## Activity: Air Hockey

Central Forces 2023 (3 years)
Students observe the motion of a puck tethered to the center of the airtable. Then they plot the potential energy for the puck on their small whiteboards. A class discussion follows based on what students have written on their whiteboards.
• groups Whole Class Activity schedule 10 min. build Individual Small Whiteboards with markers and erasers, An airtable with a pin in its center, An airtable puck tethered to the pin in the center of the table by a thread and piece of rubber band that will stretch when the string becomes taut.
• Search for related topics
central forces potential energy classical mechanics
What students learn
• observe the motion of an air hockey puck under the influence of a central force.
• explore different possible graphical representations of potential energy (vs. time, angle) (2-d vs. 3-d graphs), etc.
• discuss the conventional expert plot of potential energy showing the potential energy on the vertical axis even when the object doesn't change height.
• Media

## Instructor's Guide

Students observe the motion of a puck tethered to the center of the airtable. Then they plot the potential energy for the puck on their small whiteboards. A class discussion follows based on what students have written on their whiteboards.

### Introduction

Students begin this activity by observing the motion of the puck on the airtable. The puck is pushed so that it passes close to, but does not strike, the pin in the center of the table. Students are asked to observe the motion of the puck. Then students are asked to respond, “On your small white board, make a plot of the potential energy.” Don't give a more precise prompt.

### Student Conversations

After students have made their plots, the class discusses a variety of student responses.

This question is deliberately ambiguous to encourage a diversity of answers. This elicits responses that reveal information about what the students think is important about what they observed and what they expect to be thinking about with this type of problem.

Since the question is worded ambiguously, students draw a variety of plots including:

• The most common example of potential energy that students see in an introductory course is gravitational potential energy, for which energy and height are proportional. Some students may need to think about the fact that they can plot potential energy on the vertical axis when the puck stays on the surface of the table.
• Many students will make a plot of $U$ as a function of both $r$ and $\phi$.
• A few students will make a plot of $U$ vs. time or angle.
• A number of students may plot the canonical variables ($U$ vs. $r$) but will draw a parabola with the vertex centered at the origin, forgetting that the piece of thread means that there is constant potential energy out to some fixed radius. Only beyond that radius does the potential energy increase.
This activity is a good opportunity to remind students of the arbitrariness of the zero of potential energy.

After discussing several student responses, it is useful to ask the students whether or not they think the force acting on the puck is a central force and to justify their answer. This typically leads to a review of the properties of a central force.

### Wrap-up

While confirming that many different representations are correct, and may be useful in certain circumstances, there is a conventional representation that is commonly used, with which they should be familiar.

It is important to mention that for central force problems, physicists typically write down the potential energy as a function of $r$ since the force between the particles is only a function of the distance between them. Make sure that the students have the opportunity to see this graph and think about which parameters are shown.

• assignment Hockey

assignment Homework

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

(Synthesis Problem: Brings together several different concepts from this unit.) Use effective potential diagrams for other than $1/r^2$ forces.

Consider the frictionless motion of a hockey puck of mass $m$ on a perfectly circular bowl-shaped ice rink with radius $a$. The central region of the bowl ($r < 0.8a$) is perfectly flat and the sides of the ice bowl smoothly rise to a height $h$ at $r = a$.

1. Draw a sketch of the potential energy for this system. Set the zero of potential energy at the top of the sides of the bowl.
2. Situation 1: the puck is initially moving radially outward from the exact center of the rink. What minimum velocity does the puck need to escape the rink?
3. Situation 2: a stationary puck, at a distance $\frac{a}{2}$ from the center of the rink, is hit in such a way that it's initial velocity $\vec v_0$ is perpendicular to its position vector as measured from the center of the rink. What is the total energy of the puck immediately after it is struck?
4. In situation 2, what is the angular momentum of the puck immediately after it is struck?
5. Draw a sketch of the effective potential for situation 2.
6. In situation 2, for what minimum value of $\vec v_0$ does the puck just escape the rink?

• assignment Effective Potentials: Graphical Version

assignment Homework

##### Effective Potentials: Graphical Version
Central Forces 2023 (3 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 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 2023 (3 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.
• assignment Ice calorimetry lab questions

assignment Homework

##### Ice calorimetry lab questions
This question is about the lab we did in class: Ice Calorimetry Lab.
1. Plot your data I Plot the temperature versus total energy added to the system (which you can call $Q$). To do this, you will need to integrate the power. Discuss this curve and any interesting features you notice on it.
2. Plot your data II Plot the heat capacity versus temperature. This will be a bit trickier. You can find the heat capacity from the previous plot by looking at the slope. \begin{align} C_p &= \left(\frac{\partial Q}{\partial T}\right)_p \end{align} This is what is called the heat capacity, which is the amount of energy needed to change the temperature by a given amount. The $p$ subscript means that your measurement was made at constant pressure. This heat capacity is actually the total heat capacity of everything you put in the calorimeter, which includes the resistor and thermometer.
3. Specific heat From your plot of $C_p(T)$, work out the heat capacity per unit mass of water. You may assume the effect of the resistor and thermometer are negligible. How does your answer compare with the prediction of the Dulong-Petit law?
4. Latent heat of fusion What did the temperature do while the ice was melting? How much energy was required to melt the ice in your calorimeter? How much energy was required per unit mass? per molecule?
5. Entropy of fusion The change in entropy is easy to measure for a reversible isothermal process (such as the slow melting of ice), it is just \begin{align} \Delta S &= \frac{Q}{T} \end{align} where $Q$ is the energy thermally added to the system and $T$ is the temperature in Kelvin. What is was change in the entropy of the ice you melted? What was the change in entropy per molecule? What was the change in entropy per molecule divided by Boltzmann's constant?
6. Entropy for a temperature change Choose two temperatures that your water reached (after the ice melted), and find the change in the entropy of your water. This change is given by \begin{align} \Delta S &= \int \frac{{\mathit{\unicode{273}}} Q}{T} \\ &= \int_{t_i}^{t_f} \frac{P(t)}{T(t)}dt \end{align} where $P(t)$ is the heater power as a function of time and $T(t)$ is the temperature, also as a function of time.
• keyboard Electrostatic potential of four point charges

keyboard Computational Activity

120 min.

##### Electrostatic potential of four point charges
Computational Physics Lab II 2023 (2 years)

Students write python programs to compute and visualize the potential due to four point charges. For students with minimal programming ability and no python experience, this activity can be a good introduction to writing code in python using numpy and matplotlib.

computer Mathematica Activity

30 min.

Static Fields 2023 (7 years)

Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.
• computer Visualization of Quantum Probabilities for the Hydrogen Atom

computer Mathematica Activity

30 min.

##### Visualization of Quantum Probabilities for the Hydrogen Atom
Central Forces 2023 (3 years) Students use Mathematica to visualize the probability density distribution for the hydrogen atom orbitals with the option to vary the values of $n$, $\ell$, and $m$.
• computer Visualizing Combinations of Spherical Harmonics

computer Mathematica Activity

30 min.

##### Visualizing Combinations of Spherical Harmonics
Central Forces 2023 (3 years) Students observe three different plots of linear combinations of spherical combinations with probability density represented by color on the sphere, distance from the origin (polar plot), and distance from the surface of the sphere.
• keyboard Mean position

keyboard Computational Activity

120 min.

##### Mean position
Computational Physics Lab II 2023 (2 years)

Students compute probabilities and averages given a probability density in one dimension. This activity serves as a soft introduction to the particle in a box, introducing all the concepts that are needed.
• assignment Free energy of a two state system

assignment Homework

##### Free energy of a two state system
Helmholtz free energy entropy statistical mechanics Thermal and Statistical Physics 2020
1. Find an expression for the free energy as a function of $T$ of a system with two states, one at energy 0 and one at energy $\varepsilon$.

2. From the free energy, find expressions for the internal energy $U$ and entropy $S$ of the system.

3. Plot the entropy versus $T$. Explain its asymptotic behavior as the temperature becomes high.

4. Plot the $S(T)$ versus $U(T)$. Explain the maximum value of the energy $U$.

Learning Outcomes