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Chapter 21: Problem 25

Density is mass divided by volume. If the volume \(V\) is temperature dependent, so is the density \(\rho\). Show that the change in density \(\Delta \rho\) with change in temperature \(\Delta T\) is given by $$\Delta \rho=-\beta \rho \Delta T$$ where \(\beta\) is the coefficient of volume expansion. Explain the minus sign.

Short Answer

Expert verified
The change in density with temperature \(\Delta \rho\) is given by \(\Delta \rho=-\beta \rho \Delta T\). The minus sign indicates that the density decreases as the temperature increases, assuming a positive coefficient of volume expansion.

Step by step solution

01

Understanding the Basics

The basic formula for density is \(\rho=\frac{m}{V}\) where \(m\) is mass, \(V\) is volume, and \(\rho\) is density. Since the mass \(m\) does not change with temperature, we only need to consider volume \(V\) which is affected by temperature.
02

Incorporating Coefficient of Volume Expansion

The formula that incorporates the change in volume with temperature is \(V=V_0(1+\beta \Delta T)\) where \(V_0\) is the initial volume, \(\beta\) is the coefficient of volume expansion, and \(\Delta T\) is the change in temperature. If we substitute this back into the original density equation, we get \(\rho=\frac{m}{V_0(1+\beta \Delta T)}\). In other words, density decreases when volume increases due to increase in temperature.
03

Finding the Change in Density with Respect to Temperature

To find the change in density \(\Delta \rho\) with change in temperature \(\Delta T\), we differentiate the density equation with respect to \(T\). This gives us \(\Delta \rho=-\frac{m \beta}{V_0(1+\beta \Delta T)^2}\Delta T=-\beta \rho \Delta T\).
04

Explaining the Minus Sign

The negative sign in \(\Delta \rho=-\beta \rho \Delta T\) signifies that the density decreases as the temperature increases, given a positive value of \(\beta\). This is because as temperature increases, volume increases leading to decrease in density.

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Most popular questions from this chapter

When the temperature of a copper penny (which is not pure copper) is raised by \(100 \mathrm{C}^{\circ}\), its diameter increases by \(0.18 \%\). Find the percent increase in \((a)\) the area of a face, \((b)\) the thickness, ( \(c\) ) the volume, and \((d)\) the mass of the penny. \((e)\) Calculate its coefficient of linear expansion.

(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.

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