Question

If $L,M,N$ are physical quantities with different dimensions, then which of the following combinations can never be a meaningful quantity?

• A. $\frac{LM}{N}$
• B. $LN-M$
• C. $\frac{LN-M^2}{N}$
• D. $\frac{L-M}{N}$

Answer 1

The correct option is D $\frac{L-M}{N}$

$\text{Step 1: Check Option A}$

As quantities with different dimensions can be multiplied or divided

$\Rightarrow \frac{LM}{N}$ will be meaningful

$\text{Step 2: Check Option B}$

We can always add or subtract two quantities with the same dimensions.

$LN$ can have dimensions the same as of $M$

For example : $L=$ mass, $N=$ velocity, $M=$ momentum.

$\Rightarrow LN-M$ can be meaningful.

$\text{Step 3: Check Option C}$

$LN$ can have dimension of $M^2$

For example : $L=$ mass, $N=$ kinetic energy, $M=$ momentum.

$\Rightarrow \frac{LN-M^2}{N}$ can be meaningful.

$\text{Step 4: Check Option D}$

As $L$ & $M$ have different dimensions, we can never subtract two quantities with different dimensions.

$\Rightarrow \frac{L-M}{N}$ can never be a meaningful quantity.

Hence correct option is $\left(D\right)$.