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Chapter 9: Problem 9

A \(200-\mathrm{kg}\) load is hung on a wire of length \(4.00 \mathrm{~m}\), cross-sectional area \(0.200 \times 10^{-4} \mathrm{~m}^{2}\), and Young's modulus \(8.00 \times 10^{10} \mathrm{~N} / \mathrm{m}^{2}\). What is its increase in length?

Short Answer

Expert verified
\(\Delta L = 0.0098 \: m\) or \(9.8 \:mm\).

Step by step solution

01

Understand the Problem

The task is to find the increase in length - often called elongation - of a wire when a certain load is applied. The properties of the wire, including its original length, cross-sectional area, and elasticity (Young's modulus) are given, as well as the weight that is hung on it. The formula that connects all these values is that for Young's modulus, \(Y = \frac{F}{A}/\frac{\Delta L}{L}\), where \(F\) is the force on the wire, \(A\) is the cross-sectional area of the wire, \(\Delta L\) is the change in length (elongation), and \(L\) is the original length. Rearrange the formula, isolating \(\Delta L\), to solve for elongation.
02

Substitute the values into the formula

After rearranging the formula for Young's modulus, you get \(\Delta L = \frac{FL}{YA}\). Substitute the given values into the formula: \( F = 200 kg \times 9.81 m/s虏 = 1962 N\) (force equals mass times gravity), \( L = 4 m \), \( Y = 8.00 \times 10^{10} N/m虏 \) and \( A = 0.200 \times 10^{-4} m虏 \).
03

Calculation

Now perform the multiplication and division operations as indicated by the formula. This operation should yield the increase in length of the wire, \(\Delta L\), when a 200-kg weight is hung from it.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Elasticity
Elasticity refers to a material鈥檚 ability to return to its original shape after being stretched or compressed. It's like a rubber band snapping back after you pull on it. In physics, elasticity is quantified using Young's modulus, a constant that measures how much a material will deform under stress. A high Young's modulus means the material is rigid, while a low value indicates flexibility. Consider Young's modulus as a stiffness indicator that provides insight into how different materials behave when forces are applied.
Stress and Strain
Stress and strain are central to understanding how materials react to forces. Stress is the force exerted on a material per unit area, calculated as \( \text{Stress} = \frac{F}{A} \), where \( F \) is the force on the object and \( A \) is the cross-sectional area. Strain, on the other hand, is the deformation or elongation of the material, determined by \( \text{Strain} = \frac{\Delta L}{L} \), where \( \Delta L \) is the change in length, and \( L \) is the original length.

These two concepts help predict how much a material will stretch or compress when a force is applied. Using Young's modulus, we can link stress and strain with the equation: \( Y = \frac{\text{Stress}}{\text{Strain}} \). This relationship is key in solving physics problems related to material deformation.
Physics Problem Solving
Solving physics problems requires a clear understanding of the concepts and how to apply given formulas. Let's explore a typical approach using the example of calculating the elongation of a wire. Start by understanding what is known:

  • Original length of the wire \( (L) \)
  • Cross-sectional area \( (A) \)
  • Applied force \( (F) \)
  • Young's modulus \( (Y) \)
Next, rearrange the Young's modulus formula to solve for the unknown, typically the change in length \( (\Delta L) \): \( \Delta L = \frac{FL}{YA} \). Substitute the known values into the equation and perform the calculations step-by-step.

Check your work to ensure logical conclusions, validating that the numerical results make sense given the physical context. This practical problem-solving strategy keeps complicated tasks manageable and clear.

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Most popular questions from this chapter

Old Faithful geyser in Yellowstone Park erupts at approximately 1-hour intervals, and the height of the fountain reaches \(40.0 \mathrm{~m}\) (Fig. P9.57). (a) Consider the rising stream as a series of separate drops. Analyze the free-fall motion of one of the drops to determine the speed at which the water leaves the ground. (b) Treat the rising stream as an ideal fluid in streamline flow. Use Bernoulli's equation to determine the speed of the water as it leaves ground level. (c) What is the pressure (above atmospheric pressure) in the heated underground chamber \(175 \mathrm{~m}\) below the vent? You may assume the chamber is large compared with the geyser vent.

A certain fluid has a density of \(1080 \mathrm{~kg} / \mathrm{m}^{3}\) and is observed to rise to a height of \(2.1 \mathrm{~cm}\) in a \(1.0\)-mm-diameter tube. The contact angle between the wall and the fluid is zero. Calculate the surface tension of the fluid.

An object weighing \(300 \mathrm{~N}\) in air is immersed in water after being tied to a string connected to a balance. The scale now reads \(265 \mathrm{~N}\). Immersed in oil, the object appears to weigh \(275 \mathrm{~N}\). Find (a) the density of the object and (b) the density of the oil.

A \(62.0-\mathrm{kg}\) survivor of a cruise line disaster rests atop a block of Styrofoam insulation, using it as a raft. The Styrofoam has dimensions \(2.00 \mathrm{~m} \times 2.00 \mathrm{~m} \times 0.0900 \mathrm{~m}\). The bottom \(0.024 \mathrm{~m}\) of the raft is submerged. (a) Draw a force diagram of the system consisting of the survivor and raft. (b) Write Newton's second law for the system in one dimension, using \(B\) for buoyancy, \(w\) for the weight of the survivor, and \(w_{r}\) for the weight of the raft. (Set \(a=0\).) (c) Calculate the numeric value for the buoyancy, \(B\). (Seawater has density \(1025 \mathrm{~kg} / \mathrm{m}^{3}\).) (d) Using the value of \(B\) and the weight \(w\) of the survivor, calculate the weight \(w_{r}\) of the Styrofoam. (e) What is the density of the Styrofoam? (f) What is the maximum buoyant force, corresponding to the raft being submerged up to its top surface? (g) What total mass of survivors can the raft support?

A stainless-steel orthodontic wire is applied to a tooth, as in Figure P9.14. The wire has an unstretched length of \(3.1 \mathrm{~cm}\) and a radius of \(0.11 \mathrm{~mm}\). If the wire is stretched \(0.10 \mathrm{~mm}\), find the magnitude and direction of the force on the tooth. Disregard the width of the tooth and assume Young's modulus for stainless steel is \(18 \times 10^{10} \mathrm{~Pa}\).
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