Download GRADE 12A: Physics 4

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 = r2 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 = r2 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 = r2.
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 = –r2 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
 = 2f 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 mA22 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