Dimensional Analysis Notes PDF

Topic 14: Dimensional Analysis Lesson 14.0 Introduction to Dimensional Analysis Lesson 14.1 Dimensional Analysis Lesson 14.2 Common Units Lesson 14.3 Harder Examples  Lesson 14.1 Dimensional Analysis Dimensional Analysis Dimensional Analysis is a way to check formula...

Topic 14: Dimensional Analysis Lesson 14.0 Introduction to Dimensional Analysis Lesson 14.1 Dimensional Analysis Lesson 14.2 Common Units Lesson 14.3 Harder Examples  Lesson 14.1 Dimensional Analysis Dimensional Analysis Dimensional Analysis is a way to check formulae and equations using units (dimensions). SI Base Units Symbol Name Quantity s second time m meter length kg kilogram mass  A radian is a ratio so does not have a unit. Anything that is unitless can be left out during dimensional analysis. The coefficients of friction (μ) and restitution (e) are also ratios, and can also be ignored. Trigonometric Ratios and Logarithms: If any formula being used in dimensional analysis involves any of the trigonometric ratios (sin, cos or tan), logarithms (log) again these can be ignored to.  How to conduct dimensional analysis 1. Replace the variables with their units (constants and other unitless terms can be ignored). 2. You can only add/subtract the same units, and your answer will be in the same units, e.g 4m + 5m= 9m so metres plus metres is still metres. 3. You can multiply and divide by different units to tidy up the expression/equation. Worked example: find the unit of Volume:  RECTANGLE Width Length Area of rectangle = length x width  Lesson 14.2 Common Units When finding the units from a formula, simply substitute the measured quantity for each variable. worked example:  Speed/Velocity distance Speed = time  Acceleration Final Vel - Initial Vel acc = time  Force Force = mass x acceleration  Momentum/Impulse Momentum = Mass x Velocity  Power Power = Force x Velocity  Energy/ Work P.E = Mass x Gravity x Height  1 K.E = Mass x (velocity)² 2 Work = Force x Distance Angular Velocity Radians ω= Second  Unit Unit Velocity ms⁻¹ ms⁻¹ (Speed) Acceleration ms⁻² ms⁻² Force N Kg ms⁻² Momentum/Impulse Ns kg ms⁻¹ Energy/Work J Kg m² s⁻² Power W Kg m² s⁻³ Angular velocity Rad s⁻¹ Rad s⁻¹ Extra excercises:  Lesson 14.3 Harder Examples When validating an equation the units of the Left hand side must be equal to the units on the right hand side. i.e LHS=RHS All constants are omitted e.g Worked Example : Solution  Question 1 Use the method of dimensions to check whether the following formulae are dimensionally valid where u and v are velocities, g is an acceleration, s is a distance and t is a time. (a) v² = u² + 2gs (b) v = u - gt (c) s = 0.5 gt² + ut (d) t = 2s g  Question 2 A piano wire has a mass m, length l, and tension F. Which of the formulae for the period of vibration, t, could be correct? ml F (a) t = 2π (b) t = 2π F ml  Question 3 The viscosity of a fluid n is related to the force F required to move nAs it at speed s by the formula F = where A is an area and y is y a distance. Find the unit of viscocity. 4)A student cannot remember the formula for maximum range R of a projectile on to a horizontal plane. He knows the formula only involves the initial speed u and the acceleration due to gravity g and has no dimensionless constants. Find the formula of R using dimensional analysis.