A brass wire is to withstand a tensile force of 350 N without breaking. What minimum diameter must the wire have?

Verified step by step guidance

1. Determine the tensile strength of brass, which is the maximum stress it can withstand without breaking. This value can be found in material property tables or relevant textbooks.
2. Understand that tensile stress (\(\sigma\)) is calculated using the formula \(\sigma = \frac{F}{A}\), where \(F\) is the force applied and \(A\) is the cross-sectional area of the wire.
3. Since the wire is cylindrical, the cross-sectional area (\(A\)) can be expressed in terms of the diameter (\(d\)) using the formula \(A = \frac{\pi d^2}{4}\).
4. Substitute the expression for \(A\) into the tensile stress formula to get \(\sigma = \frac{4F}{\pi d^2}\).
5. Rearrange the formula to solve for the diameter \(d\) in terms of the force \(F\) and the tensile strength \(\sigma\) of brass. This will give you the minimum diameter needed to ensure the wire does not break under the given tensile force.

Key Concepts

Tensile Strength

Tensile strength is the maximum amount of tensile (pulling) stress that a material can withstand before failure. It is typically measured in pascals (Pa) and is crucial for determining how much force a material can handle without breaking. For brass, this value varies but is generally high, making it suitable for applications requiring strength.

Stress and Strain

Stress is defined as the force applied per unit area within materials, while strain is the deformation experienced by the material due to that stress. The relationship between stress and strain is described by Hooke's Law, which states that, within the elastic limit, stress is directly proportional to strain. Understanding these concepts is essential for calculating the required diameter of the wire to ensure it can handle the specified tensile force.

Cross-Sectional Area

The cross-sectional area of a wire is the area of a slice taken perpendicular to its length. For a circular wire, this area can be calculated using the formula A = π(d/2)², where d is the diameter. The cross-sectional area is critical in determining how much stress the wire can endure; a larger area can distribute the force more effectively, reducing the likelihood of breaking under tension.