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Dimensions in Mechanics Physical quantities have dimensions. These quantities are the basic dimensions: - mass, length, time with dimension symbols M, L, T Other quantities’ dimensions are more complex: Dimensions, Units, and Problem Solving Strategies - [velocity] = length/time = LT-1 - [force] = (mass)(length) /(time)2 = MLT-2 - [any mechanical quantity] = Mα Lβ Tγ 8.01 W01D3 - where α β, and γ can be negative and/or non-integer 1 Base Quantities Name Length Symbol for quantity l 2 SI Base Units: Symbol for dimension L SI base unit meter Time t T second Mass m M kilogram Electric current I I ampere Thermodynamic Temperature T Θ kelvin Amount of substance n N mole Luminous intensity IV J candela 3 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 kg 4 1 Speed of Light In 1983 the General Conference on Weights and Measures defined the speed of light to be the best measured value at that time: c = 299, 792, 458 meters/second This had the effect that length became a derived quantity, but the meter was kept around for practicality Fundamental and Derived Quantities: Dimensions and Units The dimensions of (new) physical quantities follow from the equations that involve them F = ma implies that [Force] = M1 L1 T −2 = M L T −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. 5 Table Problem: Dimensions Solution Table Problem: Dimensions Determine the dimensions of the following mechanical quantitites: 1. [momentum] 2. [pressure] 3. [kinetic energy] 4. [work] 5. [power] 6 Determine the dimensions of the following mechanical quantitites: [momentum] = (mass)(velocity) = M L T -1 [pressure] = [force/area]=M L T-2 / L2 = M L-1 T -2 [kinetic energy] = [(mass)(velocity) 2 ] = M( L T -1 ) 2 = M L2 T -2 [work] = [(force)(distance)] = (M L T -2 )(L) = M L2 T -2 [power] = [(work)/(time)]=M L2 T -2 / T =M L2 T -3 7 8 2 Dimensional Analysis: Strategy 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 Since: [desired quantity] = Mα Lβ Tγ where α β, and γ are known Combine the given quantities correctly so that: Table Problem: Dimensional Analysis The speed of a sail-boat or other craft that does not plane is limited by the wave it makes – it can’t climb uphill over the front of the wave. What is the maximum speed you’d expect? Hint: relevant quantities might be the length l of the boat, the density ρ of the water, and the gravitational acceleration g. [desired quantity] = Mα Lβ Tγ = (given1)X (given2)Y (given3)Z vboat = l X ρ Y - solve for X, Y, Z to match correct dimensions of desired quantity g Z 9 Dimensions of quantities that may describe the maximum speed for boat Name of Quantity Symbol Dimension Maximum speed v LT-1 density ρ ML-3 Gravitational acceleration g LT-2 Length l L 10 Problem Solving Measure understanding in technical and scientific courses Problem solving requires factual and procedural knowledge, knowledge of numerous schema, plus skill in overall problem solving. Schema is loosely defined as a “specific type of problem” such as principal, rate, and interest problems, onedimensional kinematic problems with constant acceleration, etc.. 11 12 3 Understand: Get it in Your Head Four Stages of Attack 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 13 Plan your Approach 14 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 contains the essential elements of the real problem. Examine equations of the models, constraints Is all/enough understanding embodied in Eqs.? count equations and unknowns Build on familiar models 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 15 16 4 Stuck? Review Represent the Problem in New Way a. Does the solution make sense? – Graphical – Pictures with descriptions – Pure verbal – Equations Check units, special cases, scaling. Is your answer reasonably close to a simple estimation? b. If it seems wrong, review the whole process. Could 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? c. If it seems right, review the pattern and models used, note the approximations, 17 Estimation Problems tricky/helpful math steps. 18 Estimations Strategies Identify a set of quantities that can be estimated or calculated. Quantitative estimation of phenomena 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 19 20 5 Concept Question: Estimation Volume of Earth’s Atmosphere What is your best estimate for the volume of the earth’s lower atmosphere that contains two thirds of the air molecules? 1) between 101 and 105 cubic meters 3) between and 1015 5) between and 1025 Volume of atmosphere: Vol = 4π Re 2t (4π )(6 × 106 m) 2 (1× 104 m)=((4 ×1018 m3 ) 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. at STP (Standard Temperature and Pressure): one mole of an ideal gas occupies 22.4 L = 22.4 x 10-3 m3. cubic meters 4) between 1015 and 1020 cubic meters 1020 of atmosphere. Estimate thickness of atmosphere as 10 km. Number of moles in atmosphere = (volume of atmosphere)/(volume of mole) 2) between 105 and 1010 cubic meters 1010 Number of Molecules in Atmosphere Estimation quantity is a scalar: N number of molecules and is related to volume Number of molecules in atmosphere: N = (6 × 1023 molecules/mole)(4 × 1018 m 3 ) / (22.4 × 10−3 m 3 / mole) cubic meters = 1 × 1044 molecules 6) between 1025 and 1030 cubic meters 21 22 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 of the earth is weight of a column of air per area. Patm = 1 x 105 N m-2. mass / area ( Pressure / g ) = (1×105 N m-2 ) /(10 m s-2 ) = 104 kg m-2 Total mass equals surface area of earth times mass per square meter. 2 mass = (mass / area)(4π Rearth ) (104 kg m-2 )(4π )(6 × 106 m)2 = (4 × 1018 kg) Number of moles in atmosphere = (mass of atmosphere)/(mass per mole) -3 -1 mass per mole (30 ×10 kg mole ) # moles (4 ×1018 kg)/(30 ×10-3 kg mole-1 ) 1.3 ×1020 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. N = (6 × 1023 molecules/mole)(1.3 × 1020 mole) = 1× 1044 molecules 23 6