Dimensional Analysis Post Test

  1. Given \(mv = Ft\), where \(m\) is mass, \(v\) is speed, \(F\) is force, and \(t\) is time, what are the dimensions of each side of the equation? Is the equation dimensionally correct?
  2. Given \(H = mC\Delta T\), where \(H\) is in joules, \(m\) in kilograms, and \(\Delta T\) in kelvin, what are the SI units and dimensions of \(C\)?
  3. Given \(P = kA\Delta T/\ell\), where \(A\) is the area, \(\Delta T\) is difference in temperature, \(\ell\) is length, and \(k\) is a constant with SI units of watts per (metre·kelvin), what are the SI units for \(P\) (rate of thermal energy flow)?
  4. Given \(E = a\ell \sin (bt)\), where \(E\) is energy, \(\ell\) is length and \(t\) is time: 

    (a) What are the dimensions and SI units of \(b\)?
    (b) What are the dimensions and SI units of \(a\)?
  1. [left side] = \(M\cdot L/T\)
    [right side] = \(M\cdot L/T\)
    Therefore the equation is dimensionally correct.
  2. Since \(C = H/(m\Delta T)\), the SI units are \(J\cdot kg^{-1} \cdot K^{-1}.\).
    \([C] = (M\cdot L^2 \cdot T^{-2})\cdot M^{-1}\cdot \theta^{-1} = L^2 \cdot T^{-2} \cdot \theta^{-1}.\)
  3. Recall that watt \((W)\) is joules per second, so \([k] = M\cdot L \cdot T^{-3} \cdot \theta^{-1}.\)
    \([A] = L^2, [\Delta T] = \theta,\) , and \([\ell] = L\)
    [right side] = \(M\cdot L^2 /T^3\)
    Therefore, \([P] = M\cdot L^2/T^3\), and SI units are  \(kg\cdot m^2/s^3, \), or \(J/s.\)
  4. (a) \([b] = T^{-1}\)
    Remember that the argument of the sine function must be dimensionless. Since the argument in this case is an unknown \((b)\) multiplied by time \((t)\), then \(b\) must have dimensions of inverse time.  The SI units of "\(b\)" are \(s^{-1}.\)

    (b) \([a] = [E/ \ell ] = M \cdot L/T^2\) since sine is dimensionless.  The SI units of "are" are \(kg\cdot m/s^2\), or newton.