent fro the equilibriu position Acceleration! The force described by Hooke s Law is the net force in Newton s Second Law F Hooke x = F! kx = a a Newton x k =! x The Acceleration is! Proportional to the displaceent of the block! Has direction opposite the direction of the displaceent fro equilibriu! Thus, this is Siple Haronic Motion! It is not constant a = " k x! Thus, the kineatic equations (i.e. v=u+at etc.) for constant acceleration cannot be applied A block on the end of a spring is pulled to position x = A and released fro rest. In one full cycle of its otion, through what total distance does it travel? 1. A/. A 3. A 4. 4A

3 PHYSCLIPS : Waves and Sound 1.1 Oscillations Vertical Orientation! When the block is hung fro a vertical spring, its weight will cause the spring to stretch! If the resting position of the spring is defined as y = 0, the sae analysis as was done for the horizontal spring will apply to the vertical spring-block syste! Here y = x - x equilibriu where g = k x equilibriu! See Movie HMM03AN1! SHM with vertical spring! 19sec Siple Haronic Motion Matheatical Representation! Model the block as a particle! Choose x as the axis along which the oscillation occurs! Apply Newton s nd Law:! Acceleration d x k a = =! x! We let k! =! Then a = -! x dt Siple Haronic Motion Graphical Representation! A solution is x(t) = A cos (!t + ")! A,!, " are all constants! Differentiate, to yield: v = -!A sin (!t + ")! a =-! A cos (!t + ")!! So that a = -! x as for SHM! Siilarly x(t) = A sin (!t + ") is also a solution

4 Siple Haronic Motion the constants of otion! A is the aplitude of the otion! This is the axiu position of the particle in either the positive or negative direction!! is called the angular frequency! Units are radian/s! " is the phase constant or the initial phase angle! Siple Haronic Motion, cont! Since x(t) = A cos (!t + "), then A and " are deterined uniquely by the position and velocity of the particle at t = 0! e.g. If the particle is at x = A at t = 0, then " = 0! The phase of the otion is (!t + ")! x (t) is periodic and its value is the sae each tie!t increases by " radians! PHYSCLIPS : Waves and Oscillations 1. The Equations Mh11: Siple Haronic Motion air track & springs No friction!

5 Consider a graphical representation of siple haronic otion. When the object is at point A on the graph, what can you say about its position and velocity? 1. The position and velocity are both positive.. The position and velocity are both negative. 3. The position is positive, and its velocity is zero. 4. The position is negative, and its velocity is zero. 5. The position is positive, and its velocity is negative. 6. The position is negative, and its velocity is positive. The figure shows two curves representing objects undergoing siple haronic otion. The correct description of these two otions is that the siple haronic otion of object B is: 1. of larger angular frequency and larger aplitude than that of object A.. of larger angular frequency and saller aplitude than that of object A. 3. of saller angular frequency and larger aplitude than that of object A. 4. of saller angular frequency and saller aplitude than that of object A. Period and Frequency! The period, T, is the tie interval required for the particle to go through one full cycle of its otion! i.e. x(t)=x(t+t) and v(t)=v(t+t) with T="/#! The inverse of the period is called the frequency! It represents the nuber of oscillations the particle undergoes per unit tie interval! f = 1/T = #/"! Units are cycles per second = hertz (Hz)! Since! = then k 1 T =! ƒ = k! k Period and Frequency: II! Frequency and period depend only on the ass of the particle and the force constant of the spring! See Active Figure 15.06! Frequency is larger for a stiffer spring (large values of k) and decreases with increasing ass of the particle f = 1 " k

6 Suary: Motion Equations for Siple Haronic Motion x( t) = A cos (! t + ") dx v = = #! Asin (! t + ") dt d x a = = #! A cos(! t + ") dt! Reeber, siple haronic otion is not uniforly accelerated otion Maxiu Values of v and a! Because the sine and cosine functions oscillate between ±1, then the axiu values of velocity and acceleration for an object in SHM are k vax =! A = A k aax =! A = A All three together HMA03AN3: SHM for x, v, a Phase of x, v, a! The graphs show:! (a) displaceent as a function of tie! (b) velocity as a function of tie! (c ) acceleration as a function of tie! Velocity is 90 o out of phase with the displaceent! Acceleration is 180 o out of phase with the displaceent 3 seconds

7 PHYSCLIPS : Waves and Oscillations 1.3 Forces & Energy An object of ass is hung fro a spring and set into oscillation. The period of the oscillation is easured and recorded as T. The object of ass is reoved and replaced with an object of ass. When this object is set into oscillation, what is the period of the otion? 1. T. 3. T T 4. T / 5. T/ SHM Exaple 1 SHM Exaple! x(t) = A cos (!t + ")! Initial conditions at t = 0:! x (0)= A! v (0) = 0! Thus " = 0! The acceleration reaches extrees of ±! A! The velocity reaches extrees of ±!A! x(t) = A cos (!t + ")! Initial conditions at t = 0:! x (0)=0! v (0) = v ax! Thus " = #/! i.e. x(t) = A sin (!t)! Graph is shifted one-quarter cycle to the right copared to graph when x(0) = A\$

8 Energy of the SHM Oscillator: I! Assue a spring-ass syste is oving on a frictionless surface! This tells us the total energy is constant! The kinetic energy can be found by! K =! v =!! A sin (!t + ")! i.e. K =! k A sin (!t + ") as # =k/! The elastic potential energy can be found by! U = " Fdx = " kxdx = 1 kx! i.e. U =! kx =! ka cos (!t + ")! Thus, the total energy is K + U =! ka Energy of the SHM Oscillator: II! See Active Figure 15.09! The total echanical energy is constant =! ka! The total echanical energy is proportional to the square of the aplitude! Energy is continuously being transferred between potential energy stored in the spring and the kinetic energy of the block Suary of Energy in SHM Active Figure Mh7: Transfer of Energy Coupled pendula

9 Molecular Model of SHM! If the atos in the olecule do not ove too far, the force between the can be odeled as if there were springs between the atos! The potential energy acts siilar to that of the SHM oscillator Mh3: Siple Haronic Motion and Circular Motion Connection between circular otion and SHM can be seen on the overhead projector SHM and Circular Motion: I! See Active Figure 15.13! An overhead view which shows the relationship between SHM and circular otion! As the ball rotates with constant angular velocity, its shadow oves back and forth in siple haronic otion SHM and Circular Motion: II! The particle oves along the circle with constant angular velocity!! OP akes an angle % with the x axis! At tie t the angle between OP and the x axis will be % =!t + "\$! So x(t) = A cos (!t + ") and v = -!A sin (!t + "), a =! A cos (!t + ")! Thus a =! x i.e. SHM!

10 SHM and Circular Motion: III! Siple Haronic Motion along a straight line can be represented by the projection of unifor circular otion along the diaeter of a reference circle! Unifor circular otion can be considered a cobination of two siple haronic otions! One along the x-axis! The other along the y-axis! The two differ in phase by 90 o The figure shows the position of an object in unifor circular otion at t = 0. A light shines fro above and projects a shadow of the object on a screen below the circular otion. What are the correct values for the aplitude and phase constant (relative to an x axis to the right) of the siple haronic otion of the shadow? and and and " and " Mh5: Siple Haronic Motion the siple pendulu Siple, but a classic! The Siple Pendulu! See Active Figure 15.16! Exhibiting periodic otion! The otion occurs in the vertical plane and is driven by the gravitational force! The otion is very close to that of the SHM oscillator for sall angles

11 Siple Pendulu II! The forces acting on the bob are T and g! T is the force exerted on the bob by the string! g is the gravitational force! The tangential coponent of the gravitational force is a restoring force! This is -g sin(%)! The arc length is s = L %! So ds dt = L d" and d s dt dt = L d " dt Siple Pendulu III! In the tangential direction, F t = "gsin# = d s dt = L d # dt! The length, L, of the pendulu is constant, and so, for sall values of % d! g g = " sin! = "! dt L L! Thus the otion is SHM with # =g/l Siple Pendulu IV! The function % can be written as % = % ax cos (!t + ")! The angular frequency is! = g L! The period is Mh1: The Pendulu period and length Half class tie top pendulu, half class tie botto pendulu. Is T \$%L? T! = =! " L g

12 A grandfather clock depends on the period of a pendulu to keep correct tie. Suppose a grandfather clock is calibrated correctly and then a ischievous child slides the bob of the pendulu downward on the oscillating rod. The grandfather clock will run: A grandfather clock depends on the period of a pendulu to keep correct tie. Suppose a grandfather clock is calibrated correctly at sea level and is then taken to the top of a very tall ountain. The grandfather clock will now run: 1. slow.. fast. 3. correctly. 1. slow.. fast. 3. correctly. Daped Oscillations: I! See Active Figure 15.1! In any real systes, non-conservative forces are present! Friction is a coon non-conservative force! In this case, the echanical energy of the syste diinishes in tie, the otion is said to be daped Daped Oscillations: II! A graph for a daped oscillation! The aplitude decreases with tie! The blue dashed lines represent the envelope of the otion

13 Daped Oscillation, Exaple! One exaple of daped otion occurs when an object is attached to a spring and suberged in a viscous liquid! The retarding force can be expressed as R = - b v where b is a positive constant! b is called the daping coefficient! Fro Newton s Second Law the equation of otion becoes F x = -k x bv x = a x Types of Daping Not for exaination " b # = 0 \$ % &!! ' ( k!! is also called the natural 0 = frequency of the syste! If R ax = bv ax < ka, the syste is said to be underdaped! When b reaches a critical value b c such that b c / =! 0, the syste will not oscillate! The syste is said to be critically daped! If R ax = bv ax > ka and b/ >! 0, the syste is said to be overdaped Types of Daping: II PHYSCLIPS : Sound and Waves 1.4 Non-linearity and Daping! Graphs of position versus tie for! (a) an underdaped oscillator! (b) a critically daped oscillator! (c) an overdaped oscillator! For critically daped and overdaped there is no angular frequency

14 Mh13: Double Pendulu Forced Oscillations Forced Oscillations: I! It is possible to copensate for the loss of energy in a daped syste by applying an external force! e.g. pushing child on a swing at angular frequency!! In general F = F 0 sin(!t) bv kx! The aplitude of the otion reains constant if the energy input per cycle exactly equals the decrease in echanical energy in each cycle that results fro resistive forces Forced Oscillations: II! Motion is x (t) = A cos (!t + ") where the aplitude of the driven oscillation is A = F 0 " b! # \$ + % & ' ( (!! 0 ) Mh: Resonance Pendulu Large ball will cause all balls to oscillate but only the ball of the sae natural frequency will oscillate at large aplitude. Try and predict beforehand.!! 0 is the natural frequency of the undaped oscillator

15 Resonance: I! When the frequency of the driving force is near the natural frequency (! &! 0 ) an increase in aplitude occurs! This draatic increase in the aplitude is called resonance! The natural frequency! 0 is also called the resonance frequency of the syste Resonance: II! Resonance (axiu peak) occurs when the driving frequency equals the natural frequency! The aplitude increases with decreased daping! The curve broadens as the daping increases! The shape of the resonance curve depends on b PHYSCLIPS : Waves and Sound 1.5 Resonance Resonance: Tacoa Narrows Bridge Noveber 7, 1940, Washington State, USA

16 PHYSCLIPS : Waves and Sound Diensions

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