## Activity: Mass is not Conserved

Theoretical Mechanics 2021 (2 years)

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?

What students learn
• Mass is energy.
• Energy is conserved, but mass alone is not.

## Prerequisite Knowledge

• Definitions of the relativistic momentum 2-vector, relativistic momentum conservation, & rest energy.

## Student Conversations

1. We're not changing reference frames here, so there is no Lorentz transformation. Students may not realize that the resulting lump is at rest.
2. Many students will recognize that this is a totall inelastic collision. Students will not really be comfortable with the idea that energy is conserved in this collision. Remind them that in classical physics kinetic energy is conserved for elastic collisions, but for inelastic collisions, the total energy (taking into account the thermal energies, rest energies, etc) is conserved (the universe doesn't lose energy in the interaction).
3. The big take-home message is that the total mass in the collision is not conserved. The mass of final lump is not $M=2m$. Some of the kinetic energy of the two initial lumps transforms into rest mass of the final lump.
4. In this problem, the two lumps are moving at the same speed, so the $\gamma$ factor is the same for the two initial lumps. $\gamma = 1$ for the final lump. In general, each lump will have it's own $\gamma$ factor based on it's speed relative to the lab frame.

## Wrap-Up

• Ask several students to present/defend their responses.

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 Heat pump

assignment Homework

##### Heat pump
Carnot efficiency Work Entropy Heat pump Thermal and Statistical Physics 2020
1. Show that for a reversible heat pump the energy required per unit of heat delivered inside the building is given by the Carnot efficiency: \begin{align} \frac{W}{Q_H} &= \eta_C = \frac{T_H-T_C}{T_H} \end{align} What happens if the heat pump is not reversible?

2. Assume that the electricity consumed by a reversible heat pump must itself be generated by a Carnot engine operating between the even hotter temperature $T_{HH}$ and the cold (outdoors) temperature $T_C$. What is the ratio $\frac{Q_{HH}}{Q_H}$ of the heat consumed at $T_{HH}$ (i.e. fuel burned) to the heat delivered at $T_H$ (in the house we want to heat)? Give numerical values for $T_{HH}=600\text{K}$; $T_{H}=300\text{K}$; $T_{C}=270\text{K}$.

3. Draw an energy-entropy flow diagram for the combination heat engine-heat pump, similar to Figures 8.1, 8.2 and 8.4 in the text (or the equivalent but sloppier) figures in the course notes. However, in this case we will involve no external work at all, only energy and entropy flows at three temperatures, since the work done is all generated from heat.

• assignment Energy, Entropy, and Probabilities

assignment Homework

##### Energy, Entropy, and Probabilities
Energy Entropy Probabilities Thermodynamic identity

The goal of this problem is to show that once we have maximized the entropy and found the microstate probabilities in terms of a Lagrange multiplier $\beta$, we can prove that $\beta=\frac1{kT}$ based on the statistical definitions of energy and entropy and the thermodynamic definition of temperature embodied in the thermodynamic identity.

The internal energy and entropy are each defined as a weighted average over microstates: \begin{align} U &= \sum_i E_i P_i & S &= -k_B\sum_i P_i \ln P_i \end{align}: We saw in clase that the probability of each microstate can be given in terms of a Lagrange multiplier $\beta$ as \begin{align} P_i &= \frac{e^{-\beta E_i}}{Z} & Z &= \sum_i e^{-\beta E_i} \end{align} Put these probabilities into the above weighted averages in order to relate $U$ and $S$ to $\beta$. Then make use of the thermodynamic identity \begin{align} dU = TdS - pdV \end{align} to show that $\beta = \frac1{kT}$.

• assignment Energy, Entropy, and Probabilities

assignment Homework

##### Energy, Entropy, and Probabilities
Thermal and Statistical Physics 2020

The goal of this problem is to show that once we have maximized the entropy and found the microstate probabilities in terms of a Lagrange multiplier $\beta$, we can prove that $\beta=\frac1{kT}$ based on the statistical definitions of energy and entropy and the thermodynamic definition of temperature embodied in the thermodynamic identity.

The internal energy and entropy are each defined as a weighted average over microstates: \begin{align} U &= \sum_i E_i P_i & S &= -k_B\sum_i P_i \ln P_i \end{align} We saw in clase that the probability of each microstate can be given in terms of a Lagrange multiplier $\beta$ as \begin{align} P_i &= \frac{e^{-\beta E_i}}{Z} & Z &= \sum_i e^{-\beta E_i} \end{align} Put these probabilities into the above weighted averages in order to relate $U$ and $S$ to $\beta$. Then make use of the thermodynamic identity \begin{align} dU = TdS - pdV \end{align} to show that $\beta = \frac1{kT}$.

• group de Broglie wavelength after freefall

group Small Group Activity

30 min.

##### de Broglie wavelength after freefall
Contemporary Challenges 2022 (3 years)

In this activity students combine energy conservation with the relationship between the de Broglie wavelength and momentum to find the wavelength of atoms that have been dropped a given distance.
• group Electric Field of Two Charged Plates

group Small Group Activity

30 min.

##### Electric Field of Two Charged Plates
• Students need to understand that the surface represents the electric potential in the center of a parallel plate capacitor. Try doing the activity “Electric Potential of a Parallel Plate Capacitor” before this activity.
• Students should know that
1. objects with like charge repel and opposite charge attract,
2. object tend to move toward lower energy configurations
3. The potential energy of a charged particle is related to its charge: $U=qV$
4. The force on a charged particle is related to its charge: $\vec{F}=q\vec{E}$
• group Number of Paths

group Small Group Activity

30 min.

##### Number of Paths

Student discuss how many paths can be found on a map of the vector fields $\vec{F}$ for which the integral $\int \vec{F}\cdot d\vec{r}$ is positive, negative, or zero. $\vec{F}$ is conservative. They do a similar activity for the vector field $\vec{G}$ which is not conservative.
• group Survivor Outer Space: A kinesthetic approach to (re)viewing center-of-mass

group Small Group Activity

10 min.

##### Survivor Outer Space: A kinesthetic approach to (re)viewing center-of-mass
Central Forces 2022 (2 years) A group of students, tethered together, are floating freely in outer space. Their task is to devise a method to reach a food cache some distance from their group.
• face Central Forces Introduction: Lecture Notes

face Lecture

5 min.

##### Central Forces Introduction: Lecture Notes
Central Forces 2022
• assignment Mass of a Slab

assignment Homework

##### Mass of a Slab
Static Fields 2022 (4 years)

Determine the total mass of each of the slabs below.

1. A square slab of side length $L$ with thickness $h$, resting on a table top at $z=0$, whose mass density is given by \begin{equation} \rho=A\pi\sin(\pi z/h). \end{equation}
2. A square slab of side length $L$ with thickness $h$, resting on a table top at $z=0$, whose mass density is given by \begin{equation} \rho = 2A \Big( \Theta(z)-\Theta(z-h) \Big) \end{equation}
3. An infinitesimally thin square sheet of side length $L$, resting on a table top at $z=0$, whose surface density is given by $\sigma=2Ah$.
4. An infinitesimally thin square sheet of side length $L$, resting on a table top at $z=0$, whose mass density is given by $\rho=2Ah\,\delta(z)$.
5. What are the dimensions of $A$?
6. Write several sentences comparing your answers to the different cases above.

• group Vector Integrals (Contour Map)

group Small Group Activity

30 min.

##### Vector Integrals (Contour Map)

Learning Outcomes