4.1 KINEMATICS OF SIMPLE HARMONIC MOTION 4.2 ENERGY CHANGES DURING SIMPLE HARMONIC MOTION 4.3 FORCED OSCILLATIONS AND RESONANCE Notes

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4.1 KINEMATICS OF SIMPLE HARMONIC MOTION 4.2 ENERGY CHANGES DURING SIMPLE HARMONIC MOTION 4.3 FORCED OSCILLATIONS AND RESONANCE Notes I. DEFINING TERMS A. HOW ARE OSCILLATIONS RELATED TO WAVES? II.

FORCED OSCILLATIONS AND RESONANCE B. WHEN RESONANCE IS USEFUL AND WHEN IT ISN T I. DEFINING TERMS A. HOW ARE OSCILLATIONS RELATED TO WAVES?

1 4.1 KINEMATICS OF SIMPLE HARMONIC MOTION 4.2 ENERGY CHANGES DURING SIMPLE HARMONIC MOTION 4.3 FORCED OSCILLATIONS AND RESONANCE Notes I. DEFINING TERMS A. HOW ARE OSCILLATIONS RELATED TO WAVES? II. EQUATIONS OF SHM A. DISPLACEMENT AND VELOCITY B. ACCELERATION - THE DEFINING EQUATION OF SHM III. ENERGY IN SHM IV. DAMPING A. ENERGY LOSS IN SHM B. DEGREES AND TYPES OF DAMPING V. RESONANCE A. FORCED OSCILLATIONS AND RESONANCE B. WHEN RESONANCE IS USEFUL AND WHEN IT ISN T I. DEFINING TERMS A. HOW ARE OSCILLATIONS RELATED TO WAVES? DEFINE: Oscillation = repetitive motion in time around a central (equilibrium) value. Examples: masses vibrating on a spring, the pendulum, vibrating tuning forks, water in a U-tube, block of wood bobbing on water, etc. DEMO ALL and show how they can be traced as a sine wave on a moving surface. CONSIDER: a mass on a spring as shown below. When the mass is attached to the spring the spring stretches until the equilibrium position is reached. If the spring is pushed up a further distance and then released the graph on the right of the motion would be obtained (assuming no friction). DEMO Source: Physics, 8 Note that this is a wave th Ed, Cutnell & Johnson DEFINE: 1. Displacement (x) = distance from the equilibrium position. It varies constantly. 2. Amplitude (A) = maximum displacement; in this case 2 units. 3. Periodic time (T) = time taken for one complete oscillation; here about 63 units. 1

4. Phase difference (Φ) = table-top: Source: Physics, 8 th Ed, Cutnell & Johnson Graph traces a sine wave when a pen is attached to it, and a rolling paper is set behind it!

2 4. Phase difference (Φ) = table-top: Source: Physics, 8 th Ed, Cutnell & Johnson Graph traces a sine wave when a pen is attached to it, and a rolling paper is set behind it! Total time to get back to the same spot = period = T. DEFINE: Simple Harmonic Motion (SHM) = when a body moves in such a way that its acceleration is directed towards a fixed point in its path and is directly proportional to its distance from that point. The fixed point in reality is nearly always the equilibrium position. NOTE: letting this becomes defining eqn of SHM Conditions for SHM: 1. A fixed equilibrium position 2. a -x Again consider the mass hanging on a spring. The physics when the mass is going up is a little bit different than the physics when it is going down: 2

3 Source: Dave Simms GOING UP: Note that the acceleration is always directed towards the equilibrium position. Also note that the further away from the equilibrium position the greater the acceleration. GOING DOWN: Note that the acceleration is always directed towards the equilibrium position. Also note that the further away from the equilibrium position the greater the acceleration. 3

II. EQUATIONS OF SHM A. DISPLACEMENT AND VELOCITY A ball moving in circular motion will cast a shadow on a moving strip behind it as shown. Note how the graph traces a sine wave!

4 II. EQUATIONS OF SHM A. DISPLACEMENT AND VELOCITY A ball moving in circular motion will cast a shadow on a moving strip behind it as shown. Note how the graph traces a sine wave! Again circular motion related to a wave. If the point P rotates at a constant rate, then the projection (shadow) of point P on the x axis undergoes SHM. DEFINE: ω = angular frequency = amount of angle in radians subtended per unit time = 2π/T Units = Hz Source: Physics, 8 th Ed, Cutnell & Johnson REMEMBER: letting this becomes Now this means that more on this later. By using calculus (Tsokos pp ), we get the following results: RADIANS! DISPLACEMENT VELOCITY Source: Physics for the IB Diploma, 5 th Ed, Tsokos 4

5 EXAMPLE 1 A particle undergoes SHM with an amplitude of 4.0 cm and an angular frequency of 2.0 s -1. At t = 0, the displacement is (4.0)(2) -1/2 mm. Write down the equation giving the displacement for this motion. [x(t) = 4.0 cos (2.0t + π/4)] B. ACCELERATION - THE DEFINING EQUATION OF SHM Using 2 cases for displacement (either x = 0 at t = 0, or x = a at t = 0) When x = x 0 = A at t = 0 and when x = 0 at t = 0 x = A cos (ωt) x = A sin (ωt) Source: Physics for the IB Diploma, 5 th Ed, Tsokos Therefore: when x = x 0 = A at t = 0 cos (ωt) a = -ω 2 A when x = 0 at t = 0 a = -ω 2 A sin (ωt) The ve sign indicates that the acceleration is always in the opposite direction to the displacement (as ω 2 is always +ve) i.e. the acceleration is always directed toward the equilibrium position. In some texts the ve is ignored in all of these equations; remember this is just about direction. Use common sense and draw a picture of the situation when solving these problems! 5

6 EXAMPLE 2 A particle undergoes SHM with an amplitude of 8.00 cm and an angular frequency of s -1. At t = 0, the velocity is 1.24 cm s -1. Determine: a) The equations for displacement and velocity of the motion. [x(t) = 8.00cos(0.250t 0.669), v(t) = -2.00sin(0.250t 0.669)] b) The initial displacement of the particle. [x 0 = cm] c) The first time at which the particle is as x = 2.0 cm and x = cm. [7.95 s, 9.97 s] THE SIMPLE PENDULUM revisited.. Mass attached to a string. Draw a FBD for the mass when pulled to the side Force pushing mass back to equilibrium = - mg sin θ = ma So a = -g sin θ Distance along arc from equilibrium position is x = L θ so Then, ( ) If x small compared to L, then ( ) Source: Physics for the IB Diploma, 5 th Ed, Tsokos So if ( ) 6

7 IV. ENERGY IN SHM AGAIN consider a EPE stored in spring TOTAL ENERGY OF SYSTEM: E = Ep + Ek = ½ kx 2 + ½ mv 2 (no friction). Then E = constant and if released from rest when x 0 = A, Source: Physics, 8 th Ed, Cutnell & Johnson solving for v gives: again, vmax = ωa EXAMPLE 3 The graph below shows variation with the square of the displacement (x 2 ) of the potential energy of a particle of mass 40 g that is executing SHM. Using the graph, determine the period of oscillation and the maximum speed of the particle. [ω= 100 s -1, v max = 2.0 m s -1 ] EXAMPLE 4 A particle of mass 0.50 kg undergoes SHM with angular frequency ω = 9.0 s -1 and amplitude 3.0 cm. For this particle, determine: a) v max [0.27 ms -1 ] b) v and a when displacement is 1.5 cm and moves towards the equilibrium position at x = 3.0 cm. [v = ms -1, a = -1.2 ms -2 ] c) The total energy of the motion. [18 mj] 7

Source: Physics, 8 th Ed, Cutnell & Johnson Source: Physics for the IB Diploma, 5 th Ed, Tsokos V. DAMPING A. ENERGY LOSS IN SHM B. DEGREES AND TYPES OF DAMPING 1.

8 Source: Physics, 8 th Ed, Cutnell & Johnson Source: Physics for the IB Diploma, 5 th Ed, Tsokos V. DAMPING A. ENERGY LOSS IN SHM B. DEGREES AND TYPES OF DAMPING 1. Critical Damping Large resistance forces no oscillations. 2. Over-Damping Large resistance forces no oscillations. System returns to equilibrium much slower than in critical damping. EXAMPLE: shock absorber on a car. 8

9 Source: Dave Simms VI. RESONANCE A. FORCED OSCILLATIONS AND RESONANCE If we apply an external force at same frequency that something is oscillating at anyway, the displacements will get BIGGER (constructive interference). Let f 0 = natural frequency (f of system naturally) f D = driving frequency (f of external force applied) Resulting oscillations forced oscillations After some time, the system will oscillate at f D. If f 0 = f D, then amplitude will be BIG and system is in resonance. If f 0 f D by lots, then amplitude will be small. DEMO: Barton s Pendulum B. WHEN RESONANCE IS USEFUL AND WHEN IT ISN T GOOD RESONANCE: MUSICAL INSTRUMENTS: In wind instruments, resonance occurs in the air column to naturally amplify the sound. The resonance occurs when the frequency of the vibrating reed (being driven by the moving air from the musician s mouth) matches the vibration of air within the column of the instrument. QUARTZ OSCILLATORS: This is an electronic circuit that uses the mechanical resonance of a vibrating quartz crystal to create an electrical signal with a very precise frequency. This frequency is commonly used to keep track of time (as in quartz wristwatches), to provide a stable clock signal for digital integrated circuits, and to stabilize frequencies for radio transmitters. 9

10 RADIO RECEIVERS: Electric circuits are designed to have their own natural frequency of oscillation, which is adjustable (by adjusting a dial). Resonance occurs when a radio stations emitted waves have a frequency equal to the natural frequency of the radio. The increase in amplitude results in that radio station s waves being louder than all others. Electrical resonance occurs when a radio circuit is tuned by making its natural frequency. MICROWAVE OVENS: A microwave oven works by passing radiation (electromagnetic waves), usually at a frequency of 2.45 GHz (a wavelength of cm), through food. This is the range of natural frequency of water and fat molecules. Therefore these molecules start to vibrate due to the driving force of the microwaves, and their kinetic energy gets greater (they get hot!) BAD RESONANCE: VIBRATIONS IN MACHINERY: Moving parts of machines provide driving forces on the entire system. If the frequency of these driving forces equals the natural frequency of the machine, dangerously high amplitudes can result. Think of a car vibrating at certain speeds, or a rattling rear-view mirror, or wings on an airplane VIBRATIONS IN STUCTURES: If the driving force frequency on a structure matches the natural frequency, there can be a disaster. The most famous example is The Tacoma Narrows Bridge disaster. This was caused by the wind producing an oscillating resultant force in resonance with a natural frequency of the bridge. An oscillation of large amplitude built up and destroyed the bridge. Earthquakes also cause vibrations within structures. 10

HOMEWORK ANSWERS. Lesson 4.1: Simple Harmonic Motion

HOMEWORK ANSWERS. Lesson 4.1: Simple Harmonic Motion DEVIL PHYSICS HOMEWORK ANSWERS Tsokos, Chapter 3 Test Lesson 4.1: Simple Harmonic Motion 1. Objectives. By the end of this class you should be able to: a) Understand that in simple harmonic motion there