DOC

unit_magnetism - ISLE Physics Network

By Anne Gonzalez,2014-11-12 08:05
14 views 0
unit_magnetism - ISLE Physics Network

    LUIS FERNANDO LARGO-GUTIÉRREZ

    MAGNETISM

    UNIT PLAN

    NJ STANDARDS

     5.1.12 A. Habits of mind

     5.1.12 B. Inquiry and problem solving

     5.1.12 C. Safety

     5.2.12 A. Cultural contributions

     5.2.12 B. Historical perspectives

     5.3.12 A. Numerical operations

     5.3.12 C. Patterns and algebra

     5.3.12 D. Data analysis and probability

     5.4.12 C. Technological design

    WHAT STUDENTS SHOULD KNOW TO BE SUCCESSFUL Prior to this unit students should have knowledge to be successful as it shows in the follow:

    ; Kinematics

    ; Newtonian Dynamics

    ; Circular Motion

    ; Electric charge: Force and Energy

    ; Electric fields

    OBJECTIVES

    CONCEPTUAL GOALS

    FUNDAMENTAL IDEAS THAT STUDENTS NEED TO CONSTRUCT AT THE END OF THE UNIT.

    GOAL ASSESSMENT Be able to explain why we believe that magnetic

    interactions are different from electrostatic Homework, Exam, Laboratory

    interactions.

     Concept building, Homework, Exam, Laboratory Understand that a magnetic field interacts with

    moving electrically charged particles and wires

    with electric currents.

     Concept building, Homework, Laboratory Understand the difference between sources of

    magnetic field and test objects in a magnetic field.

     Concept building, Homework, Laboratory Learn how to describe magnetic interactions

    quantitatively.

    QUANTITATIVE GOALS

    PHYSICAL QUANTITIES AND MATHEMATICAL RELATIONSHIPS THAT STUDENTS NEED TO UNDERSTAND, REPRESENT IN DIFFERENT WAYS AND BE ABLE TO APPLY PRODUCTIVELY.

     Magnetic Field: A physical quantity characterizes the direction and strength (magnitude of the magnetic field B

    at a point. Qualitatively, the direction of the field at that point is the direction in which the north pole of a

    compass needle points at that point

     Magnetic Field Lines: Magnetic field lines represent the magnetic field created by a magnetic field source. The

    direction of the magnetic field at a point is tangent to the direction of the magnetic field lime at or near that

    point. The magnitude of the field is proportional to the separation of the lines in that region, the more closely

    they are spaces, the stronger the field.

Magnetic Force on a current-carrying wire: the force that the field exerts on the current-carrying wire is B

    perpendicular to both the direction of the field and to the direction of the current. The magnitude of the B

    magnetic force F that a magnetic field exerts on a wire of length L with an electric current I flowing through Bm

    it is: , where is the angle between the directions of the field and the current. The direction of FBILsinm

    the force is given by right hand rule # 1.

     Right Hand Rule # 1 Direction of the magnetic force: point the fingers of F m

    your open right hand in the direction of the magnetic field. Orient your hand so that

    your right thumb points along the direction of the current. Then the direction of the

    magnetic force on the current carrying wire is the direction in which your open palm Bwould push perpendicular to both the direction of the current and the direction of the

    Ifield. B

     Magnetic force on a charged particle: the magnitude of the force that a magnetic field exerts on a particle

    with electric charge q moving at velocity in a magnetic field is: , where is the angle vBFqvBsinm

    between the direction of and the direction of . Use the right hand rule # 1 to find the direction of the force. vB

    This time put your thumb in the direction of the velocity instead of the direction of the electric current. The

    direction of the force is perpendicular to the plane in which and are located. The direction of the force on a vB

    negatively charged particle is in the opposite direction.

     Magnetic torque on a current carrying coil: a magnetic field can exert a torque ( on a coil of wire. The B

    (NBAIsinmagnitude of the torque is: , where N is the number of loops in the coil, I is the electric current

    in the coil, B is the magnitude of the magnetic field, A is the area of the coil, and is the angle between the coil’s

    area vector and the direction of the magnetic field.

    I Right Hand Rule # 2: to find the direction of the magnetic field lines caused by a

    current carrying wire, imagine that you grasp the current carrying wire with your right

    hand. Orient the hand so that your thumb points in the direction of the current. Your F mfour fingers will wrap around the wire in the direction of the magnetic field lines.

     B

     PROCEDURAL GOALS

    SPECIFIC SCIENCE SKILLS THAT STUDENTS NEED TO BUILD IN THIS UNIT AND THE SCIENTIFIC ABILITIES THAT YOU

    WANT YOUR STUDENTS TO STRENGTH OR DEVELOP.

    GOAL ASSESSMENT

     Be able to analyze qualitative information and data to devise Concept building, Homework, Exam,

    a rule for the direction and magnitude of a magnetic force. Laboratory

     Be able to find the direction of a magnetic field created by a Concept building, Homework, Exam,

    current-carrying wire at any given point. Laboratory

     Be able to evaluate somebody else’s reasoning. Concept building, Homework, Exam,

     Laboratory

    SCIENTIFIC ABILITIES

     Is able to evaluate the consistency of different Concept building, Homework, Exam,

    representations and modify then when necessary Laboratory

     Is able to use representations to solve problems Homework, Exam, Laboratory

     Is able to decide what is to be measured and identify Homework, Exam, Laboratory

    independent and dependent variables.

     Is able to use equipment to make measurements. Laboratory

     Is able to identify the shortcomings in an experimental design Laboratory

    and suggest improvements.

     Is able to identify the relationship or explanation to be tested Laboratory

     Is able to make a reasonable judgment about the relationship Laboratory

    or explanation. Is able to identify the assumptions made in using the Laboratory, Homework

    mathematical procedure. Is able to determine specifically the way in which Laboratory

    assumptions might affect the results. Is able to communicate the details of an experimental Laboratory

    procedure clearly and completely. Is able to identify sources of experimental uncertainty. Laboratory Is able to evaluate specifically how experimental uncertainties Laboratory

    may affect the data.

    EPISTEMOLOGICAL GOALS

    UNDERSTANDING OF THE CONSTRUCTION OF KNOWLEDGE THAT YOU WANT YOUR STUDENTS TO ACQUIRE. How do you know right hand rule?

     Why do you believe that magnetic interactions are different from electric interactions?

     What is the difference between field vectors and field lines?

    LENGTH TOTAL AND SUBUNITS

     CLASS 1. Magnetic Fields

    o Class activities

    ; Students work with ALG 17.1.1

    ; Students work with ALG 17.1.2

    ; Students work with ALG 17.1.3

    ; Students work with ALG 17.1.4

    o Homework

    ; ALG 17.1.5

    ; Problems from text book

     CLASS 2. Magnetic Fields & Right Hand Rule # 1

    o Class activities

    ; Students work with ALG 17.1.6

    ; Students work with ALG 17.1.8

    ; Students work with ALG 17.1.10

    ; Students work with ALG 17.1.11

    ; Discussion right hand rule # 1

    ; Watch this video: Click here (Physics Videos)

    o Homework

    ; Problems from text book

     CLASS 3. Magnetic Force on a moving charge & Right Hand Rule # 2

    o Class activities

    ; Students work with ALG 17.2.1

    ; Students work with ALG 17.2.3

    ; Discussion right hand rule # 2

    ; Students work with ALG 17.2.4

    o Homework

    ; Finish ALG 17.2.1

    ; ALG 17.2.2

    ; ALG 17.2.5

     CLASS 4. Magnetic force on a current-carrying wire

    o Class activities

    ; Students work with ALG 17.2.6.

    ; Students work with ALG 17.2.8.

    ; Students work with ALG 17.2.9.

    o Homework

    ; ALG 17.2.7

    ; Finish ALG 17.2.9

    ; ALG 17.2.10

     CLASS 5. Mathematical representation for magnetic forces

    o Class activities

    ; Students work with ALG 17.3.1

    ; Students work with ALG 17.3.2

    ; Watch this video: Click here (Youtube Video)

    ; Students work with ALG 17.4.1

    o Homework

    ; Problems from text book

     LABORATORY

    ; Watch this video: Click here (Physics Video)

    ; Watch this video: Click here (Physics Video)

     CLASS 6. Quantitative reasoning

    o Class activities

    ; Students work with ALG 17.4.2

    ; Students work with ALG 17.4.5

    ; Students work with ALG 17.4.6

    ; Students work with ALG 17.4.7

    o Homework

    ; ALG 17.4.3

    ; ALG 17.4.4

     CLASS 7. Quantitative reasoning

    o Class activities

    ; Students work with ALG 17.4.8

    ; Students work with ALG 17.4.9

    ; Students work with ALG 17.4.10

    ; Students work with ALG 17.4.11

    ; Watch this video: click here (Youtube Video)

    ; Watch this video: click here (Youtube Video)

    o Homework

    ; Problems from text book

     CLASS 8. Review

    o Review Session

     CLASS 9. Exam

    o Summative Assessment

    STUDENT DIFFICULTIES

     Students might think that magnetic poles are the same as electric charges.

     Students might think that magnetic forces are the same as electric forces. Students might think that magnetic fields are “magnetic force fields”. Students might find difficult to figure out what the direction of the current is.

     Students might find difficult to figure out the direction of the magnetic field is.

     Students might find difficult to figure out hot to set up right hand rule number one

     Students might choose the wrong system

     Students might find difficult to translate Newton’s 2nd and 3rd law into magnetism.

RELEVANCE TO STUDENTS’ LIVES

Magnets are used in every day’s life when:

     Using a computer, a hard drive relies on magnets to store data, and some monitors use magnets to

    create images on the screen.

     Doorbells, uses an electromagnet to drive a noisemaker.

     Magnets are also vital components in CRT televisions, speakers, microphones, generators, transformers,

    electric motors, burglar alarms, cassette tapes, compasses and car speedometers.

     Magnets have numerous amazing properties. They can induce current in wire and supply torque for

    electric motors.

     A strong enough magnetic field can levitate small objects or even small animals.

     Maglev trains use magnetic propulsion to travel at high speeds.

     Magnetic fluids help fill rocket engines with fuel.

     The Earth's magnetic field, known as the magnetosphere, protects it from the solar wind.

     Magnetic Resonance Imaging (MRI) machines use magnetic fields to allow doctors to examine patients'

    internal organs. Doctors also use pulsed electromagnetic fields to treat broken bones that have not

    healed correctly. This method, can mend bones that have not responded to other treatment. Similar

    pulses of electromagnetic energy may help prevent bone and muscle loss in astronauts who are in zero-

    gravity environments for extended periods.

TEXTS AND MATERIALS PLANNED TO USE IN THE UNIT

     1Magnetic levitation transport, or maglev, is a form of transportation that suspends guides and propels vehicles via electromagnetic force. This method can be faster than wheeled mass transit systems, potentially reaching velocities

    comparable to turboprop and jet aircraft (500 to 580 km/h).

    The world's first commercial application of a high-speed maglev line is the IOS (initial operating segment) demonstration line in Shanghai, China that transports people 30 km (18.6 miles) to the airport in just 7 minutes 20 seconds (top speed of 431 km/h or 268 mph, average speed 250 km/h or 150 mph). Other maglev projects worldwide are being studied for feasibility. However, scientific, economic and political barriers and limitations have hindered the widespread adoption of the technology.

    Maglev technology has minimal overlap with wheeled train technology and is not compatible with conventional railroad tracks. Because they cannot share existing infrastructure, maglevs must be designed as complete transportation

    systems. The term "maglev" refers not only to the vehicles, but to the railway system as well, specifically designed for magnetic levitation and propulsion.

Technology

    There are two primary types of maglev technology:

     Electromagnetic suspension (EMS) uses the attractive magnetic force of a magnet beneath a rail to lift the train

    up.

     Electrodynamic suspension (EDS) uses a repulsive force between two magnetic fields to push the train away

    from the rail.

     1 http://en.wikipedia.org/wiki/Maglev_train

Electromagnetic suspension

    In current EMS systems, the train levitates above a steel rail while electromagnets, attached to the train, are oriented toward the rail from below. The electromagnets use feedback control to maintain a train at a constant distance from a

    track.

Electrodynamic suspension

    In Electrodynamic suspension (EDS), both the rail and the train exert a

    magnetic field, and the train is levitated by the repulsive force between these

    magnetic fields. The magnetic field in the train is produced by either

    superconducting electromagnets (as in JR-Maglev) or by an array of

    permanent magnets (as in Inductrack). The repulsive force in the track is

    created by an induced magnetic field in wires or other conducting strips in

    the track.

At slow speeds, the current induced in these coils and the resultant magnetic flux is not large enough to support the

    weight of the train. For this reason the train must have wheels or some other form of landing gear to support the train

    until it reaches a speed that can sustain levitation.

Propulsion coils on the guideway are used to exert a force on the magnets in the train and make the train move

    forwards. The propulsion coils that exert a force on the train are effectively a linear motor: An alternating current flowing through the coils generates a continuously varying magnetic field that moves forward along the track. The

    magnets on the train line up with this field, and the train moves.

Pros and cons of different technologies

    Each implementation of the magnetic levitation principle for train-type travel involves advantages and disadvantages. Time will tell as to which principle, and whose implementation, wins out commercially.

    TECHNOLOGY PROS CONS

    Magnetic fields inside and outside the The separation between the vehicle and

    vehicle are insignificant; proven, the guideway must be constantly

    commercially available technology that monitored and corrected by computer EMS (Electromagnetic) can attain very high speeds (500 km/h); systems to avoid collision due to the

    no wheels or secondary propulsion unstable nature of electromagnetic

    system needed attraction.

    Powerful onboard superconducting Strong magnetic fields onboard the train

    magnets and large margin between rail make the train inaccessible to passengers

    and train enable highest recorded train with pacemakers or magnetic data

    speeds (581 km/h) and heavy load storage media such as hard drives and Superconducting EDS capacity; has recently demonstrated (Dec credit cards, necessitating the use of (Electrodynamic) 2005) successful operations using high magnetic shielding; vehicle must be

    temperature superconductors (HTS) in its wheeled for travel at low speeds; system

    onboard magnets, cooled with per mile cost still considered prohibitive;

    inexpensive liquid nitrogen the system is not yet out of prototype

    phase.

    Failsafe Suspension - no power required Requires either wheels or track segments

    to activate magnets; Magnetic field is that move for when the vehicle is

    localized below the car, can generate stopped. New technology that is still

    enough force at low speeds (around 5 under development (as of 2006) and has Inductrack System (Permanent km/h) to levitate maglev train; in case of as yet no commercial version or full scale Magnet EDS) power failure cars slow down on their system prototype.

    own in a safe, steady and predictable

    manner before coming to a stop; Halbach

    arrays of permanent magnets may prove

    more cost-effective than electromagnets

    Neither Inductrack nor the Superconducting EDS are able to levitate vehicles at a standstill, although Inductrack provides levitation down to a much lower speed. Wheels are required for both systems. EMS systems are wheel-less.

    The German Transrapid, Japanese HSST (Linimo), and Korean Rotem EMS maglevs levitate at a standstill, with electricity extracted from guideway using power rails for the latter two, and wirelessly for Transrapid. If guideway power is lost on the move, the Transrapid is still able to generate levitation down to 10 km/h speed, using the power from onboard batteries. This is not the case with the HSST and Rotem systems.

PROPULSION

    An EMS system can provide both levitation and propulsion using an onboard linear motor. EDS systems can only levitate

    the train using the magnets onboard, not propel it forward. As such, vehicles need some other technology for

    propulsion. A linear motor (propulsion coils) mounted in the track is one solution. Over long distances where the cost of propulsion coils could be prohibitive, a propeller or jet engine could be used.

STABILITY

    Static magnetic bearings using only electromagnets and permagnets are unstable, as explained by Earnshaw's theorem.

    EMS systems rely on active electronic stabilization. Such systems constantly measure the bearing distance and adjust

    the electromagnet current accordingly. As all EDS systems are moving systems (i.e. no EDS system can levitate the train unless it is in motion), Earnshaw's theorem does not apply to them.

PROS AND CONS OF MAGLEV VS. CONVENTIONAL TRAINS

    Due to the lack of physical contact between the track and the vehicle, there is no rolling friction, leaving only air

    resistance (although maglev trains also experience electromagnetic drag, this is relatively small at high speeds).

    Maglevs can handle high volumes of passengers per hour (comparable to airports or eight-lane highways) and do it

    without introducing air pollution along the right of way. Of course, the electricity has to be generated somewhere, so

    the overall environmental impact of a maglev system is dependent on the nature of the grid power source.

    The weight of the large electromagnets in EMS and EDS designs are a major design issue. A very strong magnetic field is required to levitate a massive train. For this reason one research path is using superconductors to improve the efficiency of the electromagnets.

    Due to its high speed and shape, the noise generated by a maglev train is similar to a jet aircraft, and is considerably more disturbing than standard steel on steel intercity train noise. A study found the difference between disturbance levels of maglev and traditional trains to be 5dB (about 78% noisier).

ECONOMICS

    The Shanghai maglev cost 9.93 billion yuan (US$1.2 billion) to build. This total includes infrastructure capital costs such as manufacturing and construction facilities, and operational training. At 50 yuan per passenger and the current 7,000

    passengers per day, income from the system is incapable of recouping the capital costs (including interest on financing) over the expected lifetime of the system, even ignoring operating costs.

    China aims to limit the cost of future construction extending the maglev line to approximately 200 million yuan (US$24.6 million) per kilometer. These costs compare competitively with airport construction (e.g., Hong Kong Airport cost US$20

    billion to build in 1998) and eight-lane Interstate highway systems that cost around US$50 million per mile in the US.

    While high-speed maglevs are expensive to build, they are less expensive to operate and maintain than traditional high-

    speed trains, planes or intercity buses. Data from the Shanghai maglev project indicates that operation and maintenance costs are covered by the current relatively low volume of 7,000 passengers per day. Passenger volumes on the Pudong International Airport line are expected to rise dramatically once the line is extended from Longyang Road metro station all the way to Shanghai's downtown train depot.

The proposed Chūō Shinkansen line is estimated to cost approximately US$82 billion to build.

    The only low-speed maglev (100 km/h) currently operational, the Japanese Linimo HSST, cost approximately US$100

    million/km to build. Besides offering improved O&M costs over other transit systems, these low-speed maglevs provide

    ultra-high levels of operational reliability and introduce little noise and zero air pollution into dense urban settings.

    As maglev systems are deployed around the world, experts expect construction costs to drop as new construction methods are perfected.

LABORATORY

    1. TESTING EXPERIMENT: FORCE ON A MAGNET DUE TO A CURRENT CARRYING WIRE (RIGHT HAND RULE #1)

    Think of how you can use this equipment to test the right hand rule for the direction of the force exerted by a magnetic field on a current carrying wire. Also, think of what physical quantities you could determine using the scale.

     A3, A5, C1, C2, C3, C4 RUBRICS:

    A horseshoe magnet whose poles are known (Red: North, White: South), a scale, a wire through which a Available equipment:

    current can flow, a voltage source, and connecting wires.

Warning: Do not leave the voltage source on after you finish the measurements.

    a) First, recall the right hand rule for the magnetic force. Write what quantities it relates and express it with a picture or using

    words. Consider the available equipment and how you could use it to achieve the goal of the experiment. Brainstorm and write

    down your ideas including what you could measure and sketches of possible experimental setups).

    b) Describe your procedure. The description should contain a labeled sketch of your experimental set-up, an outline of what you

    plan to do, what you will measure, how you will measure it. Explain how you will use the reading of the scale to determine the

    force exerted by the wire on the magnet, and the force exerted by the magnet on the wire. To help, use free-body diagram(s)

    and Newton’s second and third laws.

    c) Make a qualitative prediction for the reading of the scale (more than some value, less than some value) for your particular

    arrangement. Show the reasoning used to make the prediction with free-body diagrams. Call your TA over once you have done

    this but before you turn on the current. Then perform the experiment and record the outcome.

    d) Did the outcome match your prediction? If not, list possible reasons.

    e) Based on your prediction and the experimental outcome, make a judgment about the right-hand rule.

2. TESTING EXPERIMENT: DOES A COIL BEHAVE LIKE A MAGNET?

    Your friend Jim has an idea that a coil of wire with current flowing in it behaves like a bar magnet whose poles can be determined using right hand rule #2. Design an experiment to test his idea.

RUBRICS: A7

    A bar magnet (poles known), a long wire that you can use to make a coil with several turns, alligator clips, AVAILABLE EQUIPMENT:

    a swivel, a voltage source.

    a) First recall what right hand rule #2 says. Decide how you can apply it to determine the shape of the magnetic field of a current

    carrying coil.

    b) Design an experiment to test Jim’ idea. Describe your procedure and draw a sketch of your experimental setup.

    c) Using Jim’ idea, make a prediction of the outcome of the experiment. Explain the reasoning used to make the prediction in

    detail. What assumption do you need to make the prediction?

    d) Use the wire to make a coil with several turns. Conduct the experiment and record the outcome.

    e) Did the outcome match the prediction?

    f) What is your judgment about Jim’ idea?

3. WHY DID WE DO THIS LAB?

     What was the purpose of using free-body diagrams in this lab? Describe the instances when the diagrams helped you make

    decisions related to the collection of your data and the analysis.

     Why was it important to consider the assumption you made in experiment III?

RUBRIC A

Ability to represent information in multiple ways

    Needs some Scientific Ability Missing Inadequate Adequate improvement Representations created At least one representation agree with each other but Is able to evaluate the is made but there are major may have slight All representations, both consistency of different No representation is made A3 discrepancies between the discrepancies with the given created and given, are in representations and modify to evaluate the consistency. constructed representation representation. Can be seen agreement with each other. them when necessary and the given one. that modifications were made to a representation.

    Representations students can make

    Needs some Scientific Ability Missing Inadequate Adequate improvement FBD is constructed but FBD contains no errors in contains major errors such vectors but lacks a key The diagram contains no as incorrect mislabeled or feature such as labels of errors and each force is No representation is not labeled force vectors, A5 Free-Body Diagram forces with two subscripts labeled so that it is clearly constructed. length of vectors, wrong or vectors are not drawn understood what each force direction, extra incorrect from single point or axes represents. vectors are added, or are missing. vectors are missing. Picture is drawn but it is incomplete with no physical Picture has no incorrect quantities labeled, or information but has either Picture contains all key No representation is important information is no or very few labels of A7 Picture items with the majority of constructed. missing, or it contains a given quantities. Majority of labels present. wrong information, or key items are drawn in the coordinate axes are picture. missing. RUBRIC C

    Ability to design and conduct a testing experiment (testing an idea/hypothesis/explanation or mathematical relation)

    Needs some Scientific Ability Missing Inadequate Adequate improvement

    An attempt is made to The hypothesis to be identify the hypothesis tested is described but Is able to identify the No mention is made of The hypothesis is clearly C1 to be tested but is there are minor hypothesis to be tested a hypothesis. stated. described in a confusing omissions or vague manner. details.

    The experiment tests the The experiment tests the The experiment tests the hypothesis, but due to hypothesis, but due to hypothesis and has a Is able to design a the nature of the design The experiment does the nature of the design high likelihood of C2 reliable experiment that there is a moderate not test the hypothesis. it is likely the data will producing data that will tests the hypothesis chance the data will lead lead to an incorrect lead to a conclusive to an inconclusive judgment. judgment. judgment.

    A prediction is made and A prediction is made, is No prediction is made. is distinct from the distinct from the Is able to distinguish A prediction is made but The experiment is not hypothesis but does not hypothesis, and C3 between a hypothesis it is identical to the treated as a testing describe the outcome of describes the outcome and a prediction hypothesis. experiment. the designed of the designed experiment. experiment

    A prediction is made that A prediction is made that A prediction is made that Is able to make a follows from the No attempt to make a is distinct from the follows from the C4 reasonable prediction hypothesis and prediction is made. hypothesis but is not hypothesis but does not based on a hypothesis incorporates based on it. incorporate assumptions assumptions.

FINAL SUMMATIVE ASSESSMENT

    1. Two parallel wires carry electric current in the same direction. Does the moving charge in one wire cause a

    magnetic force to be exerted on the moving charge in the other wire: If so, in what direction is the force relative

    to the wires? Explain. Repeat for currents moving in opposite directions.

    2. Which of the following pictures best represents the magnetic field lines created by a wire with a current flowing in

    it? The current is flowing into the page.

    A B C D E

    None of these

    IIBBII

    BB

    Solution

Use right hand rule # 2

Answer is A.

    3. A positively charged particle is at rest in a plane above and between two bars magnets, as shown in the figure.

    Which choice below best represents the resulting magnetic force exerted by the magnets on the charged particle?

    SNSN

    A B C D E

     Zero

Solution

FqvBsinm

    mv0 s

    F0Nm

Answer is D.

     -274. An electron moves in a circle perpendicular to a 2.2x10 T magnetic field. If the electron’s speed is 1.5x10 m/s,

    what is the radius of the circle?

    A B C D E

     -3-3-2-3-42.2x10 m 3.9x10 m 1.5x10 m 1.5x10 m 3.9x10 m

Solution

Report this document

For any questions or suggestions please email
cust-service@docsford.com