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 x 10¹¹ Pa). (Ans : 1.25 x10⁻³, 7.5 x 10⁻⁴, 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 = D2D2 = 2.0x10⁻³m2 = 1.0 x10⁻³ m
Length of composite wire = L' = 3.003 m
Young's modulus of copper wire = YC = 1.2 x 10¹¹ Pa
Young's modulus of steel wire = YS = 2.0 x 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
Or
FA = YC x ΔLCL ---------(b)
For both copper and steel wire the magnitude of stress = FA will be same So,
for steel wire
FA = YS x ΔLSL ---------(c)
By comparing eqn (b) and (c) we get
The quantity L will be canceled from both sides thus
YC x ΔLC = YS x ΔLS
By putting the value of ΔLS from equation (a)
By putting all the corresponding values
1.2 x 10¹¹ Pa x ΔLC = 2.0 x 10¹¹ Pa x (0.003 m - ΔLC)
1.2 x 10¹¹ Pa x ΔLC = 0.006 x 10¹¹ m Pa - 2.0 x 10¹¹ Pa x ΔLC
1.2 x 10¹¹ Pa x ΔLC + 2.0 x 10¹¹ Pa x ΔLC = 0.006 x 10¹¹ m Pa
3.2 x 10¹¹ Pa x ΔLC = 0.006 x 10¹¹ m Pa
or
ΔLC = 0.006x10¹¹mPa3.2x10¹¹Pa
ΔLC = 0.001875 m --------------(d)
again using eqn (a)
ΔLS = 0.003 m - ΔLC = 0.003 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
Or
FA = YC 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"
************************************
************************************
Shortcut Links For
1. Website for School and College Level Physics 2. Website for School and College Level Mathematics 3. Website for Single National Curriculum Pakistan - All Subjects Notes
© 2022-Onwards by Academic Skills and Knowledge (ASK)
Note: Write me in the comment box below for any query and also Share this information with your class-fellows and friends.
1. Website for School and College Level Physics
2. Website for School and College Level Mathematics
3. Website for Single National Curriculum Pakistan - All Subjects Notes
© 2022-Onwards by Academic Skills and Knowledge (ASK)
Note: Write me in the comment box below for any query and also Share this information with your class-fellows and friends.
0 Comments
If you have any QUESTIONs or DOUBTS, Please! let me know in the comments box or by WhatsApp 03339719149