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GRADE 12A: Physics 4 UNIT 12AP.4 9 hours Oscillations About this unit Previous learning Resources This unit is the fourth of seven units on physics for Grade 12 advanced. To meet the expectations of this unit, students should already understand the concepts of displacement, speed, velocity and acceleration, represent them graphically and interpret graphs that represent them. They should know and be able to use the terms amplitude, phase difference, period, frequency. They should be familiar with angular displacement and angular velocity expressed using radians and should be able to use the expressions v = r, a = r2 and a = v2 ⁄ r. They should know that a force applied to an object can cause deformation (which can often be described by Hooke’s law). They should be able to define kinetic and potential energy, and describe the principle of energy conservation and apply it to simple examples. They should have a qualitative knowledge of frictional and viscous forces, including air and water resistance. The main resources needed for this unit are: The unit is designed to guide your planning and teaching of physics lessons. It provides a link between the standards for science and your lesson plans. The teaching and learning activities should help you to plan the content and pace of lessons. Adapt the ideas to meet your students’ needs. For consolidation activities, look at the scheme of work for Grade 11A. You can also supplement the activities with appropriate tasks and exercises from your school’s textbooks and other resources. Introduce the unit to students by summarising what they will learn and how this builds on earlier work. Review the unit at the end, drawing out the main learning points, links to other work and real world applications. Expectations By the end of the unit, students solve mathematical problems in simple harmonic motion and explain practical examples of resonance, critically and non-critically damped oscillations and forced oscillations. Students who progress further use calculus and graphical methods to deduce equations for simple harmonic motion. They derive and use expressions for kinetic and potential energy during the motion. • film or video clip of the Tacoma Narrows bridge collapse • battery-operated buzzer or electric toothbrush • xenon strobe lamp • vibration generator • flexible track (e.g. curtain track) bent into parabolic and semicircular shapes and mounted on a board • guitar or sonometer • transparent U-shaped tube • shock absorber from a car • Barton’s pendulums demonstration • hacksaw blades Key vocabulary and technical terms Students should understand, use and spell correctly: • amplitude, period, frequency, angular frequency, phase angle, phase difference • oscillation, vibration, simple harmonic motion • free oscillation, restoring force • damping, critical damping, under-damping, over-damping • forced oscillation, natural frequency, resonance 533 | Qatar science scheme of work | Grade 12 advanced | Unit 12AP.4 | Physics 4 © Education Institute 2005 Standards for the unit 9 hours 5 hours Oscillations 2 hours SUPPORTING STANDARDS 10A.28.3 Know and use the terms ... displacement, amplitude, phase difference, period, frequency ... Unit 12AP.4 CORE STANDARDS Grade 12 standards EXTENSION STANDARDS 12A.28.1 Describe examples of free oscillations and understand and use the terms amplitude, period, frequency, angular frequency and phase difference. Express the period in terms of both frequency and angular frequency. 12A.25.1 Express angular displacement in radians ... Energy in oscillations 2 hours Forced oscillations and resonance 12A.25.2 Understand and use the concept of angular velocity to solve problems in various situations using the formulae v = r, a = r2 and a = v2 ⁄ r. 10A.26.1 Understand the concepts of displacement, speed, velocity and acceleration, represent them graphically and interpret graphs that represent them. 12A.28.2 Deduce, by calculus or graphical methods, and use the equations for expressing the displacement, period, velocity and acceleration in simple harmonic motion. 10A.26.3 Know that a force acting on an object can cause deformation ... 11A.27.2 Define kinetic and potential energy ... Describe the principle of energy conservation and apply it to simple examples. 10A.26.5 Show a qualitative knowledge of frictional and viscous forces including air and water resistance ... 10A.28.8 ... illustrate the phenomenon of resonance with particular reference to vibrating stretched strings and air columns. 12A.28.3 Describe, using graphical illustrations, the changes in displacement, velocity and acceleration during simple harmonic motion. Describe the changes between kinetic and potential energy during the motion. 12A.28.4 Describe and explain practical examples of critically and non-critically damped oscillations. 12A.28.5 Describe practical examples of forced oscillations and resonance and show how the amplitude of a forced oscillation changes with frequency near to the natural frequency of the system. 12A.28.6 Describe circumstances in which resonance is desirable and others when it should be avoided. 534 | Qatar science scheme of work | Grade 12 advanced | Unit 12AP.4 | Physics 4 © Education Institute 2005 Activities Unit 12AP.4 Objectives Possible teaching activities 5 hours Oscillations Oscillations Show the class several examples of oscillating objects with a wide range of sizes and frequencies. Examples should include a video or film clip of the Tacoma Narrows bridge collapse, a string attached to a vibration generator, and a small battery-driven oscillator such as a buzzer or electric toothbrush. Describe examples of free oscillations and understand and use the terms amplitude, period, frequency, angular frequency and phase difference. Express the period in terms of both frequency and angular frequency. Deduce, by calculus or graphical methods, and use the equations for expressing the displacement, period, velocity and acceleration in simple harmonic motion. Describe, using graphical illustrations, the changes in displacement, velocity and acceleration during simple harmonic motion... Notes School resources Use this column to note your own school’s resources, e.g. textbooks, worksheets. Use quick-fire oral questions to remind students of the terms period, amplitude and frequency from earlier grades. Introduce, or remind students of, the terms oscillation and vibration, each meaning a regular repeating to-and-fro motion. Establish that oscillation is a type of motion found in many different situations. Ask students to suggest how the frequency of an oscillation can be measured. For long-period oscillations, such as those of the Tacoma Narrows bridge, they should be able to suggest that the period, and hence the frequency, can readily be deduced using freeze-frame video. Demonstrate the use of a xenon stroboscope to determine the frequency of a string attached to a vibration generator. By comparing settings of the strobe and the generator, establish that there are several strobe frequencies that ‘freeze’ the motion, and that the vibration frequency must be the highest of these. Safety: Stroboscopes can be hazardous to people with epilepsy Divide the class in half, and each half into small groups. Ask students in one half of the class to use digital freeze-frame video to find the frequency of the Tacoma Narrows bridge oscillations, and ask those in the other half to use a xenon strobe to find the frequency of a buzzer, electric toothbrush or similar high-frequency vibration. The two halves should then change over so that all students experience both techniques. ICT opportunity: Use of digital video. Enquiry skills 12A.4.1, 12A.4.2 SHM or not SHM? Set up a circus of oscillating objects. Aim for a wide variety and choose some examples where period is independent of amplitude and there is a sinusoidal variation of displacement with time (i.e. SHM) and some that behave in other ways. Suitable examples of oscillators include: Ask students to work in pairs to explore each station of the circus in turn. Tell them to experiment and note whether the period of each oscillation depends on the amplitude. Where possible, they should also obtain a record showing how displacement of the oscillator varies with time. • ball-bearing rolling on a semi-circular track; Discuss students’ findings with the whole class. Establish that many different types of oscillator behave in a similar way in that the period of oscillation is independent of amplitude and a graph of displacement against time is sinusoidal. Tell students that such oscillations are called simple harmonic motion (SHM) and will be the main subject of this unit. • bouncing ball; Also establish that, while many oscillators perform SHM, not all oscillations follow this pattern and the analysis developed in this unit is not applicable to them. Students might have reached differing conclusions concerning the oscillation of pendulums: by suitable questioning, establish that at small amplitudes these perform SHM, but at large amplitudes the period increases noticeably. 535 | Qatar science scheme of work | Grade 12 advanced | Unit 12AP.4 | Physics 4 • dynamics trolley tethered between two springs; • ball-bearing rolling on a parabolic track; • vibrating guitar or sonometer string; • liquid in a U-shaped tube; • simple pendulum; • rigid pendulum suspended from potentiometer shaft (connect the potentiometer to a DC power supply in series with a fixed resistor; connect a CRO across the potentiometer so that the displayed voltage trace indicates the angular displacement of the pendulum). Enquiry skills 12A.1.2–12A.1.4 © Education Institute 2005 Objectives Possible teaching activities Notes School resources SHM and circular motion Set up the following demonstration to show students the relationship between circular motion and SHM. Suspend a tennis ball (or similar) from a long string to make a pendulum. Mount another tennis ball on a turntable that can spin slowly about a vertical axis. Place both behind a translucent screen and arrange a light source to cast shadows of both balls onto the screen. Adjust the length of the string and/or the rotation speed of the turntable so that both motions have the same period, and the shadows on the screen oscillate in phase and with similar amplitude. Students should observe that SHM can be treated as a projection of circular motion. Ask each student to draw, on graph paper, a circle with a radius of a few centimetres, centred on an intersection of grid lines. Tell them to mark the circumference of the circle at regular intervals (e.g. every 30°) to represent successive positions of an object moving around the circle at constant speed and observed at regular time intervals. They should then note the x (or y) coordinate of each position and plot a graph showing how this coordinate varies with time: the graphs will be sinusoidal. Let students explore this projected motion using an appropriate Java applet. ICT opportunity: Use of Java applets. Equations of SHM On the board or OHP, show students how to derive equations for SHM using a projection of circular motion. Mathematics. Trigonometry and a knowledge of angles measured in radians are required. Draw a large diagram showing an object moving anticlockwise around a circle of radius A, starting from the x-axis at time t = 0. By suitable questioning, remind students of the meaning of angular velocity, , and establish that, expressed in radians, angular displacement = t. Also remind them of the relationships v = r and a = v2 ⁄ r = r2. Choose one point on the circumference of the circle and use trigonometry to show students that the displacement in the x direction is x = r cos ( t ). Draw a velocity vector at the same point and show that its x component is v = r sin ( t ). Draw a vector representing the centripetal acceleration at the same point and show that its x component is a = –r2 cos ( t ). Establish that these equations describe the displacement, velocity and acceleration (respectively) of an object oscillating with SHM along the x-axis. Students should be able to identify r with the amplitude, A, of the oscillation and rewrite the equations as x = A cos ( t ) and so on. Explain that, when dealing with SHM (rather than circular motion), is usually called the angular frequency (rather than angular velocity). Ask students to derive the relationships = 2f and T = 2 ⁄ where f is the frequency and T the period of the oscillation. Now draw another similar diagram, this time showing an object starting with an angular displacement when t = 0. Ask students to work individually or in pairs to derive expressions for the x components of displacement, velocity and acceleration (i.e. x = A cos ( t + ) and so on). Establish that the angle is called the phase angle, and that the phase difference between two oscillations can be denoted by the difference in their phase angles. Remind students that such angles are conventionally expressed in radians (e.g. two oscillations in antiphase have a phase difference of ). 536 | Qatar science scheme of work | Grade 12 advanced | Unit 12AP.4 | Physics 4 © Education Institute 2005 Objectives Possible teaching activities Notes School resources Give students plenty of opportunities to practise drawing and interpreting graphs showing how displacement, velocity and acceleration vary with time for SHM. Mathematical derivation of SHM equations Discuss the equations derived above and establish that, regardless of the amplitude and phase, the acceleration is always proportional to the displacement and in the opposite direction (i.e. a = –2x). Mathematics: A knowledge of calculus is required. Show that this expression can be rewritten as d2x ⁄ dt2 = –2x and point out that it is a differential equation in x. By suitable questioning and discussion, show that x = A cos ( t ), x = A sin ( t ) and x = A cos ( t + ) are all solutions to the equation. Also show that corresponding expressions for velocity can be obtained by differentiating the expressions for displacement. Point out that the values of phase angle and amplitude are independent of the angular frequency and can, in principle, take any value, depending on the particular oscillator being described. Emphasise that this is a characteristic of SHM as noted earlier: the period is independent of amplitude. Give students plenty of algebraic and numerical examples that allow them to practise using equations of SHM. Measuring SHM Ask students, in pairs or small groups, to explore in detail one example of an oscillator performing SHM. As far as apparatus permits, each pair or small group should explore a different oscillator. Tell students to devise and use methods for obtaining an accurate record of its motion, showing how the displacement, velocity and acceleration vary with time. The oscillators themselves should be fairly simple, and students should be encouraged to use a range of methods and instruments for recording their motion, such as sensors, dataloggers and digital video cameras. Review work on linear motion from earlier grades with students and make sure they recall that velocity can be deduced from the gradient of a displacement–time graph, and acceleration from the gradient of a velocity–time graph. Depending on how the initial records have been produced and stored, gradients can be found either from hand-drawn tangents or by using suitable software. Ask students to summarise their results in the form of labelled graphs. These can be photocopied and distributed to the whole class. Suitable examples include: • a dynamics trolley tethered between two springs; • a magnet on a spring oscillating in and out of a coil (the induced emf will indicate velocity); • a rigid pendulum suspended from a potentiometer shaft; • a simple pendulum. (The amplitude of the pendulum oscillations must be small.) Enquiry skills 12A.1.1, 12A.1.3, 12A.1.4, 12A.1.5, 12A.3.1, 12A.3.2, 12A.3.4, 12A.4.1 ICT opportunity: Use of dataloggers and digital video. Force and SHM Write the SHM equation a = –2x on the board or OHP. Establish that this relationship implies that the oscillating object must be experiencing a restoring force F that is proportional to its displacement and in the opposite direction: F = – kx. By means of suitable questioning, remind students of Hooke’s law and the behaviour of springs. Establish that, if a mass is suspended from a spring and displaced from equilibrium by a distance x, it will experience a restoring force F = –kx and must therefore perform SHM when released. 537 | Qatar science scheme of work | Grade 12 advanced | Unit 12AP.4 | Physics 4 © Education Institute 2005 Objectives Possible teaching activities Notes School resources Point out that, if this restoring force is the only force acting on an oscillator, it will perform socalled free oscillations. (Later in this unit there are examples of oscillations where other forces act.) Ask students to deduce an expression for the acceleration of a mass m suspended from a spring of stiffness k (i.e. a = –kx ⁄ m). Then compare the two acceleration equations and identify 2 = k ⁄ m. Establish that the angular frequency is = (k ⁄ m) and hence the period and frequency of the SHM can be deduced. Emphasise that any system in which there is a restoring force proportional to displacement will oscillate with SHM, and that the angular frequency can always be deduced from the relationship between displacement and acceleration. Provide several examples that allow students to practise relating frequency and period of SHM to physical parameters of an oscillator. Ask students to work in pairs to measure k for a spring and hence predict the frequency of oscillations performed by a known mass m attached to the spring. Then ask them to determine the frequency of the SHM performed by the mass suspended from the spring and to compare their result with their prediction. Enquiry skills 12A.1.1, 12A.1.2, 12A.4.1, 12A.4.2 On the board or OHP, show a large diagram of a simple pendulum, length l, displaced through a small angle . By resolving forces into components show that, provided is small and sin , the restoring force is proportional to displacement and hence the pendulum will perform SHM with angular frequency = (g ⁄ l) where g is the gravitational field strength. Mathematics: A knowledge of the small angle approximation is required. Then ask students to work in pairs to determine a value of g by timing oscillations of a pendulum. They should consider how best to design their experiment and process their results in order to ensure precision and accuracy and should quantify the uncertainty in their final result. Tell them to produce a brief written report of their work. 2 hours Energy and SHM Energy in oscillations Set up a demonstration of a dynamics trolley oscillating horizontally between two springs. Ask the class to describe, qualitatively, the energy of the system. By suitable questioning, establish that, when displacement is maximum, kinetic energy is zero and potential energy is maximum, while at the mid-point of the oscillation kinetic energy is maximum and potential energy is zero. Students should be able to draw on their previous experience of energy transformation and conservation, and appreciate that, provided there is minimal dissipation to the surroundings, the total energy of the oscillator remains constant. ... Describe the changes between kinetic and potential energy during the [simple harmonic] motion. Describe and explain practical examples of critically and non-critically damped oscillations. This activity also relates to Standard 10A.25.2 Enquiry skills 12A.1.1, 12A.1.3, 12A.1.5 Ask students to speculate about the likely shapes of graphs showing how the kinetic (or potential) energy of the oscillator varies with time and with position. Ask a few students to draw rough sketches on the board showing their ideas, and to explain their thinking to the rest of the class. Some students might use equations of SHM to deduce the shapes of the graphs, while others might reason qualitatively. 538 | Qatar science scheme of work | Grade 12 advanced | Unit 12AP.4 | Physics 4 © Education Institute 2005 Objectives Possible teaching activities Notes School resources Show, on the board or OHP, how to manipulate the equations of SHM to derive expressions for the kinetic energy of the oscillating trolley, i.e. Ek = ½ mv2 = ½ mA22 sin2( t ) = ½ kA2 sin2( t ) Then get students to work in pairs to deduce similar expressions for potential energy, starting from the expression for the potential energy in a stretched spring: Ep = ½ kx2. Check that students have a correct record of the equations for Ek and Ep, then use them to establish that the total energy of the oscillator is ½ kA2. Provide plenty of numerical and algebraic examples that allow students to practise using the energy equations for SHM. Damped oscillations Ask each pair of students to suspend a mass from a spring and observe its oscillations, first in air and then with the mass immersed in a beaker of water. Tell them to describe their observations as fully as possible. Prompt them with questions such as ‘What is the period of oscillation in each case?’ and ‘How many cycles take place before the oscillations cease?’. It is important to ensure that students appreciate that the damping being considered here is not damping due to water. Discuss students’ observations with the whole class. Establish that drag forces between the mass and the surrounding air or water always act to reduce the speed of the moving mass and dissipate energy. • producing a note on a guitar string; Introduce the term damping. Establish that damping reduces the energy and hence the amplitude of the oscillations but has little effect on the frequency. • oscillation of a car as it drives along a rough road surface; Divide students into small groups and ask them to hold a brief brainstorming session. They should first try to think of examples of everyday examples of oscillations and then, for each example, say whether damping of the oscillations is desirable or not. Ask a representative of each group to write their examples in two lists on the board or OHP. Discuss some of the examples in which damping is desirable with the whole class, and ask them to suggest how it is achieved in practice. Show students the construction of a shock absorber from a car – a piston immersed in oil ensures that oscillations are damped and that the car’s occupants have a comfortable ride. Choose a suitable example to illustrate the meaning of critical damping. This is best defined by comparison with under-damping and over-damping. Examples where damping is undesirable include: • using a pendulum to regulate a clock. Examples where damping is desirable include: • earthquake-induced oscillation of a building. Critical damping can be explained with reference to a swing door. If under-damped, the door swings to and fro many times when released, but if over-damped it is difficult to open and takes a long time slowly to swing shut. If the door is well designed, so that its oscillations are critically damped, it will easily swing shut but without over-shooting. A useful working definition of critical damping is that it occurs when the oscillator returns to its equilibrium position in a time approximately equal to the period of the undamped oscillation. If the time to return to equilibrium is considerably longer than the period, the oscillator is overdamped, and if the oscillator overshoots the equilibrium position and oscillations persist for several cycles, it is under-damped. Point out that, in most situations where damping is desirable, it is usually critical damping that is required. 539 | Qatar science scheme of work | Grade 12 advanced | Unit 12AP.4 | Physics 4 © Education Institute 2005 Objectives Possible teaching activities 2 hours Forced oscillations Forced oscillations and resonance Remind students of the examples of oscillators they saw at the start of this unit. Point out that the Tacoma Narrows bridge, the buzzer or toothbrush, and the vibrating string are all examples of forced oscillations in which energy is continuously supplied to the system to maintain the oscillations. Contrast these with free oscillations in which a system is disturbed then allowed to move freely under the influence of only its own restoring force and any damping forces. Describe practical examples of forced oscillations and resonance and show how the amplitude of a forced oscillation changes with frequency near to the natural frequency of the system. Describe circumstances in which resonance is desirable and others when it should be avoided. Notes School resources Resonance Demonstrate Barton’s pendulums to show how the frequency of a periodic driving force affects the amplitude of the driven oscillator. Introduce and define the terms natural frequency and driving frequency. Establish that, when an oscillator is driven with a frequency that is close to its own natural frequency, there is a large transfer of energy and the oscillations build up to large amplitude. Introduce the term resonance. Show how damping affects the response of an oscillator to a periodic driving force. (Weighting the paper cones in the Barton’s pendulum demonstration reduces the effect of air resistance on their motion and hence reduces damping.) Point out to more advanced students the relationship between the phases of the pendulums: at resonance, there is a phase difference of a quarter of a cycle between the driver and the driven pendulum. Remind students that they have already seen examples of resonance in an earlier unit when they studied the vibration of strings and air columns. Ask students to work in pairs or small groups to explore the resonance of a vibrating hacksaw blade using the apparatus shown in the schematic diagram. They should plot a graph showing how the amplitude, A, of the forced vibration varies with driving frequency. Safety: Ensure that all parts are fixed firmly together. Wear eye protection. Do not put fingers near the vibrating blade. Enquiry skills: 12A.1.1–12A.1.4, 12A.3.1, 12A.3.2, 12A.3.4, 12A.4.1, 12A.4.2 Discuss the example of the Tacoma Narrows bridge collapse with the whole class. Explain that the gusting wind provided a driving force that matched the bridge’s own natural frequency. (Explain to more advanced students how periodic ‘vortex shedding’ allows a steady wind to produce an oscillatory force on an object.) Divide students into small groups and ask them to suggest other examples of resonance. In each case, they should say whether the effect is desirable or not. If resonance is desirable, they should suggest how it may be brought about. If it is undesirable, they should suggest how it might be reduced. Discuss students’ ideas with the whole class, and be prepared to suggest some examples yourself if they have not thought of many. Establish that resonance can be promoted by ensuring that the natural frequency of oscillation is close to that of the driving force, and by reducing damping. Conversely, resonance can be reduced by adjusting the frequencies so that they are very different, and by increasing the amount of damping. Examples include: • pushing a child on a swing; • a singer shattering a wine glass with a loud note; • tuning a string instrument so that it resonates with a tuning-fork; • vibration of parts of a car while in motion; • vibration of machinery in a factory; • absorption spectroscopy (e.g. infrared spectroscopy used to determine the structure of molecules); • magnetic resonance imaging. 540 | Qatar science scheme of work | Grade 12 advanced | Unit 12AP.4 | Physics 4 © Education Institute 2005 Assessment Examples of assessment tasks and questions Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples to incorporate in the activities. Unit 12AP.4 Notes School resources A mass on a spring performs SHM with amplitude 4 cm and period 1.5 s, starting with a displacement of 4 cm when t = 0. a. Calculate the angular frequency of the motion. b. Draw graphs showing how the displacement, velocity and acceleration of the mass vary with time. A trolley of mass 0.75 kg is tethered between two springs, and a force of 6.2 N produces a displacement of 5.0 cm. The trolley is then released and it performs SHM. a. Calculate the energy transferred to the trolley as it is displaced. b. Calculate the trolley’s kinetic energy as it passes through the mid-point of the oscillation. c. Calculate the maximum speed of the trolley. On a single set of axes, sketch graphs showing how the kinetic, potential and total energy of an oscillator vary with time as it performs one complete cycle of oscillation. Write a short article about damping and resonance. Use at least one everyday example to explain the difference between critically damped and non-critically damped oscillations. Include one example of a situation in which resonance is desirable, and one example of a situation in which it is not. Molecules of hydrogen chloride (HCl) are found to absorb electromagnetic radiation with a wavelength 3.47 × 10–6 m. The radiation makes the molecules oscillate at their own natural frequency. By assuming that the chlorine remains at rest while the hydrogen oscillates as if held by a spring, calculate the stiffness, k, of the interatomic bond. Data: speed of light c = 3.00 × 108 m s–1; mass of hydrogen atom m = 1.67 × 10–27 kg. 541 | Qatar science scheme of work | Grade 12 advanced | Unit 12AP.4 | Physics 4 © Education Institute 2005 542 | Qatar science scheme of work | Grade 12 advanced | Unit 12AP.4 | Physics 4 © Education Institute 2005