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)

1 Like

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

3 Likes