A steel rod bas a radius of 20 mm and a length of 2.0 m. A force of 62.8 kN stretches it along its length. Young's modulus of steel is 2.0 × 10^{11} N/m^2 . The longitudinal strain produced in the wire is _______ × 10^{-5}
To determine the longitudinal strain in the steel rod, we start by identifying the needed properties and formulas. The strain (ε) can be calculated using the formula: ε = ΔL / L, where ΔL is the change in length and L is the original length. According to Hooke's Law and the definition of Young's modulus (E), the relationship is given by:
ΔL = (F × L) / (A × E)
where F is the force applied, A is the cross-sectional area, and E is Young's modulus.
First, calculate the cross-sectional area (A) of the rod as it is circular: A = π × r².
Given:
r = 20 mm = 0.02 m
L = 2.0 m
F = 62.8 kN = 62,800 N
E = 2.0 × 10¹¹ N/m²
Calculating the area: A = π × (0.02)² = 1.25664 × 10⁻³ m²
Substituting values into ΔL:
ΔL = (62,800 N × 2 m) / (1.25664 × 10⁻³ m² × 2.0 × 10¹¹ N/m²)
ΔL ≈ 5.008 × 10⁻⁴ m
Now, compute the strain (ε) as:
ε = ΔL / L = 5.008 × 10⁻⁴ m / 2.0 m = 2.504 × 10⁻⁴
Converting to the requested form (× 10⁻⁵):
ε ≈ 25.04 × 10⁻⁵
Verification: The computed strain falls within the given range of 25 to 25. This confirms the correctness of our calculations and process.
