you are told that the volume of a spherer is given by V=pieD^3/3, where V is the volume and D isthediameter?

Question

Is this equation dimensionally correct? ( use the SI unit analysis to find out)

Answer 1

Yes, the equation is dimensionally correct, although the actual formula itself is incorrect. The real formula for the volume of a sphere is:

V = (π d³) / 6

Nonetheless, you are seeing if it is dimensionally correct with unit analysis, so let's use the formula you were given:

V = (π d³) / 3

Now, we know that units of volume are meters cubed, which is denoted by [m³]. Also, diameter is a length, so SI units are meters, which is [m]. Well, pi and the denominator of 3 are absolutely dimensionless, so you just have:

V = (π d³) / 3
[m³] = [m]³
[m³] = [m³]

So, you can see that it is dimensionally correct.

Answer 2

D is measured in meters, so D^3 has units of meters^3. This makes sense, because volume is measured in meters^3 or feet^3 or something similar. Hence the equation is dimensionally correct (though not actually correct).