Step 1: Understanding the Concept:
This question requires finding the dimensional formulas for each quantity in the given pairs and identifying the pair with identical dimensions. Step 2: Key Formula or Approach:
We need to derive the dimensional formula for each physical quantity from its definition or a related physical law. The fundamental dimensions are Mass [M], Length [L], and Time [T]. Step 3: Detailed Explanation:
Let's find the dimensions for each quantity in the options:
(A) Torque and momentum
- Torque ($\tau$) = Force $\times$ perpendicular distance = $[MLT^{-2}] \times [L] = [ML^2T^{-2}]$. (Same as Work/Energy)
- Momentum (p) = mass $\times$ velocity = $[M] \times [LT^{-1}] = [MLT^{-1}]$.
The dimensions are different.
(B) Surface tension and tension
- Surface Tension (S) = Force / Length = $[MLT^{-2}] / [L] = [MT^{-2}]$.
- Tension (T) is a type of force, so its dimension is $[MLT^{-2}]$.
The dimensions are different.
(C) Pressure and modulus of elasticity
- Pressure (P) = Force / Area = $[MLT^{-2}] / [L^2] = [ML^{-1}T^{-2}]$.
- Modulus of Elasticity (E) = Stress / Strain.
- Stress = Force / Area = $[ML^{-1}T^{-2}]$.
- Strain = $\Delta L / L$ = $[L]/[L] = [M^0L^0T^0]$ (dimensionless).
- So, the dimension of Modulus of Elasticity is the same as stress: $[E] = [ML^{-1}T^{-2}]$.
The dimensions of pressure and modulus of elasticity are the same.
(D) Force constant and Planck's constant
- Force Constant (k), from Hooke's Law (F=-kx), is k = Force / displacement = $[MLT^{-2}] / [L] = [MT^{-2}]$. (Same as Surface Tension)
- Planck's Constant (h), from E=hf (where f is frequency), is h = Energy / frequency = $[ML^2T^{-2}] / [T^{-1}] = [ML^2T^{-1}]$. (Same as Angular Momentum)
The dimensions are different.
Step 4: Final Answer:
The pair with the same dimensional formula is Pressure and modulus of elasticity. Therefore, option (C) is correct.
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