Question

A rtain physical quantity is calculated from the formula $\frac{\pi}{3}(a^2-b^2)hd$, where a, b and h are all lengths and d is density. The physical quantity being calculated is

• A. velocity
• B. volume
• C. mass
• D. acceleration

Hint:

Dimensional Analysis:

• Use $[M]$ for mass, $[L]$ for length, $[T]$ for time.
• Constants like $\pi$ are dimensionless.
• Combine dimensions algebraically as per multiplication/division in the formula.
• Resulting dimension identifies the physical quantity.

Answer 1

The Correct Option is C

We are given: $P = \frac{\pi}{3}(a^2-b^2)hd$
Given:

• $a, b, h$ are lengths $\Rightarrow$ dimension $[L]$
• $d$ is density $\Rightarrow$ dimension $[ML^{-3}]$

\[ [P] = [L^2] \cdot [L] \cdot [ML^{-3}] = M \cdot L^0 = [M] \] Hence, the physical quantity is mass.

Answer 2

Step 1: Analyze the given formula
The formula is \(\frac{\pi}{3}(a^2 - b^2) h d\), where \(a\), \(b\), and \(h\) are lengths and \(d\) is density.

Step 2: Understand the terms
- \((a^2 - b^2)\) involves subtraction of squares of lengths, giving an area dimension (\(L^2\)).
- Multiplying by \(h\) (a length) gives volume dimension (\(L^3\)).
- Multiplying volume by density \(d\) (mass per unit volume, \(M L^{-3}\)) gives mass (\(M\)).

Step 3: Physical interpretation
The expression \(\frac{\pi}{3}(a^2 - b^2) h\) resembles the volume of a frustum of a cone or a similar 3D shape.
Multiplying this volume by density \(d\) yields the mass of the object.

Step 4: Conclusion
Hence, the physical quantity calculated by the given formula is the mass.