The physical quantities not having the same dimensions are

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CGP EDU Academic Team · Accepted Answer

The correct answer is A. Step 1: Identify the dimensions of each quantity.Torque: Torque (τ) is defined as force multiplied by distance. Its dimension is [M][L²][T⁻²] × [L] = [M][L³][T⁻²].Work: Work (W) is defined as force times distance. Its dimension is the same as that of torque: [M][L²][T⁻²] × [L] = [M][L²][T⁻²].Both torque and work have the same dimensions.Step 2: Check the next option.Momentum: Momentum (p) is mass times velocity. Its dimension is [M][L][T⁻¹].Planck's constant: Planck’s constant (h) has the dimension of energy multiplied by time, which is [M][L²][T⁻²] × [T] = [M][L²][T⁻¹].Therefore, momentum and Planck's constant do not have the same dimensions.Step 3: Check the next option.Stress: Stress (σ) is force per unit area, so its dimension is [M][L²][T⁻²] × [L⁻²] = [M][L⁻¹][T⁻²].Young's Modulus: Young's modulus (E) has the same dimension as stress, which is [M][L⁻¹][T⁻²].Both stress and Young's modulus have the same dimensions.Step 4: Check the final option.Speed: Speed (v) has the dimension [L][T⁻¹].The quantity (\mu_0 \epsilon_0)^{-1/2}: The dimensions can be derived from electrical permeability \mu_0 and permittivity \epsilon_0. Both of these qualities have dimensions related to [M][L⁻³][T⁻⁶][I²], resulting in (\mu_0 \epsilon_0)^{-1/2} having dimensions of [L][T⁻¹] which is similar to speed.Both speed and (\mu_0 \epsilon_0)^{-1/2} have the same dimensions.Final Conclusion: The only pair that does not share the same dimensions is option B: momentum and Planck's constant.Therefore, the correct option is B.

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