## Activity: Events on Spacetime Diagrams

Theoretical Mechanics 2021
Students practice identifying whether events on spacetime diagrams are simultaneous, colocated, or neither for different observers. Then students decide which of two events occurs first in two different reference frames.
• Media
1. Which pairs of events (if any) are simultaneous in the unprimed frame?

2. Which pairs of events (if any) are simultaneous in the primed frame?

3. Which pairs of events (if any) are colocated in the unprimed frame?

4. Which pairs of events (if any) are colocated in the primed frame?

1. For each of the figures, answer the following questions:
1. Which event occurs first in the unprimed frame?

2. Which event occurs first in the primed frame?

• assignment Events on Spacetime Diagrams

assignment Homework

##### Events on Spacetime Diagrams
Special Relativity Spacetime Diagram Simultaneity Colocation Theoretical Mechanics (4 years)
1. Which pairs of events (if any) are simultaneous in the unprimed frame?

2. Which pairs of events (if any) are simultaneous in the primed frame?

3. Which pairs of events (if any) are colocated in the unprimed frame?

4. Which pairs of events (if any) are colocated in the primed frame?

1. For each of the figures, answer the following questions:
1. Which event occurs first in the unprimed frame?

2. Which event occurs first in the primed frame?

• accessibility_new Time Dilation Light Clock Skit

accessibility_new Kinesthetic

5 min.

##### Time Dilation Light Clock Skit

Students act out the classic light clock scenario for deriving time dilation.
• assignment_ind Time Dilation

assignment_ind Small White Board Question

10 min.

##### Time Dilation
Theoretical Mechanics (4 years)

• face Lorentz Transformation (Geometric)

face Lecture

30 min.

##### Lorentz Transformation (Geometric)
Theoretical Mechanics (3 years)

In this lecture, students see a geometric derivation of the Lorentz Transformation on a spacetime diagram.
• assignment Gibbs entropy is extensive

assignment Homework

##### Gibbs entropy is extensive
Gibbs entropy Probability Thermal and Statistical Physics 2020

Consider two noninteracting systems $A$ and $B$. We can either treat these systems as separate, or as a single combined system $AB$. We can enumerate all states of the combined by enumerating all states of each separate system. The probability of the combined state $(i_A,j_B)$ is given by $P_{ij}^{AB} = P_i^AP_j^B$. In other words, the probabilities combine in the same way as two dice rolls would, or the probabilities of any other uncorrelated events.

1. Show that the entropy of the combined system $S_{AB}$ is the sum of entropies of the two separate systems considered individually, i.e. $S_{AB} = S_A+S_B$. This means that entropy is extensive. Use the Gibbs entropy for this computation. You need make no approximation in solving this problem.
2. Show that if you have $N$ identical non-interacting systems, their total entropy is $NS_1$ where $S_1$ is the entropy of a single system.

##### Note
In real materials, we treat properties as being extensive even when there are interactions in the system. In this case, extensivity is a property of large systems, in which surface effects may be neglected.

• group Right Angles on Spacetime Diagrams

group Small Group Activity

30 min.

##### Right Angles on Spacetime Diagrams
Theoretical Mechanics (4 years)

Students take the inner product of vectors that lie on the spacetime axis to show that they are orthogonal. To do the inner product, students much use the Minkowski metric.
• assignment Scattering

assignment Homework

##### Scattering
Central Forces 2023 (3 years)

Consider a very light particle of mass $\mu$ scattering from a very heavy, stationary particle of mass $M$. The force between the two particles is a repulsive Coulomb force $\frac{k}{r^2}$. The impact parameter $b$ in a scattering problem is defined to be the distance which would be the closest approach if there were no interaction (See Figure). The initial velocity (far from the scattering event) of the mass $\mu$ is $\vec v_0$. Answer the following questions about this situation in terms of $k$, $M$, $\mu$, $\vec v_0$, and $b$. ()It is not necessarily wise to answer these questions in order.)

1. What is the initial angular momentum of the system?
2. What is the initial total energy of the system?
3. What is the distance of closest approach $r_{\rm{min}}$ with the interaction?
4. Sketch the effective potential.
5. What is the angular momentum at $r_{\rm{min}}$?
6. What is the total energy of the system at $r_{\rm{min}}$?
7. What is the radial component of the velocity at $r_{\rm{min}}$?
8. What is the tangential component of the velocity at $r_{\rm{min}}$?
9. What is the value of the effective potential at $r_{\rm{min}}$?
10. For what values of the initial total energy are there bound orbits?
11. Using your results above, write a short essay describing this type of scattering problem, at a level appropriate to share with another Paradigm student.

• group Mass is not Conserved

group Small Group Activity

30 min.

##### Mass is not Conserved
Theoretical Mechanics (4 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?

• assignment Effective Potentials: Graphical Version

assignment Homework

##### Effective Potentials: Graphical Version
Central Forces 2023 (2 years)

Consider a mass $\mu$ in the potential shown in the graph below. You give the mass a push so that its initial angular momentum is $\ell\ne 0$ for a given fixed value of $\ell$.

1. Give the definition of a central force system and briefly explain why this situation qualifies.
2. Make a sketch of the graph of the effective potential for this situation.
3. How should you push the puck to establish a circular orbit? (i.e. Characterize the initial position, direction of push, and strength of the push. You do NOT need to solve any equations.)
4. BRIEFLY discuss the possible orbit shapes that can arise from this effective potential. Include a discussion of whether the orbits are open or closed, bound or unbound, etc. Make sure that you refer to your sketch of the effective potential in your discussions, mark any points of physical significance on the sketch, and describe the range of parameters relevant to each type of orbit. Include a discussion of the role of the total energy of the orbit.

• group Vector Differential--Curvilinear

group Small Group Activity

30 min.

##### Vector Differential--Curvilinear
Vector Calculus II 23 (9 years)

Integration Sequence

In this small group activity, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in different coordinate systems (cartesian, cylindrical, spherical).

Use Vector Differential--Rectangular as an introduction. This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins to show the geometry.

Learning Outcomes