IMAT 2015 Q54 [Dimensional Consistency]

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)

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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