Dimensions in Mechanics Physical quantities have dimensions. These quantities are the basic dimensions: Dimensions, Units, and - mass, length, time with dimension symbols M, L, T Problem Solving Strategies Other quantities’ dimensions are more complex: - [velocity] = length/time = LT-1 8.01 - [force] = (mass)(length) /(time)2 = MLT-2 W01D3 - [any mechanical quantity] = Mα Lβ Tγ - where αβ, and γ can be negative and/or non-integer

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Base Quantities SI Base Units: Name Symbol for Symbol for SI base quantity dimension unit Second: The second is the duration of 9,192,631,770 periods of the radiation Length l L meter corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom. Time t T second

Mass m M kilogram Meter: The meter (m) is now defined as the distance traveled by a light wave in vacuum in 1/299,792,458 seconds. Electric current I I ampere Mass: The SI standard of mass is a Thermodynamic Temperature T Θ kelvin platinum-iridium cylinder assigned a mass of one 1 kg Amount of substance n N mole

Luminous intensity IV J candela 3 4

1 Speed of Light Fundamental and Derived Quantities: Dimensions and Units In 1983 the General Conference on Weights and Measures defined the speed of light to be the best measured value at that time: The dimensions of (new) physical quantities follow from the equations that involve them c = 299, 792,458 meters/second F = ma This had the effect that length became a derived quantity, but the meter was kept around for practicality implies that [Force]== M11 L T−− 2 M LT 2 However, it is impossible to measure the speed of light these days! Since we use force so often, we define new units to measure it: Newtons, Pounds, Dynes, Troy Oz.

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Table Problem: Dimensions Table Problem: Dimensions Determine the dimensions of the following mechanical Solution quantitites: Determine the dimensions of the following mechanical quantitites: 1. [momentum]

-1 2. [pressure] [momentum]== (mass)(velocity) M LT [pressure]== [force/area]=M L T-2 / L 2 M L -1 T -2 3. [kinetic energy] [kinetic energy]== [(mass)(velocity)2-122-2 ] = M( LT ) M L T 4. [work] [work]== [(force)(distance)] = (M L T-2 )(L) M L 2 T -2 5. [power] [power]= [(work)/(time)]=M L2-2 T / T =M L 2-3 T

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2 Dimensional Analysis: Table Problem: Dimensional Strategy Analysis When 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 unity The speed of a sail-boat or other craft that does not plane is limited by the wave it makes – it can’t climb Since: uphill over the front of the wave. What is the maximum speed you’d expect? [desired quantity] = Mα Lβ Tγ where αβ, and γ are known

Combine the given quantities correctly so that: Hint: relevant quantities might be the length l of the boat, the density ρ of the water, and the gravitational [desired quantity] = Mα Lβ Tγ = (given1)X (given2)Y (given3)Z acceleration g. Z - solve for X, Y, Z to match correct dimensions of desired quantity v = X Y boat l ρ g

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Dimensions of quantities that Problem Solving may describe the maximum Measure understanding in technical and scientific speed for boat courses

Name of Quantity Symbol Dimension Problem solving requires factual and procedural knowledge, knowledge of numerous schema, plus skill Maximum speed v LT-1 in overall problem solving. density ρ ML-3 Gravitational Schema is loosely defined as a “specific type of problem” g LT-2 acceleration such as principal, rate, and interest problems, one- dimensional kinematic problems with constant Length l L acceleration, etc..

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3 Four Stages of Attack Understand: Get it in Your Head What concepts are involved? 1. Understand the Problem and Models Represent problem - Draw pictures, graphs, storylines… 2. Plan your Approach – Models and Similarity to previous problems? Schema What known models/physical principles are involved? 3. Execute your plan (does it work?) e.g. motion with constant acceleartion; two bodies 4. Review - does answer make sense? Find special features, constraints e.g. different acceleration before and after some instant - return to plan if necessary in time

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Plan your Approach Execute the Plan

From general model to specific equations Model: Real life contains great complexity, so in physics you actually solve a model problem that Examine equations of the models, constraints contains the essential elements of the real problem. Is all/enough understanding embodied in Eqs.? Build on familiar models count equations and unknowns simplest way through the algebra Lessons from previous similar problems Solve analytically (numbers later) Select your system, Keep notation simple (substitute later) Pick coordinates to your advantage Keep track of where you are in your plan Are there constraints, given conditions? Check that intermediate results make sense

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4 Stuck? Review Represent the Problem in New Way a. Does the solution make sense? –Graphical Check units, special cases, scaling. Is your – Pictures with descriptions – Pure verbal answer reasonably close to a simple – Equations estimation? Could You Solve it if… b. If it seems wrong, review the whole process. – the problem were simplified? c. If it seems right, – you knew some other fact/relationship? review the pattern and models used, – You could solve any part of problem, even a note the approximations, simple one? 17 tricky/helpful math steps. 18

Estimation Problems Estimations Strategies Quantitative estimation of phenomena Identify a set of quantities that can be estimated or calculated. Order of magnitude estimations 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 problem

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5 Number of Molecules in Concept Question: Estimation Atmosphere Estimation quantity is a scalar: N number of molecules and is related to volume Volume of Earth’s Atmosphere of atmosphere. Estimate thickness of atmosphere as 10 km. Volume of atmosphere: Vol=×××4ππ R2624183 t  (4 )(6 10 m) (1 10 m)=((4 10 m ) What is your best estimate for the volume of the earth’s lower e atmosphere that contains two thirds of the air molecules? Number of molecules = (number of molecules in a mole)(number of moles in atmosphere) 1) between 101 and 105 cubic meters Number of moles in atmosphere = (volume of atmosphere)/(volume of mole)

5 10 2) between 10 and 10 cubic meters 23 Number of molecules in a mole NA = 6 x 10 molecules/mole.

3) between 1010 and 1015 cubic meters at STP (Standard Temperature and Pressure): one mole of an ideal gas occupies 22.4 L = 22.4 x 10-3 m3.

4) between 1015 and 1020 cubic meters Number of molecules in atmosphere:

23 18 3 −3 3 5) between 1020 and 1025 cubic meters N = (6 × 10 molecules/mole)(4 × 10 m ) / (22.4 × 10 m /mole) = 1× 1044 molecules 21 22 6) between 1025 and 1030 cubic meters

Molecules in Atmosphere: Pressure Estimation quantity is a scalar: N number of molecules and is related to the mass of the atmosphere. Determine mass per square meter from atmospheric pressure. Atmospheric pressure at the surface 5 -2 of the earth is weight of a column of air per area. Patm = 1 x 10 N m . mass/ area ( Pressure / g )=× (1 105-2-24 N m ) /(10ms ) = 10 kg m -2 Total mass equals surface area of earth times mass per square meter. 24-26218 mass=×=×( mass / area )(4ππ Rearth ) (10 kg m )(4 )(6 10 m) (4 10 kg) Number of moles in atmosphere = (mass of atmosphere)/(mass per mole) mass per mole  (30× 10-3 kg mole -1 ) #moles  (4×× 1018 kg)/(30 10 -3 kg mole -1 ) 1.3 × 10 20 mole Number of molecules = (number of molecules in a mole)(number of moles in atmosphere). Number of molecules in a mole NA = 6 x 1023 molecules/mole. 23 20 44 N =×(6 10 molecules/mole)(1.3 × 10 mole) =× 1 10 molecules 23

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