Which one of the following equations is dimensionally consistent (has consistent units)?
[All the symbols have their usual meanings:
v = velocity; F = force; m = mass; t = time; V = voltage; Q = charge; R_1, R_2, R_3, R_4 = resistance]
A. energy = (½mv^2) + Fv
B. resistance = R_1+ R_2 + (1/R_3) + (1/R_4)
C. temperature change = energy × m × specific heat capacity
D. acceleration = (½vt^2) + (F/m)
E. electrical current = (V/R_1) + (Q/t)
To elaborate, for one of these formulas to be dimensionally consistent, the formulas must be relative to the unit.
For A, the first part is the formula for Kinetic Energy- however our second part is clearly incorrect.
For B, no such formula exists. Simply because we know:
For parallel resistance in a circuit: \frac{1}{R_e} =\frac{1}{R_1}+\frac{1}{R_2} + \dotsc +\frac{1}{R_m}
For series resistance in a circuit: R_e = R_1+R_2+\dotsc + R_n
Making B incorrect.
For C, it is a spin on the formula: E=mc\Delta t
But if we want \Delta t to be the subject of the formula, our answer should instead be \Delta t= \frac{E}{mc}. Therefore C is also incorrect.
For D, the second part may be true- however, the first formula is an incorrect interpretation for motion with constant acceleration.
E is our correct answer. Since we know it is possible to say:
Current= Voltage/Resistance and Current= Charge/time