What stress would cause a wire to increase in length by 0.01% if Young's modulus of the wire is 12 x 10¹⁰ Pa? What force would produce this stress if the diameter of the wire is 0.56 mm? (A n s: 1.2 x 10⁷ Pa, 2.96 N)



Given Data:

Tensile Stress = Ɛ = 0.01 % = `\frac {0.01}{100}` = 10⁻⁴

Young's modulus of steel  = Y = 12 ｘ 10¹⁰ Pa

Diameter of the cylinder = D =  0.56 mm = 0.56 ｘ10⁻³  m

Radius of the cylinder = r = `\frac {D}{2}` = `\frac {0.56 ｘ10⁻³  m }{2}` = 0.28 ｘ10⁻³  m


To Find:

Tensile Stress = σ = ?

Force = F = ?



Solution:


Using Young's Modulus formula is 

Y = `\frac {Stress}{Strai n}` 

Y = `\frac {σ}{Ɛ}`

or

σ = Y Ɛ

By putting values

σ = 12 ｘ 10¹⁰ Pa ｘ  10⁻⁴

σ = 12 ｘ 10⁶ Pa 

or

σ = 1.2 ｘ 10⁷ Pa -------------Ans.1


Now using the formula for stress that is 

σ = `\frac {F}{A}`

or

F = σ A

where A = area =  ㄫ r² So

F = σ ㄫ r²

putting the corresponding values

F = 1.2 ｘ 10⁷ Pa ｘ 3.1416 ｘ (0.28 ｘ10⁻³  m)²

[ 1 Pa = 1 N m⁻² ]

F = 3.76992 ｘ 10⁷ N m⁻² ｘ 0.0784 ｘ10⁻⁶  m²

F = 0.295561728 ｘ 10 N

or

F = 2.956 N --------------Ans.2




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