A cylindrical copper wire and a cylindrical steel wire each of length 1.5 m and diameter 2.0 mm is joined at one end to form a composite wire 3.0 m long. The wire is loaded until its length becomes 3.003 m. Calculate the strain in copper and steel wires and the force applied to the wire. (Young's modulus of copper is 1.2 x 10¹¹ Pa and for steel is 2.0  10¹¹ Pa). (Ans : 1.25 x10⁻³, 7.5   10⁴,  471  N )



Data Given:


Length of copper wire = LCLC = 1.5 m 

Length of steel wire = LsLs = 1.5 m

Total length = L = 1.5 m + 1.5 m = 3.0 m

Diameter of copper wire = Diameter of steel wire = D = 2.0 mm = 2.0 x10⁻³  m

Radius = r = D2D22.010³m2 = 1.0 x10⁻³  m

Length of composite wire = L' = 3.003 m

Young's modulus of copper wire  = YC = 1.2  10¹¹ Pa

Young's modulus of steel wire  = YS 2.0  10¹¹ Pa

From the given data we can calculate:

ΔL = L' - L = 3.003 m - 3.0 m = 0.003 m

and 

Area = A = ㄫ r² = 3.1416  x (1.0 x10⁻³  m)² = 3.1416 x10⁻⁶  m² 



To Find:

Strain in copper wire = ƐC ?

Strain in steel wire = ƐS ?

Force = F = ?



Solution:


The formula for strain in copper wire is

ƐC ΔLCL ----------(1)

similarly formula for strain in steel wire

ƐS ΔLsL -------------(2)

Here we see that the value of ΔLC and ΔLS are unknown. So we have to find it first.

Let 

The elongation in the copper wire = ΔLC then

The elongation in the steel wire = ΔLS = ΔL - ΔLC 

ΔLS =  0.003 m - ΔLC --------(a)

Now using the formula of Young Modulus for copper and steel wire separately. 

For Copper wire

YC = FALΔLC


Or


FAYC x ΔLCL ---------(b)

For both copper and steel wire the magnitude of stress = FA will be same So, 

for steel wire

FAYS x ΔLSL ---------(c)


By comparing eqn (b) and (c) we get

YCΔLCL = YSΔLSL

The quantity L will be canceled from both sides thus

YC x ΔLC =   YS ΔLS

By putting the value of ΔLS from equation (a) 

YCΔLC = YS x (0.003 m - ΔLC)

By putting  all the corresponding values

1.2  10¹¹ Pa x ΔLC =   2.0  10¹¹ Pa x (0.003 m - ΔLC)

1.2  10¹¹ Pa x ΔLC =   0.006  10¹¹ m Pa - 2.0  10¹¹ Pa x ΔLC

1.2  10¹¹ Pa x ΔLC + 2.0  10¹¹ Pa x ΔLC =   0.006  10¹¹ m Pa 

3.2  10¹¹ Pa x ΔLC =   0.006  10¹¹ m Pa 

or

ΔLC =  0.00610¹¹mPa3.210¹¹Pa 

ΔLC =  0.001875 m  --------------(d)

again using eqn (a)

ΔLS =  0.003 m - ΔLC0.003 m - 0.001875 m 

ΔLS =  0.001125 m ------------(e)




Now for the calculation of strain in copper wire

Using Eqn (1)

ƐC ΔLCL

ƐC 0.001875m1.5m

ƐC 0.00125  ----------Ans (1)



for calculation of strain in steel wire 

Using Eqn (2)

ƐS ΔLSL

ƐS 0.001125m1.5m

ƐS 0.00075 ----------Ans (2)


To the value of force applied to the wire, we have the formula of young modulus for:

Let for Copper wire 

YC = FALΔLC


Or


FAYC x ΔLCL 

Or 

F = A YC x ΔLCL 


By putting values


F 3.1416 x10⁻⁶ m² x 1.2 x 10¹¹ Pa x 0.001875m1.5m

F = 3.76992 x10⁵ m² Pa x 0.00125 m

[ 1 Pa = 1 N m⁻² ]

F = 0.0047124 x10⁵ N m

or

F =471.24  N m -----------------Ans.3

Note: You can use the Young Modulus formula for steel wire with the same result for finding Fore "F"


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