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PhyEasyAdv 2023 • Paper 2

Dimensional Analysis of Young Modulus in Fundamental Quantities

Young's modulus of elasticity YY is expressed in terms of three derived quantities, namely, the gravitational constant GG, Planck's constant hh and the speed of light cc, as Y=cαhβGγY = c^\alpha h^\beta G^\gamma. Which of the following is the correct option?

Options

A

α=7,β=−1,γ=−2\alpha = 7, \beta = -1, \gamma = -2

Correct
B

α=−7,β=−1,γ=−2\alpha = -7, \beta = -1, \gamma = -2

C

α=7,β=−1,γ=2\alpha = 7, \beta = -1, \gamma = 2

D

α=−7,β=1,γ=−2\alpha = -7, \beta = 1, \gamma = -2

Topics & Concepts

Units and MeasurementsElasticity

Step-by-Step Solution

To find the exponents α\alpha, β\beta, and γ\gamma in the relation Y=cαhβGγY = c^\alpha h^\beta G^\gamma, we use dimensional analysis.

The dimensional formulas of the given physical quantities in terms of mass (M\text{M}), length (L\text{L}), and time (T\text{T}) are:

  1. Young's modulus of elasticity, YY: [Y]=[Force][Area]=MLT−2L2=M1L−1T−2[Y] = \frac{[\text{Force}]}{[\text{Area}]} = \frac{\text{M}\text{L}\text{T}^{-2}}{\text{L}^2} = \text{M}^{1}\text{L}^{-1}\text{T}^{-2}

  2. Speed of light, cc: [c]=L1T−1[c] = \text{L}^{1}\text{T}^{-1}

  3. Planck's constant, hh: [h]=[Energy]×[Time]=(ML2T−2)(T)=M1L2T−1[h] = [\text{Energy}] \times [\text{Time}] = (\text{M}\text{L}^2\text{T}^{-2})(\text{T}) = \text{M}^{1}\text{L}^{2}\text{T}^{-1}

  4. Universal gravitational constant, GG: [G]=[Force]×[Distance]2[Mass]2=(MLT−2)(L2)M2=M−1L3T−2[G] = \frac{[\text{Force}] \times [\text{Distance}]^2}{[\text{Mass}]^2} = \frac{(\text{M}\text{L}\text{T}^{-2})(\text{L}^2)}{\text{M}^2} = \text{M}^{-1}\text{L}^{3}\text{T}^{-2}

Substituting these dimensional formulas into the given relation [Y]=[c]α[h]β[G]γ[Y] = [c]^\alpha [h]^\beta [G]^\gamma: M1L−1T−2=(LT−1)α(ML2T−1)β(M−1L3T−2)γ\text{M}^{1}\text{L}^{-1}\text{T}^{-2} = (\text{L}\text{T}^{-1})^\alpha (\text{M}\text{L}^2\text{T}^{-1})^\beta (\text{M}^{-1}\text{L}^3\text{T}^{-2})^\gamma M1L−1T−2=Mβ−γLα+2β+3γT−α−β−2γ\text{M}^{1}\text{L}^{-1}\text{T}^{-2} = \text{M}^{\beta - \gamma} \text{L}^{\alpha + 2\beta + 3\gamma} \text{T}^{-\alpha - \beta - 2\gamma}

Equating the powers of M\text{M}, L\text{L}, and T\text{T} on both sides:

  1. For M\text{M}: β−γ=1  ⟹  β=γ+1— (1)\beta - \gamma = 1 \quad \implies \quad \beta = \gamma + 1 \quad \text{--- (1)}

  2. For L\text{L}: α+2β+3γ=−1— (2)\alpha + 2\beta + 3\gamma = -1 \quad \text{--- (2)}

  3. For T\text{T}: −α−β−2γ=−2  ⟹  α+β+2γ=2— (3)-\alpha - \beta - 2\gamma = -2 \quad \implies \quad \alpha + \beta + 2\gamma = 2 \quad \text{--- (3)}

Subtracting equation (3) from equation (2): (α+2β+3γ)−(α+β+2γ)=−1−2(\alpha + 2\beta + 3\gamma) - (\alpha + \beta + 2\gamma) = -1 - 2 β+γ=−3— (4)\beta + \gamma = -3 \quad \text{--- (4)}

Now, solving equations (1) and (4): Adding (1) and (4): 2β=−2  ⟹  β=−12\beta = -2 \implies \beta = -1

Substituting β=−1\beta = -1 into (1): −1−γ=1  ⟹  γ=−2-1 - \gamma = 1 \implies \gamma = -2

Substituting the values of β\beta and γ\gamma into equation (3): α+(−1)+2(−2)=2\alpha + (-1) + 2(-2) = 2 α−5=2  ⟹  α=7\alpha - 5 = 2 \implies \alpha = 7

Thus, the values are: α=7,β=−1,γ=−2\alpha = 7, \quad \beta = -1, \quad \gamma = -2

Therefore, the correct option is (A).

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