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

The dimensions of Young's modulus of elasticity (𝑌) are [𝑀][𝐿]⁻¹[𝑇]⁻², where [𝑀] represents mass, [𝐿] represents length, and [𝑇] represents time. 
Let's analyze the dimensions of each derived quantity: 
The speed of light (𝑐) has dimensions [𝐿][𝑇]⁻¹. 
Planck's constant (ℎ) has dimensions [𝑀][𝐿]²[𝑇]⁻¹. 
The gravitational constant (𝐺) has dimensions [𝑀]⁻¹[𝐿]³[𝑇]⁻².
Substituting the dimensions of 𝑐, ℎ, and 𝐺 into the expression 𝑌 = 𝑐^𝛼ℎ^𝛽𝐺^𝛾, we have: 
[𝑀][𝐿]⁻¹[𝑇]⁻² = ([𝐿][𝑇]⁻¹)^α([𝑀][𝐿]²[𝑇]⁻¹)^β([𝑀]⁻¹[𝐿]³[𝑇]⁻²)^γ. 
By equating the dimensions on both sides of the equation, we can set up the following equations: 
For mass dimension: 1 = 0 + β - γ. 
For length dimension: -1 = 1α + 2β + 3γ. 
For time dimension: -2 = -1α - β - 2γ. 
Solving these equations simultaneously will allow us to determine the values of 𝛼, 𝛽, and 𝛾. 
Solving the equations, we find that 𝛼 = 7, 𝛽 = -1, and 𝛾 = -2. 
Therefore, the correct option is (A) 𝛼 = 7, 𝛽 = -1, 𝛾 = -2.