Question W: Derive the dimension of the volume of a cylinder using height (h), radius (r), and diameter (d).

Here are the solutions to the questions:

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Question W: Derive the dimension of the volume of a cylinder using height (h), radius (r), and diameter (d).

Step 1: Recall the formula for the volume of a cylinder.
The volume of a cylinder ($V$) is given by:

$$V = \pi r^2 h$$

Step 2: Identify the dimensions of each physical quantity.

• Radius ($r$) is a length, so its dimension is $[L]$.
• Height ($h$) is a length, so its dimension is $[L]$.
• Diameter ($d$) is also a length, so its dimension is $[L]$.
• $\pi$ is a dimensionless constant.

Step 3: Substitute the dimensions into the volume formula.
The dimension of $V$ is:

$$[V] = [r]^2 [h]$$ $$[V] = [L]^2 [L]$$ $$[V] = [L^3]$$

Step 4: Express the dimension using diameter (d).
Since the radius $r$ is half the diameter $d$ ($r = \frac{d}{2}$), its dimension is still $[L]$. Therefore, using diameter instead of radius does not change the overall dimension of volume.

$$[V] = [d]^2 [h]$$ $$[V] = [L]^2 [L]$$ $$[V] = [L^3]$$

The dimension of the volume of a cylinder is $\boxed{[L^3]}$.

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