1Test One:Review Topics8.01W05D22Dimensions, Units, and Problem Solving Strategies8.01W01D33Dimensions in Mechanics :Base QuantitiesampereIIElectric currentkelvinΘTThermodynamic TemperaturemoleNnAmount of substancecandelaJIVLuminous intensityMTLSymbol for dimensionsecondtTimemeterlLengthkilogrammMassSI base unitSymbol for quantityName4SI Base Units:Second: The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.Meter: The meter (m) is now defined as the distance traveled by a light wave in vacuum in 1/299,792,458 seconds.Mass:The SI standard of mass is a platinum-iridium cylinder assigned a mass of one 1 kg5Speed of LightIn 1983 the General Conference on Weights and Measures defined the speed of light to be the best measured value at that time:This had the effect that length became a derived quantity, but the meter was kept around for practicalityHowever, it is impossible to measure the speed of light these days!c = 299,792,458 meters/second6Problem Solving Strategy: Dimensional AnalysisWhen trying to find a dimensional correct formula for a quantity from a set of given quantities, an answer that is dimensionally correct will scale properly and is generally off by a constant of order unitySince: [desired quantity] = MαLβTγwhere αβ, and γ are knownCombine the given quantities correctly so that:[desired quantity] = MαLβTγ= (given1)X (given2)Y(given3)Z- solve for X, Y, Z to match correct dimensions of desired quantity7Problem Solving: Four Stages of Attack1. Understand the Problem and Models2. Plan your Approach – Models and Schema3. Execute your plan (does it work?)4. Review - does answer make sense?- return to plan if necessary8Understand: Get it in Your HeadWhat concepts are involved? Represent problem - Draw pictures, graphs, storylines…Similarity to previous problems?What known models/physical principles are involved?e.g. motion with constant acceleartion; two bodiesFind special features, constraintse.g. different acceleration before and after some instant in time9Plan your ApproachModel: Real life contains great complexity, so in physics you actually solve a model problem that contains the essential elements of the real problem.Build on familiar modelsLessons from previous similar problemsSelect your system, Pick coordinates to your advantageAre there constraints, given conditions?10Execute the PlanFrom general model to specific equationsExamine equations of the models, constraintsIs all/enough understanding embodied in Eqs.?count equations and unknownssimplest way through the algebraSolve analytically (numbers later)Keep notation simple (substitute later)Keep track of where you are in your planCheck that intermediate results make sense11Stuck?Represent the Problem in New Way– Graphical– Pictures with descriptions– Pure verbal– EquationsCould You Solve it if…– the problem were simplified?– you knew some other fact/relationship?– You could solve any part of problem, even a simple one?12Reviewa. Does the solution make sense?Check units, special cases, scaling. Is your answer reasonably close to a simple estimation?b. If it seems wrong, review the whole process.c. If it seems right, review the pattern and models used,note the approximations,tricky/helpful math steps.13Problem Solving Strategy: EstimationIdentify a set of quantities that can be estimated or calculated.What type of quantity is being estimated?How is that quantity related to other quantities, which can be estimated more accurately? Establish an approximate or exact relationship between these quantities and the quantity to be estimated in the problem14Kinematics and One Dimensional Motion 8.01W02D115Coordinate System in One DimensionUsed to describe the position of a point in spaceA coordinate system consists of: 1. An origin at a particular point in space2. A set of coordinate axes with scales and labels3. Choice of positive direction for each axis: unit vectors 4. Choice of type: Cartesian or Polar or SphericalExample: Cartesian One-Dimensional Coordinate System16One-Dimensional Kinematics: SummaryPosition:Displacement:Average velocity:Instantaneous VelocityAverage AccelerationInstantaneous Accelerationˆ() ()txt=xiˆ()xtΔ≡Δriˆˆ() ()xxtvttΔ≡=Δvii0() limxtxdxvttdtΔ→Δ≡≡ΔˆˆxxvatΔ≡=Δaii0ˆˆ() limxxtvdvttdtΔ→Δ=≡Δaii17Instantaneous Velocityx-component of the velocity is equal to the slope of the tangent line of the graph of x-component of position vs. time at time t18Instantaneous Acceleration()()00 0ˆˆ ˆ ˆˆ() () lim lim limxxxxxxtt tvt t vtvdvtat attdtΔ→ Δ→ Δ→+Δ −Δ=≡ = = ≡ΔΔaii i iiThe x-component of acceleration is equal to the slope of the tangent line of the graph of the x-component of the velocity vs. time at time tax(t) =dvxdt19Summary: Constant AccelerationAccelerationVelocityPosition vx(t)= vx,0+axt20,01()2xxxt x v t at=+ +ax= constant20Problem Solving Strategies: One-Dimensional Kinematics21I. Understand – get a conceptual grasp of the problemQuestion 1: How many objects are involved in the problem? Question 2: How many different stages of motion occur?Question 3: For each object, how many independent directions are needed to describe the motion of that object? Question 4: What choice of coordinate system best suits the problem?Question 5: What information can you infer from the problem?22II. Devise a PlanSketch the problemChoose a coordinate systemWrite down the complete set of equations for the position and velocity functions; identify any specified quantities; clean up the equations.Finding the “missing links”: count the number of independent equations and the number of unknowns.You can solve a system of n independent equations if you have exactly n unknowns. Look for constraint conditions23III. SolveDesign a strategy for solving a system of equations.Check your algebra and dimensions.Substitute in numbers.Check your results and units.24IV. ReviewCheck your results, do they make sense (think!)Check limits of an algebraic expression (be creative)Think about how to extend model to cover more general cases (thinking outside the box)Solved problems act as models for thinking about new problems. (Mechanics provides a foundation of of solved problems.)25Non-Uniform Acceleration,Vectors,Kinematics in Two-Dimensions8.01 W02D226Summary: Time Dependent AccelerationAcceleration is a
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