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1 answer:
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Explanation:
We need to rearrange the following formula for the values given in parenthesis.
(1) x+xy = y, (x)
taking x common in LHS,
x(1+y)=y
(2) x+y = xy, (x)
Subrtacting both sides by xy.
x+y-xy = xy-xy
x+y-xy = 0
x-xy=-y
x(1-y)=-y
(3) x = y+xy, (x)
Subrating both sides by xy
x-xy = y+xy-xy
x(1-y)=y
(4) E = (1/2)mv^2-(1/2)mu^2, (u)
Subtracting both sides by (1/2)mv^2
E-(1/2)mv^2 = (1/2)mv^2-(1/2)mu^2-(1/2)mv^2
E-(1/2)mv^2 =-(1/2)mu^2
So,
(5) (x^2/a^2)-(y^2/b^2) = 1, (y)
(6) ay^2 = x^3, (y)
Hence, this is the required solution.
Answer:
a) = 133.75 Gpa
b) Fnet = 560 N
c) thermal expansion of the composite material = 14.31 / °C
Explanation:
Solution:
a) Elastic Modulus of the composite:
Area of steel wire = x (
) = 0.8 x
Area of Copper wire = x (
) - 0.8 x
Area of Copper wire = 2.4 x
Young's Modulus of Composite mixture:
=
+
Equation 1
here,
= Stress in Steel
= Stress in Copper.
We know that,
F = P/A
F is inversely proportional to Area, so if area is large, stress will less and vice versa. So, Take
Ratio for area of steel =
Ratio for area of steel = = 0.25
Similarly, for Copper,
Ratio for area of copper =
Ratio for area of copper = = 0.75
Put these values in equation 1:
=
+
= (0.25)
+ (0.75)
We are given that,
= 205 Gpa
= 110 Gpa
So,
= (0.25) (205 Gpa) + (0.75) (110 GPa)
= 51.25GPa + 82.5 Gpa
Hence, the Elastic Modulus of the composite will be:
= 133.75 Gpa
b) maximum force:
Fnet = Fst + Fcu
We know that F = (Yield Stress x Area)
F = fst x Ast + fcu x Acu
And we are given that,
Yield stress of Steel = 280 Mpa
Yield stress of Copper = 140 Mpa
And,
Ast = 0.8 x
Acu = 2.4 x
Just plugging in the values, we get:
F = (280 Mpa) (0.8 x
) + (140 Mpa) (2.4 x
)
F = 224 + 336
Fnet = 560 N ( because Mpa = N/
)
So, it means the composite will carry the maximum force of 560N
c) Coefficient of Thermal Expansion:
Strain on both material is same upon loading so,
(ΔL/L)st = (ΔL/L)cu
by thermal expansion equation:
(ΔT +
) =
ΔT +
)
Where = Coefficient of Thermal expansion
Here, fst = -fcu = F
and ΔT = 1°
So,
Plugging in the values, we get.
( 10 x x (1) +
) = ( 17 x
x (1) +
)
Solving for F, we get:
F = 0.71 N
Here,
fst = F = 0.71 N (Tension on Heating)
fcu = -F = 0.71 N ( Compression on Heating )
So, the combined thermal expansion of the composite material will be:
(ΔL/L)cu = ( 17 x x (1°) +
)
(ΔL/L)cu = ( 17 x x (1°) - 2.69 x
combined thermal expansion of the composite material = 14.31 / °C