Dimensional Analysis Post Test

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?
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 $C$ ?
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)?
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 $a$ ?

[left side] = $M\cdot L/T$
[right side] = $M\cdot L/T$
Therefore the equation is dimensionally correct.
Since $C = H/(m\Delta T)$ , the SI units are $J\cdot kg^{-1} \cdot K^{-1}.$ .
$[C] = (M\cdot L^2 \cdot T^{-2})\cdot M^{-1}\cdot \theta^{-1} = L^2 \cdot T^{-2} \cdot \theta^{-1}.$
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 $J/s.$
(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.

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