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A steel tape measure is marked in such a way that it gives accurate length measurements at a normal room temperature of \(20^{\circ} \mathrm{C}\). If this tape measure is used outdoors on a cold day when the temperature is \(0^{\circ} \mathrm{C}\), are its measurements too long, too short, or accurate?

Short Answer

Expert verified
Measurements will be too short, not accurate.

Step by step solution

01

Understanding Thermal Expansion

When materials are subjected to temperature changes, their dimensions change due to thermal expansion or contraction. Metals, including steel, will contract when cooled and expand when heated. This means at colder temperatures, the steel tape measure will undergo contraction.
02

Determining the Effect of Temperature Change

Consider the linear thermal expansion formula: \[ \Delta L = L_0 \times \alpha \times \Delta T \]where \( \Delta L \) is the change in length, \( L_0 \) is the original length, \( \alpha \) is the coefficient of linear expansion for steel, and \( \Delta T \) is the change in temperature. Here, \( \Delta T = 0^{\circ}C - 20^{\circ}C = -20^{\circ}C \), indicating contraction.
03

Calculating the Effect

Since \( \Delta T \) is negative, \( \Delta L \) will also be negative, meaning the tape measure will contract, making each marked length on the tape slightly shorter than it would be at \(20^{\circ} C\). Therefore, any measurements taken will be too short.
04

Concluding the Measurement Accuracy

On a cold day, with the steel tape measure at \(0^{\circ}C\), the measurements appear slightly shorter because the tape contracts. Therefore, outside measurements are shorter than they should be, not accurate.

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Key Concepts

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

Steel Properties
Steel is a commonly used alloy known for its strong and durable nature.
It's composed of iron, carbon, and other elements that give it unique mechanical properties.
One key property of steel is its ability to withstand thermal changes, though it does experience thermal expansion and contraction.
Some important properties about steel include:
  • Durability: Steel is resistant to wear and tear.
  • Tensile Strength: This refers to steel's ability to resist tension and being pulled apart.
  • Elasticity: Steel can return to its original shape after being stretched or compressed.
  • Thermal Resistance: Steel can handle a certain amount of thermal stress without deforming or weakening.
In regards to temperature, steel has a specific coefficient of linear expansion, meaning it will change size with temperature changes.
This makes it crucial for applications requiring precise measurements and why engineers take into account these changes during different weather conditions.
Temperature Effect on Measurement
When a steel tape measure is used to measure length, the external temperature can impact its accuracy.
As the temperature drops, steel contracts, meaning its length slightly decreases.
This is due to its linear thermal expansion properties, where materials change length in response to temperature changes. If a steel measuring tape is accurate at a baseline room temperature (like 20°C), it will not measure correctly at different temperatures.
In colder temperatures, such as 0°C, the tape contracts creating a situation where the measured lengths become shorter than they actually are.
Thus:
  • Colder environments: Measurements tend to be shorter because of contraction.
  • Warmer environments: Measurements might be longer due to expansion.
Understanding how temperature impacts measurement is crucial for tasks requiring precision.
Professionals often adjust their measurements based on known temperature effects to maintain accuracy in their work.
Linear Thermal Expansion Formula
The linear thermal expansion formula provides a mathematical way to determine how much a material's length changes with temperature shifts.
The formula is expressed as: \[ \Delta L = L_0 \times \alpha \times \Delta T \]Where:
  • \( \Delta L \) is the change in length.
  • \( L_0 \) is the original length of the material.
  • \( \alpha \) is the coefficient of linear expansion specific to the material, for steel, this is typically around \(11 \times 10^{-6} /^{\circ}C\).
  • \( \Delta T \) is the temperature change, calculated as the final temperature minus the initial temperature.
If there is a negative temperature change, like going from 20°C to 0°C, the resulting negative \( \Delta L \) indicates contraction.
Understanding and applying this formula allows for accurate predictions of how steel and other materials change with temperature, ensuring that engineers and builders can maintain precision in their projects.

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Most popular questions from this chapter

A sheet of aluminum has a circular hole with a diameter of \(10.0 \mathrm{cm} .\) A \(9.99-\mathrm{cm}\) -long steel rod is placed inside the hole, along a diameter of the circle, as shown in Figure \(16-21\). It is desired to change the temperature of this system until the steel rod just touches both sides of the circle. (a) Should the temperature of the system be increased or decreased? Explain. (b) By how much should the temperature be changed?

Find the heat that flows in 1.0 s through a lead brick \(15 \mathrm{cm}\) long if the temperature difference between the ends of the brick is \(9.5 \mathrm{C}^{\circ} .\) The cross-sectional area of the brick is \(14 \mathrm{cm}^{2}\).

Heat Transport in the Human Body The core temperature of the human body is \(37.0^{\circ} \mathrm{C}\), and the skin, with a surface area of \(1.40 \mathrm{m}^{2},\) has a temperature of \(34.0^{\circ} \mathrm{C}\). (a) Find the rate of heat transfer out of the body under the following as sumptions: (i) The average thickness of tissue between the core and the skin is \(1.20 \mathrm{cm} ;\) (ii) the thermal conductivity of the tissue is that of water. (b) Without repeating the calculation of part (a), what rate of heat transfer would you expect if the skin temperature were to fall to \(31.0^{\circ} \mathrm{C}\) ? Explain.

Lowest Temperature on Earth The official record for the lowest temperature ever recorded on Earth was set at Vostok, Antarctica, on July \(21,1983 .\) The temperature on that day fell to \(-89.2^{\circ} \mathrm{C},\) well below the temperature of dry ice. What is this temperature in degrees Fahrenheit?

A pendulum is made from an aluminum rod with a mass attached to its free end. If the pendulum is cooled, (a) does the pendulum's period increase, decrease, or stay the same? (b) Choose the best explanation from among the following: I. The period of a pendulum depends only on its length and the acceleration of gravity. It is independent of mass and temperature. II. Cooling makes everything move more slowly, and hence the period of the pendulum increases. III. Cooling shortens the aluminum rod, which decreases the period of the pendulum.

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