A force of 1.5ｘ10⁴ N causes a strain of 1.4ｘ10⁻⁴ in a steel cable of cross-sectional area 4.8ｘ10⁴ m². 

(a) What is Young's Modulus of the steel cable? 

(b) The stress-strain graph is linear for this cable. Calculate the strain energy per unit volume stored in the cable when the cable has a strain of 1ｘ10⁻⁴.  Answer((a) 2.2ｘ10¹¹ N m⁻², (b) 1.5 ｘ 10³  Jm⁻³)



Given Data:

Force = F = 1.5ｘ10⁴ N

Tensile strain in steel cable = 1.4ｘ10⁻⁴

Area of steel cable = A = 4.8 ｘ10⁻⁴ m²



To Find:

(a). Young's modulus of steel  = Y = ?

(b).  Strain Energy per unit Volume = ? (at Tensile strain = 1ｘ10⁻⁴ )


Solution:

(a). Young's modulus of steel  = Y = ?

Young's Modulus is defined as :

Y = `\frac {Stress}{Strai n}` ---------(1)


Where Stress = `\frac {F}{A}` 

Stress = `\frac {1.5ｘ10⁴ N}{ 4.8 ｘ10⁻⁴ m²}`

Stress = 3.12 ｘ10⁸ N m⁻² 

putting the values of stress and strain in equation (1)

Y = `\frac {3.12 ｘ10⁸ N m⁻² }{1.4ｘ10⁻⁴}`

by simplifying we get

Y = 2.2 ｘ10¹¹ N m⁻² ------------Ans. (1)


(b).  Strain Energy per unit Volume = ? (when Tensile strain = 1ｘ10⁻⁴ )

Strain Energy per unit Volume can be calculated as

Strain Energy per unit Volume = `\frac{1}{2}`ｘ Stress ｘ strain ----------(2)

Where the value of strain is given i.e 1ｘ10⁻⁴  and the value of stress can be calculated from the Young's Modulus formula which is 

Y = `\frac {Stress}{Strai n}` 

or 

Stress = Y ｘ Strain = 2.2 ｘ10¹¹ N m⁻² ｘ 1ｘ10⁻⁴ = 2.2 ｘ10⁷ N m⁻²


Now putting the value of the given strain and calculated stress in equation (2)


Strain Energy per unit Volume = `\frac{1}{2}` ｘ 2.2 ｘ10⁷ N m⁻² ｘ 1ｘ10⁻⁴

Strain Energy per unit Volume = 1.1 ｘ10³ N m⁻²

or

Strain Energy per unit Volume = 1.1 ｘ10³ J m⁻³  -----Ans. (2)


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