In one model of a solid, the material is assumed to consist of a regular array of atoms in which each atom has a fixed equilibrium position and is connected by springs to its neighbors. Each atom can vibrate in the $x, y$, and $z$ directions. The total energy of an atom in this model is

$$E=\frac{1}{2} m v_x^2+\frac{1}{2} m v_y^2+\frac{1}{2} m v_z^2+\frac{1}{2} k x^2+\frac{1}{2} k y^2+k z^2$$

What is the average energy of an atom in the solid when the temperature is $T$ ? What is the total energy of $1 \mathrm{~mol}$ of such a solid?

A helium balloon is used to lift a load of $110 \text{~N}$. The weight of the envelope of the balloon is $50.0 \text{~N}$ and the volume of the helium when the balloon is fully inflated is $32.0 \text{~m}^3$. The temperature of the air is $0^{\circ} \text{C}$ and the atmospheric pressure is $1.00 \text{~atm}$. The balloon is inflated with a sufficient amount of helium gas that the net upward force on the balloon and its load is $30.0$ N. Neglect any effects due to the changes of temperature as the altitude changes. $(d)$ If the answer to Part $(c)$ is "Yes," what is the maximum altitude attained by the balloon?

A helium balloon is used to lift a load of $110 \text{~N}$. The weight of the envelope of the balloon is $50.0 \text{~N}$ and the volume of the helium when the balloon is fully inflated is $32.0 \text{~m}^3$. The temperature of the air is $0^{\circ} \text{C}$ and the atmospheric pressure is $1.00 \text{~atm}$. The balloon is inflated with a sufficient amount of helium gas that the net upward force on the balloon and its load is $30.0$ N. Neglect any effects due to the changes of temperature as the altitude changes. $(c)$ Does the balloon ever reach the altitude at which it is fully inflated?

A helium balloon is used to lift a load of $110 \text{~N}$. The weight of the envelope of the balloon is $50.0 \text{~N}$ and the volume of the helium when the balloon is fully inflated is $32.0 \text{~m}^3$. The temperature of the air is $0^{\circ} \text{C}$ and the atmospheric pressure is $1.00 \text{~atm}$. The balloon is inflated with a sufficient amount of helium gas that the net upward force on the balloon and its load is $30.0$ N. Neglect any effects due to the changes of temperature as the altitude changes. $(b)$ At what altitude will the balloon be fully inflated?

The listener moves at $80 \mathrm{~m} / \mathrm{s}$ relative to still air toward the stationary source. (a) What is the wavelength of the sound in the region between the source and the listener? (b) What is the frequency heard by the listener?

A helium balloon is used to lift a load of $110 \text{~N}$. The weight of the envelope of the balloon is $50.0 \text{~N}$ and the volume of the helium when the balloon is fully inflated is $32.0 \text{~m}^3$. The temperature of the air is $0^{\circ} \text{C}$ and the atmospheric pressure is $1.00 \text{~atm}$. The balloon is inflated with a sufficient amount of helium gas that the net upward force on the balloon and its load is $30.0$ N. Neglect any effects due to the changes of temperature as the altitude changes. $(a)$ How many moles of helium gas is contained in the balloon?

The kinetic energy of a segment of length $\Delta x$ and mass $\Delta m$ of a vibrating string is given by $\Delta K=\frac{1}{2} \Delta m(\partial y / \partial t)^2=$ $\frac{1}{2} \mu(\partial y / a t)^2 \Delta x$, where $\mu=\Delta m / \Delta x$. (a) Find the total kinetic energy of the $n$th mode of vibration of a string of length $L$ fixed at both ends. (b) Give the maximum kinetic energy of the string. (c) What is the wave function when the kinetic energy has its maximum value? $(d)$ Show that the maximum kinetic energy in the $n$th mode is proportional to $n^2 A_n^2$.

A scuba diver is $40 \text{~m}$ below the surface of a lake, where the temperature is $5.0^{\circ} \text{C}$. He releases an air bubble that has a volume of $15 \text{~cm}^3$. The bubble rises to the surface, where the temperature is $25^{\circ} \text{C}$. Assume that the air in the bubble is always in thermal equilibrium with the surrounding water, and assume that there is no exchange of molecules between the bubble and the surrounding water. What is the volume of the bubble right before it breaks the surface? Hint: Remember that the pressure also changes.

Power is to be transmitted along a stretched wire by means of transverse harmonic waves. The wave speed is $10 \mathrm{~m} / \mathrm{s}$ and the linear mass density of the wire is $0.01 \mathrm{~kg} / \mathrm{m}$. The power source oscillates with an amplitude of $0.50 \mathrm{~mm}$.(c) Which of the quantities would probably be the easiest to vary?

Calculate the mass density of air at a temperature of $24^{\circ} \mathrm{C}$ and a pressure of $1 \mathrm{~atm}\left(1.01 \times 10^5 \mathrm{~N} / \mathrm{m}^2\right)$ using the ideal-gas law. Air is roughly 74 percent $\mathrm{N}_2$ and 26 percent $\mathrm{O}_2$.

An automobile tire is filled to a gauge pressure of $200 \text{kPa}$ when its temperature is $20^{\circ} \text{C}$. (Gauge pressure is the difference between the actual pressure and atmospheric pressure.) After the car has been driven at high speeds, the tire temperature increases to $50^{\circ} \text{C}$. $(b)$ Calculate the gauge pressure if the tire expands so the volume of the enclosed air increases by 10 percent.

An automobile tire is filled to a gauge pressure of $200 \text{kPa}$ when its temperature is $20^{\circ} \text{C}$. (Gauge pressure is the difference between the actual pressure and atmospheric pressure.) After the car has been driven at high speeds, the tire temperature increases to $50^{\circ} \text{C}$. $(a)$ Assuming that the volume of the tire does not change and that air behaves as an ideal gas, find the gauge pressure of the air in the tire.

An acrobat is walking on a tightrope of length $L=20.1 \mathrm{~m}$ attached to supports $A$ and $B$ at a distance of $20.0 \mathrm{~m}$ from each other. The combined weight of the acrobat and his balancing pole is $800 \mathrm{~N}$, and the friction between his shoes and the rope is large enough to prevent him from slipping. Neglecting the weight of the rope and any elastic deformation, write a computer program to calculate the deflection $y$ and the tension in portions $A C$ and $B C$ of the rope for values of $x$ from $0.5 \mathrm{~m}$ to $10.0 \mathrm{~m}$ using $0.5 \mathrm{~m}$ increments. From the data obtained, determine $(a)$ the maximum deflection of the rope, $(b)$ the maximum tension in the rope, $(c)$ the smallest values of the tension in portions $A C$ and $B C$ of the rope.

A plot of volume V versus absolute temperature T for a process that takes a fixed amount of an ideal gas from point A to point B. What happens to the pressure of the gas during this process?