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Chapter 21: Problem 21

A rod is measured to be \(20.05 \mathrm{~cm}\) long using a steel ruler at a room temperature of \(20^{\circ} \mathrm{C}\). Both the rod and the ruler are placed in an oven at \(270^{\circ} \mathrm{C}\), where the rod now measures \(20.11 \mathrm{~cm}\) using the same rule. Calculate the coefficient of thermal expansion for the material of which the rod is made.

Short Answer

Expert verified
The coefficient of linear thermal expansion for the material of rod at the given conditions is \(1.2 \times 10^{-5}~K^{-1}\).

Step by step solution

01

Calculate the change in length and temperature

The change in length \(\Delta L\) of the rod is the final length subtracted by the initial length: \(\Delta L = L_{final} - L_{initial} = 20.11~cm - 20.05~cm = 0.06~cm\). The change in temperature \(\Delta T\) is also the final temperature subtracted by the initial temperature: \(\Delta T = T_{final} - T_{initial} = 270^{\circ} C - 20^{\circ} C = 250^{\circ} C\).
02

Conversion of units

Although not strictly necessary, to maintain unit consistency, it is better to convert temperatures from Celsius to Kelvin. In this case, because we're dealing with a temperature difference, the numeric value remains the same, since degrees Celsius and Kelvin have the same magnitude. It is just \(\Delta T = 250 K\). Furthermore, convert the length from cm to m, so \(\Delta L = 0.0006~m\) and \(L_{initial} = 0.2005~m\).
03

Calculate the coefficient of linear thermal expansion

The coefficient of linear thermal expansion \(\alpha\) can be calculated from the equation \(\alpha = \frac{\Delta L}{L_{initial} \cdot \Delta T} = \frac{0.0006~m}{0.2005~m \cdot 250~K} \)

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Most popular questions from this chapter

(a) Using the ideal gas law and the definition of the coefficient of volume expansion (Eq. \(21-12\) ), show that \(\beta=1 / T\) for an ideal gas at constant pressure. \((b)\) In what units must \(T\) be expressed? If \(T\) is expressed in those units, can you express \(\beta\) in units of \(\left(\mathrm{C}^{\circ}\right)^{-1} ?(c)\) Estimate the value of \(\beta\) for an ideal gas at room temperature.

(a) Prove that the change in rotational inertia \(I\) with temperature of a solid object is given by \(\Delta I=2 \alpha I \Delta T .(b)\) A thin uniform brass rod, spinning freely at 230 rev/s about an axis perpendicular to it at its center, is heated without mechanical contact until its temperature increases by \(170 \mathrm{C}^{\circ}\). Calculate the change in angular velocity.

An air bubble of \(19.4 \mathrm{~cm}^{3}\) volume is at the bottom of a lake \(41.5 \mathrm{~m}\) deep where the temperature is \(3.80^{\circ} \mathrm{C}\). The bubble rises to the surface, which is at a temperature of \(22.6^{\circ} \mathrm{C}\). Take the temperature of the bubble to be the same as that of the surrounding water and find its volume just before it reaches the surface.

The best vacuum that can be attained in the laboratory corresponds to a pressure of about \(10^{-18} \mathrm{~atm}\), or \(1.01 \times 10^{-13} \mathrm{~Pa}\). How many molecules are there per cubic centimeter in such a vacuum at \(22^{\circ} \mathrm{C} ?\)

An aluminum cup of \(110 \mathrm{~cm}^{3}\) capacity is filled with glycerin at \(22^{\circ} \mathrm{C}\). How much glycerin, if any, will spill out of the cup if the temperature of the cup and glycerin is raised to \(28^{\circ} \mathrm{C}\) ? (The coefficient of volume expansion of glycerin is \(5.1 \times\) \(\left.10^{-4} / \mathrm{C}^{\circ} .\right)\)
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