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 = 𝑐𝛼ℎ𝛽𝐺𝛾. 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

Answer: (A)

𝛼 = 7, 𝛽 = −1, 𝛾 = −2

Answer 2

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 = 𝑐𝛼ℎ𝛽𝐺𝛾, 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.