Question

If a physical quantity ‘Q’ is defined in terms of moment of inertia ‘I’, force ‘F’, velocity ‘V’, work ‘W’ and length ‘L’ as \[ Q = \frac{IFV^2}{WL^3} \] then ‘Q’ is

• A. stress
• B. modulus of elasticity
• C. specific heat capacity
• D. surface tension

Hint:

If dimensional analysis gives \([MT^{-2}]\), it often indicates surface tension (force per unit length).

Answer 1

The Correct Option is D

We are given: \[ Q = \frac{IFV^2}{WL^3} \] Let’s derive the dimensional formula of each component:
- Moment of inertia \(I\): \([ML^2]\)
- Force \(F\): \([MLT^{-2}]\)
- Velocity \(V\): \([LT^{-1}]\), so \(V^2 = [L^2T^{-2}]\)
- Work \(W\): \([ML^2T^{-2}]\)
- Length \(L^3 = [L^3]\)
Now substitute all into the equation: \[ Q = \frac{[ML^2] \cdot [MLT^{-2}] \cdot [L^2T^{-2}]}{[ML^2T^{-2}] \cdot [L^3]} = \frac{M^2L^5T^{-4}}{ML^5T^{-2}} = [MT^{-2}] \] The dimensional formula \([MT^{-2}]\) corresponds to surface tension, which is force per unit length.
Hence, the correct answer is surface tension.

Answer 2

Step 1: Write down the given expression for \(Q\)
\[ Q = \frac{I F V^2}{W L^3} \]

Step 2: Identify the dimensions of each quantity
- Moment of inertia, \(I\): \([M L^2]\)
- Force, \(F\): \([M L T^{-2}]\)
- Velocity, \(V\): \([L T^{-1}]\)
- Work, \(W\): \([M L^2 T^{-2}]\)
- Length, \(L\): \([L]\)

Step 3: Find the dimension of numerator
\[ I F V^2 = [M L^2] \times [M L T^{-2}] \times [L T^{-1}]^2 = [M L^2] \times [M L T^{-2}] \times [L^2 T^{-2}] \]
\[ = M \times M \times L^2 \times L \times L^2 \times T^{-2} \times T^{-2} = M^2 L^{5} T^{-4} \]

Step 4: Find the dimension of denominator
\[ W L^3 = [M L^2 T^{-2}] \times [L^3] = M L^{5} T^{-2} \]

Step 5: Find the dimension of \(Q\)
\[ Q = \frac{M^2 L^{5} T^{-4}}{M L^{5} T^{-2}} = M^{2-1} L^{5-5} T^{-4+2} = M^{1} L^{0} T^{-2} \]
So, dimension of \(Q = [M T^{-2}]\).

Step 6: Identify the physical quantity with dimension \([M T^{-2}]\)
Surface tension has dimension of force per unit length:
\[ \text{Surface tension} = \frac{\text{Force}}{\text{Length}} = \frac{M L T^{-2}}{L} = M T^{-2} \]

Step 7: Final Conclusion
Therefore, \(Q\) represents surface tension.