Question

A \(2.00-\mathrm{m}\) - long steel wire in a musical instrument has a radius of \(0.300 \mathrm{~mm}\). When the wire is under a tension of \(90.0 \mathrm{~N}\), how much does its length change?

Short answer

Answer: The change in length of the steel wire is approximately 3.18 mm.

Step-by-step solution

Identify the known variables

The known variables in this problem are the length (\(L\)) of the steel wire, its radius (\(r\)), and the tension force applied to it (\(F\)). We have: Length, \(L = 2.00\,\mathrm{m}\) Radius, \(r = 0.300\,\mathrm{mm} = 3.0 \times 10^{-4}\,\mathrm{m}\) Tension force, \(F = 90.0\,\mathrm{N}\)

Calculate the cross-sectional area of the wire

To find the change in length, we need the cross-sectional area (\(A\)) of the wire. For a cylindrical wire, this area can be calculated using the formula: $$ A = \pi r^2 $$ Substituting the given radius: $$ A = \pi (3.0 \times 10^{-4}\,\mathrm{m})^2 \approx 2.827 \times 10^{-7}\,\mathrm{m}^2 $$

Calculate the stress in the wire

The stress (\(\sigma\)) in the wire due to the tension force can be calculated using the formula: $$ \sigma = \frac{F}{A} $$ Substituting the given tension force and the calculated cross-sectional area: $$ \sigma = \frac{90.0\,\mathrm{N}}{2.827 \times 10^{-7}\,\mathrm{m}^2} \approx 3.184 \times 10^8\,\mathrm{Pa} $$

Determine the strain and Young's modulus for steel

In order to find the change in length, we need to determine the strain (\(\epsilon\)) in the wire. This can be done using the stress-strain relationship: $$ \sigma = E \epsilon $$ Where \(E\) is Young's modulus of the material (steel in this case), and \(\epsilon\) is the strain in the wire. The Young's modulus for steel is approximately \(2.0 \times 10^{11}\,\mathrm{Pa}\), so we can solve for the strain: $$ \epsilon = \frac{\sigma}{E} = \frac{3.184 \times 10^8\,\mathrm{Pa}}{2.0 \times 10^{11}\,\mathrm{Pa}} \approx 1.592 \times 10^{-3} $$

Calculate the change in length

Finally, we can use the strain to calculate the change in length (\(\Delta L\)) using Hooke's Law: $$ \Delta L = \epsilon \cdot L $$ Substituting the calculated strain and the given length: $$ \Delta L = (1.592 \times 10^{-3}) \cdot 2.00\,\mathrm{m} \approx 3.18 \times 10^{-3}\,\mathrm{m} $$ Therefore, the length of the steel wire changes by approximately \(3.18 \times 10^{-3}\,\mathrm{m}\) or \(3.18\,\mathrm{mm}\) under a tension of \(90.0\,\mathrm{N}\).

Key concepts

Stress and Strain

Understanding the concepts of stress and strain is fundamental in the study of materials and their mechanical properties. Stress is a measure of the force applied to a material over the area it acts upon. It's calculated with the formula \[\sigma = \frac{F}{A}\]where \(\sigma\) is the stress, \(F\) is the force applied, and \(A\) is the cross-sectional area. Stress is typically expressed in pascals (Pa).Strain, on the other hand, describes how much a material deforms under stress. It is a dimensionless quantity that measures the deformation as a ratio of the original length, using the formula \[\epsilon = \frac{\Delta L}{L}\]where \(\epsilon\) is the strain, \(\Delta L\) is the change in length, and \(L\) is the initial length. Strain is a measure of how much material stretches or compresses due to the applied force.

Young's Modulus

Moving to Young's modulus, this is a measure of the stiffness of a material. It shows the relationship between stress and strain and is defined by the straight-line region of the stress-strain curve for a material when it's deformed elastically. The formula for Young's modulus is\[E = \frac{\sigma}{\epsilon}\]where \(E\) represents Young's modulus, \(\sigma\) is the stress, and \(\epsilon\) is the strain. Young's modulus is a constant for a given material and is crucial for determining how much a material will deform when a certain stress is applied. For steel, which is often used in construction due to its considerable stiffness, Young's modulus is approximately \(2.0 \times 10^{11}\mathrm{Pa}\).

Hooke's Law

Delving into Hooke's Law, it's a principle that states that, within the elastic limit of a solid material, the strain in the material is directly proportional to the applied stress. The law can be expressed as\[F = kx\]where \(F\) is the force applied, \(k\) is the spring constant, and \(x\) is the displacement or change in length. This principle helps understand how materials deform under mechanical forces and is the cornerstone of elasticity theory. By rearranging Hooke's Law, it's also possible to find the change in length when the material is under tension or compression, applying the formula\[\Delta L = \epsilon \cdot L\].

Cross-sectional Area

Considering the cross-sectional area is vital when analyzing forces and resulting stress on a material. It is the area of an object as seen from a cross-section, perpendicular to the direction of the force applied. For example, in the case of a cylindrical wire, the cross-sectional area is obtained through the formula\[A = \pi r^2\]where \(r\) stands for the radius of the cylinder. A material's cross-sectional area will affect how it distributes and withstands stress. A larger cross-sectional area generally means that stress is spread out over a greater area, often allowing the material to withstand a larger force without failing.