Question

Young’s modulus of elasticity 𝑌 is expressed in terms of three derived quantities, namely, the gravitational constant 𝐺, Planck’s constant ℎ and the speed of light 𝑐, as 𝑌 = 𝑐 ^𝛼ℎ ^𝛽𝐺 ^𝛾 . Which of the following is the correct option?

• A. 𝛼 = 7, 𝛽 = −1, 𝛾 = −2
• B. 𝛼 = −7, 𝛽 = −1, 𝛾 = −2
• C. 𝛼 = 7, 𝛽 = −1, 𝛾 = 2
• D. 𝛼 = −7, 𝛽 = 1, 𝛾 = −2

Answer 1

The Correct Option is A

To determine the correct exponents 𝛼, 𝛽, and 𝛾 for expressing Young’s modulus of elasticity, 𝑌, in terms of the gravitational constant 𝐺, Planck’s constant ℎ, and the speed of light 𝑐, we must equate the dimensions on both sides of the equation: 𝑌 = 𝑐^𝛼ℎ^𝛽𝐺^𝛾. The dimension of Young’s modulus 𝑌 is [ML^-1T^-2]. 

Let's consider the dimensions of each of the constants:

• Gravitational constant 𝐺: [M^-1L³T^-2]
• Planck's constant ℎ: [ML²T^-1]
• Speed of light 𝑐: [LT^-1]

The equation becomes: [ML^-1T^-2] = [LT^-1]^𝛼[ML²T^-1]^𝛽[M^-1L³T^-2]^𝛾.

Equating dimensions on both sides:

• For mass (M): 1 = 𝛽 - 𝛾
• For length (L): -1 = 𝛼 + 2𝛽 + 3𝛾
• For time (T): -2 = -𝛼 - 𝛽 - 2𝛾

Solving these equations:

• From 1^st equation: 𝛽 = 𝛾 + 1
• Substitute 𝛽 in 2^nd equation: -1 = 𝛼 + 2(𝛾 + 1) + 3𝛾 => 𝛼 = -7, 𝛼 = 𝛾
• Substitute 𝛽 and 𝛼 in 3^rd equation: -2 = -(-7) - (𝛾 + 1) - 2𝛾 => 𝛾 = -7 / 2

This provides the correct exponents: α = 7, β = -1, γ = -2, confirming the correct Option:

𝛼 = 7, 𝛽 = −1, 𝛾 = −2.

Answer 2

The given equation is:

\(Y = c^\alpha h^\beta G^\gamma\)

We are also given the following dimensional relations:

\([M L^{-1} T^{-2}] = [M^0 L^1 T^{-1}]^\alpha [M L^2 T^{-1}]^\beta [M^{-1} L^3 T^{-2}]^\gamma\)

Equating the powers of \( M \), \( L \), and \( T \), we get the following system of equations:

\(1 = \beta - \gamma\)

\(-1 = \alpha + 2\beta + 3\gamma\)

\(-2 = -\alpha - \beta - 2\gamma\)

Now, solving this system of equations:

From the first equation: \( \beta = 1 + \gamma \)

Substitute this into the second and third equations:

\(-1 = \alpha + 2(1 + \gamma) + 3\gamma\)

\(-2 = -\alpha - (1 + \gamma) - 2\gamma\)

Solving these equations results in:

\(\alpha = 7, \quad \beta = -1, \quad \gamma = -2\)

Thus, the correct option is (A): \( \alpha = 7 \), \( \beta = -1 \), \( \gamma = -2 \).