Question

What are the dimensions of density, which is mass per volume?

Short answer

The dimension of density is ${\mathrm{M}}^1 {\mathrm{L}}^{-3} {\mathrm{T}}^0$.

Step-by-step solution

Step 1. Pointing out the definition of density

You have already studied that density is the amount of mass per unit volume.

Given data:

$\rho =\frac{M}{V}$

where $\rho$is the density, M is the mass, and V is the volume.

Step 2. Determining the dimensional formula for different quantities

You know that the unit of mass is kg, and the unit of volume is ${\mathrm{m}}^3$. Thus, the unit of density will be $\mathrm{kg}·{\mathrm{m}}^{-3}$.

Similarly, the dimensional formula for mass is, localid="1654696905824" $\left[\mathrm{mass}\right]={\mathrm{M}}^1$.

The dimensional formula for volume is, localid="1654696917042" $\left[\mathrm{Volume}\right]={\mathrm{L}}^3$.

Step 3. Determining the dimensional formula of the density

The dimensional formula for density is:

$\begin{array}{c}\left[\mathrm{Density}\right]=\frac{\left[\mathrm{Mass}\right]}{\left[\mathrm{Volume}\right]}\\ =\frac{{\mathrm{M}}^1}{{\mathrm{L}}^3}\\ ={\mathrm{M}}^1{\mathrm{L}}^{-3}{\mathrm{T}}^0\end{array}$