Question

If Y, K and η are the values of Young's modulus, bulk modulus and modulus of rigidity of any material respectively. Choose the correct relation for these parameters.

विकल्प

• K =`("Y"eta)/(9eta - 3"Y")`N / m²

• K =`(3"YK")/(9"K" + "Y")`N / m²

• K =`(9"K"eta)/(3"K" - eta)`N / m²

• K =`(9"K"eta)/(2eta + 3"K")`N / m²

Answer 1

`bb("K" =("Y"eta)/(9eta - 3"Y")"N" // "m"^2)`

Explanation:

Given,

Young's modulus = Y

Bulk modulus = K

Modulus of rigidity = 11

We know, relation between young modulus and bulk modulus,

Y = 3K (1 - 2 σ)

⇒ 1 - 2 σ = `"Y"/(3"K")`

⇒ σ = `1/2 (1-"Y"/(3"K"))`       ...(i)

The relationship between the young modulus and the rigidity modulus,

Y = 2η (1 + σ)

 ⇒ σ = `"Y"/(2eta)-1`       ...(ii)

From equation (i) and (ii)

`1/2(1-"Y"/(3"K")) = "Y"/(2eta) - 1`

⇒ `"Y"/(2eta) + "Y"/(6"K") = 1/2 +1`

⇒ `"Y"/(2eta) + "Y"/(6"K") = 3/2`

⇒ `1/(6"K") = 3/(2"Y") - 1/(2eta)`

⇒ `1/(6"K") = (3eta-"Y")/(2"Y"eta)`

⇒ K = `(2"Y"eta)/(6(3eta - "Y")) = ("Y"eta)/(9eta -3"Y")`

⇒ K = `("Y"eta)/(9eta - 3"Y")"N"//"m"^2`