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Units and Dimensionality

Units and Dimensionality. Systems of Units. SI or metric system US or Conventional system. SI or Metric System. Basic Units Length meter Mass kilogram Time second Charge Coulomb Temperature K. Conventional/US System. Basic Units Length foot Force pound Time second

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Units and Dimensionality

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  1. Units and Dimensionality

  2. Systems of Units • SI or metric system • US or Conventional system

  3. SI or Metric System • Basic Units • Length meter • Mass kilogram • Time second • Charge Coulomb • Temperature K

  4. Conventional/US System • Basic Units • Length foot • Force pound • Time second • Charge Coulomb • Temperature F

  5. Derived Units/SI • Force Newton • Work Newton meter • Energy Newton meter, Joule • Power Watt • Velocity meter/sec • Acceleration meter/sec/sec

  6. Derived Units/US • Mass slugs • Work foot pounds, BTU • Energy foot pounds, BTU • Power Horsepower • Velocity ft/sec • Acceleration ft/sec/sec

  7. Dimensionality • Mass M • Length L • Time T

  8. Dimensionality/Force • F = ma Newton’s second law • [F] square brackets means dimensionality • [F] = [m][a] • [m] = M • [a] = L/T2 • [F] = M L/T2 • Does not matter whether SI or US system!

  9. Dimensionality/Work • W = F x distance • [W] = [F] [distance] • [F] = M L/T2 • [distance] = L • [W] = ML2/T2 • Does not matter which system • [W] = [Energy]

  10. Dimensionality/Power • P = Work/time = Energy/time • [P] = [W]/[time] • [W] = ML2/T2 • [time] = T • [P] = ML2/T3 • Does not matter which system.

  11. Dimensionality/Velocity • v = distance/time • [v] = L/T

  12. Dimensionality/Acceleration • a = distance/sec/sec • [a] = L/T2

  13. Functions and Arguments • Consider the function f(x). • The function is f • The argument of the function is x • Consider the function sin Θ. • The function is sin • The argument of the sin is Θ

  14. Consider the function ln x • The function is ln (the natural logarithm) • The argument of ln is x. • Consider the function ex or exp x • The function is e or exponential • The argument is x

  15. Rules • The argument of a trigonometric, logarithmic, exponential, etc. function must be dimensionless! • Sin Θ • Θ must be dimensionless • Angles in radians or in degrees are dimensionless • Sin αt • [αt] = 1, so [α][t] = (1/T)(T) = 1

  16. Log x • x must be dimensionless • Log ax • ax must be dimensionless • If [x] = L • Then [a] = 1/L • So that [ax] = (1/L)(L) = 1

  17. Exp (- Q/RT) = e-(Q/RT) • [Q/RT] = 1 • If T is temperature and [T] = K • And Q is expressed in Joules/mole • Then R must be expressed in Joules/mole/K

  18. Dimensionally Homogeneous Equations • Every term on both the left hand side and the right hand side of an equation must have the same dimensionality. • Consider the equation • E = 0.5mv2 + mgh • E = kinetic energy plus potential energy • [E] = ML2/T2 • [0.5mv2] = M(L/T)2 [mgh] = M(L/T2)(L) • So all terms have the same dimensionality • And equation is dimensionally homnogeneous!

  19. Consider the equation for the distance y • y = 0.5mgh • [y] = L • [mgh] = (M)(L/T2)(L) • [LHS] = L • [RHS] = ML2/T2 • This equation is NOT dimensionally homgeneous!

  20. Class Problem • Consider the equation for total energy G • G = (4π/3)R3 g + (4π)R2σ • Where R is the radius of a sphere • g is the energy per unit volume of the sphere • σ is the surface energy of the sphere • Find [g], [σ] and determine if the equation is dimensionally homogeneous.

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