Question

Which pair of physical quantities have same dimensional formula

• A. Torque and momentum
• B. Surface tension and tension
• C. Pressure and modulus of elasticity
• D. Force constant and Planck's constant

Hint:

Remember that quantities like Pressure, Stress, and any Modulus of Elasticity (Young's, Bulk, Shear) all share the same dimension of Force/Area, which is \([ML^{-1}T^{-2}]\). This is because strain is always dimensionless.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to check the dimensional formulas for each pair of physical quantities listed in the options and find the pair with identical dimensions.

Step 2: Key Formula or Approach:
We will derive the dimensional formula for each quantity based on its physical definition or formula. The fundamental dimensions are Mass (M), Length (L), and Time (T).

Step 3: Detailed Explanation:
Let's analyze each option:

(A) Torque and momentum:
- Torque (\(\tau\)) = Force \(\times\) perpendicular distance = \([MLT^{-2}] \times [L] = [ML^2T^{-2}]\).
- Momentum (\(p\)) = mass \(\times\) velocity = \([M] \times [LT^{-1}] = [MLT^{-1}]\).
The dimensions are not the same.

(B) Surface tension and tension:
- Surface Tension = Force per unit length = \([MLT^{-2}] / [L] = [MT^{-2}]\).
- Tension is a type of force, so its dimension is \([MLT^{-2}]\).
The dimensions are not the same.

(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 = \([MLT^{-2}] / [L^2] = [ML^{-1}T^{-2}]\).
- Strain = Change in dimension / Original dimension = \([L]/[L] = [M^0L^0T^0]\) (dimensionless).
- Therefore, the dimension of Modulus of Elasticity is the same as Stress: \([ML^{-1}T^{-2}]\).
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) = Force / distance = \([MLT^{-2}] / [L] = [MT^{-2}]\).
- Planck's constant (\(h\)) from (E=h\(\nu\)) = Energy / frequency = \([ML^2T^{-2}] / [T^{-1}] = [ML^2T^{-1}]\).
The dimensions are not the same.

Step 4: Final Answer:
The pair with the same dimensional formula is Pressure and modulus of elasticity.