A steel wire of length 4.7 m and cross-sectional area 3.0 × 10...
A steel wire of length 4.7 m and cross-sectional area 3.0 × 10...
Given:
Length of steel wire, L1 = 4.7 m
Cross-sectional area of steel wire, A1 = 3.0 × 10^5 m^2
Length of copper wire, L2 = 3.5 m
Cross-sectional area of copper wire, A2 = 4.0 × 10^5 m^2
To find:
Ratio of Young's modulus of steel to that of copper
Formula:
Young's modulus (Y) = Stress / Strain
Stress = Force / Area
Strain = Change in length / Original length
Calculation:
Let the force applied on both wires be F.
For steel wire:
Stress1 = F / A1
Strain1 = Change in length1 / Original length1 = x / L1
For copper wire:
Stress2 = F / A2
Strain2 = Change in length2 / Original length2 = x / L2
Since both wires stretch by the same amount under the given load, we can equate the strains.
Strain1 = Strain2
x / L1 = x / L2
Cross multiplying, we get:
x * L2 = x * L1
L2 = L1
Therefore, the ratio of the lengths of the two wires is 3.5 / 4.7 = 1.4.
Now, substituting the values in the stress equation:
Stress1 = F / A1
Stress2 = F / A2
Since the forces applied are the same for both wires, we can equate the stresses.
Stress1 = Stress2
F / A1 = F / A2
Cross multiplying, we get:
F * A2 = F * A1
A2 = A1
Therefore, the ratio of the cross-sectional areas of the two wires is 4.0 × 10^5 / 3.0 × 10^5 = 1.33.
Now, substituting the values in the Young's modulus equation:
Young's modulus of steel (Y1) = Stress1 / Strain1 = (F / A1) / (x / L1)
Young's modulus of copper (Y2) = Stress2 / Strain2 = (F / A2) / (x / L2)
We can cancel out the applied force and the change in length:
Y1 = 1 / (A1 / L1)
Y2 = 1 / (A2 / L2)
Substituting the values:
Y1 = 1 / (3.0 × 10^5 / 4.7)
Y2 = 1 / (4.0 × 10^5 / 3.5)
Calculating the ratios:
Y1 / Y2 = [(4.7 × 10^5) / 3.0] / [(3.5 × 10^5) / 4.0]
= (4.7 × 10^5 × 4.0) / (3.0 × 3.5 × 10^5)
= 1.8
Therefore, the ratio of the Young's modulus of steel to that of copper is 1.8.
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