A circular steel wire 2.00 m long must stretch no more than 0.25 cm when a tensile force of 700 N is applied to each end of the wire. What minimum diameter is required for the wire?

Verified step by step guidance

1. Identify the relevant properties and constants for steel, such as Young's modulus (E). For steel, E typically ranges around 200 GPa.
2. Convert all units to the International System of Units (SI). For example, convert the length change from cm to meters (0.25 cm = 0.0025 m) and ensure the length of the wire is in meters (2.00 m).
3. Use Hooke's Law for stretching, which relates the force (F), the cross-sectional area (A), Young's modulus (E), the original length (L), and the change in length (\( \Delta L \)). The formula is given by \( F = \frac{EA \Delta L}{L} \).
4. Rearrange the formula to solve for the cross-sectional area (A) of the wire: \( A = \frac{FL}{E \Delta L} \).
5. Since the wire is circular, relate the cross-sectional area (A) to the diameter (d) using the area formula for a circle, \( A = \frac{\pi d^2}{4} \). Solve for the diameter (d) by rearranging the formula: \( d = \sqrt{\frac{4A}{\pi}} \).

Key Concepts

Hooke's Law

Hooke's Law states that the force exerted by a spring (or elastic material) is directly proportional to the amount it is stretched or compressed, as long as the material's elastic limit is not exceeded. This relationship can be expressed as F = kx, where F is the force, k is the spring constant, and x is the displacement. In this context, it helps determine how much the wire will stretch under a given tensile force.

Young's Modulus

Young's Modulus is a measure of the stiffness of a material, defined as the ratio of tensile stress to tensile strain. It is expressed as E = stress/strain, where stress is the force per unit area and strain is the relative change in length. This concept is crucial for understanding how much a material will deform under stress, which is necessary for calculating the required diameter of the wire to limit its stretch.

Tensile Stress and Strain

Tensile stress is the force applied per unit area of a material, while tensile strain is the deformation experienced by the material relative to its original length. These concepts are essential for analyzing how the wire will respond to the applied tensile force. By calculating the tensile stress and relating it to the allowable strain (based on the maximum stretch), one can determine the minimum diameter needed to meet the requirements.