- heat entropy ideal gas
*assignment*Bottle in a Bottle*assignment*Homework##### Bottle in a Bottle

irreversible helium internal energy work first law Energy and Entropy 2021 (2 years)The internal energy of helium gas at temperature \(T\) is to a very good approximation given by \begin{align} U &= \frac32 Nk_BT \end{align}

Consider a very irreversible process in which a small bottle of helium is placed inside a large bottle, which otherwise contains vacuum. The inner bottle contains a slow leak, so that the helium leaks into the outer bottle. The inner bottle contains one tenth the volume of the outer bottle, which is insulated. What is the change in temperature when this process is complete? How much of the helium will remain in the small bottle?*assignment*Directional Derivative*assignment*Homework##### Directional Derivative

Static Fields 2022 (5 years)You are on a hike. The altitude nearby is described by the function \(f(x, y)= k x^{2}y\), where \(k=20 \mathrm{\frac{m}{km^3}}\) is a constant, \(x\) and \(y\) are east and north coordinates, respectively, with units of kilometers. You're standing at the spot \((3~\mathrm{km},2~\mathrm{km})\) and there is a cottage located at \((1~\mathrm{km}, 2~\mathrm{km})\). You drop your water bottle and the water spills out.

- Plot the function \(f(x, y)\) and also its level curves in your favorite plotting software. Include images of these graphs. Special note: If you use a computer program written by someone else, you must reference that appropriately.
- In which direction in space does the water flow?
- At the spot you're standing, what is the slope of the ground in the direction of the cottage?
- Does your result to part (c) make sense from the graph?

*group*Mass is not Conserved*group*Small Group Activity30 min.

##### Mass is not Conserved

Theoretical Mechanics (4 years)energy conservation mass conservation collision

Groups are asked to analyze the following standard problem:

Two identical lumps of clay of (rest) mass m collide head on, with each moving at 3/5 the speed of light. What is the mass of the resulting lump of clay?

*assignment*Free Expansion*assignment*Homework##### Free Expansion

Energy and Entropy 2021 (2 years)The internal energy is of any ideal gas can be written as \begin{align} U &= U(T,N) \end{align} meaning that the internal energy depends only on the number of particles and the temperature, but not the volume.

The ideal gas law \begin{align} pV &= Nk_BT \end{align} defines the relationship between \(p\), \(V\) and \(T\). You may take the number of molecules \(N\) to be constant. Consider the free adiabatic expansion of an ideal gas to twice its volume. “Free expansion” means that no work is done, but also that the process is also neither quasistatic nor reversible.^{*}What is the change in entropy of the gas? How do you know this?

- What is the change in temperature of the gas?

*assignment*Adiabatic Compression*assignment*Homework##### Adiabatic Compression

ideal gas internal energy engine Energy and Entropy 2020A diesel engine requires no spark plug. Rather, the air in the cylinder is compressed so highly that the fuel ignites spontaneously when sprayed into the cylinder.

In this problem, you may treat air as an ideal gas, which satisfies the equation \(pV = Nk_BT\). You may also use the property of an ideal gas that the internal energy depends only on the temperature \(T\), i.e. the internal energy does not change for an isothermal process. For air at the relevant range of temperatures the heat capacity at fixed volume is given by \(C_V=\frac52Nk_B\), which means the internal energy is given by \(U=\frac52Nk_BT\).

**Note: in this problem you are expected to use only the equations given and fundamental physics laws. Looking up the formula in a textbook is***not*considered a solution at this level.If the air is initially at room temperature (taken as \(20^{o}C\)) and is then compressed adiabatically to \(\frac1{15}\) of the original volume, what final temperature is attained (before fuel injection)?

- By what factor does the pressure increase?

*group*Projectile with Linear Drag*group*Small Group Activity120 min.

##### Projectile with Linear Drag

Theoretical Mechanics (4 years)Projectile Motion Drag Forces Newton's 2nd Law Separable Differential Equations

Students consider projectile motion of an object that experiences drag force that in linear with the velocity. Students consider the horizontal motion and the vertical motion separately. Students solve Newton's 2nd law as a differential equation.*assignment*Icecream Mass*assignment*Homework##### Icecream Mass

Static Fields 2022 (5 years)Use integration to find the total mass of the icecream in a packed cone (both the cone and the hemisphere of icecream on top).

*group*Applying the equipartition theorem*group*Small Group Activity30 min.

##### Applying the equipartition theorem

Contemporary Challenges 2022 (4 years) Students count the quadratic degrees of freedom of a few toy molecules to predict their internal energy at temperature \(T\).*assignment*Center of Mass for Two Uncoupled Particles*assignment*Homework##### Center of Mass for Two Uncoupled Particles

Central Forces 2023 (3 years)Consider two particles of equal mass \(m\). The forces on the particles are \(\vec F_1=0\) and \(\vec F_2=F_0\hat{x}\). If the particles are initially at rest at the origin, find the position, velocity, and acceleration of the center of mass as functions of time. Solve this problem in two ways, with or without theorems about the center of mass motion. Write a short description comparing the two solutions.

*grading*Free expansion*grading*Quiz60 min.

##### Free expansion

Energy and Entropy 2021 (2 years) Students will determine the change in entropy (positive, negative, or none) for both the system and surroundings in three different cases. This is followed by an active whole-class discussion about where the entropy comes from during an irreversible process.-
Energy and Entropy 2021 (2 years)
Consider the bottle in a bottle problem in a previous problem set, summarized here.

A small bottle of helium is placed inside a large bottle, which otherwise contains vacuum. The inner bottle contains a slow leak, so that the helium leaks into the outer bottle. The inner bottle contains one tenth the volume of the outer bottle, which is insulated.The volume of the small bottle is 0.001 m

^{23}and the volume of the big bottle is 0.01 m^{3}. The initial state of the gas in the small bottle was \(p=106\) Pa and its temperature \(T=300\) K. Approximate the helium gas as an ideal gas of equations of state \(pV=Nk_BT\) and \(U=\frac32 Nk_BT\).How many molecules of gas does the large bottle contain? What is the final temperature of the gas?

Compute the integral \(\int \frac{{\mathit{\unicode{273}}} Q}{T}\) and the change of entropy \(\Delta S\) between the initial state (gas in the small bottle) and the final state (gas leaked in the big bottle).

- Discuss your results.