Student explore the properties of an orthonormal basis using the Cartesian and \(S_z\) bases as examples.
Students generate a list of properties a glass of water might have. The class then discusses and categorizes those properties.
Students, pretending they are point charges, move around the room acting out various prompts from the instructor regarding charge densities, including linear \(\lambda\), surface \(\sigma\), and volume \(\rho\) charge densities, both uniform and non-uniform. The instructor demonstrates what it means to measure these quantities. In a remote setting, we have students manipulate 10 coins to model the prompts in this activity and we demonstrate the answers with coins under a doc cam.
Students, pretending they are point charges, move around the room so as to make an imaginary magnetic field meter register a constant magnetic field, introducing the concept of steady current. Students act out linear \(\vec{I}\), surface \(\vec{K}\), and volume \(\vec{J}\) current densities. The instructor demonstrates what it means to measure these quantities by counting how many students pass through a gate.
A student is invited to “act out” motion corresponding to a plot of effective potential vs. distance. The student plays the role of the “Earth” while the instructor plays the “Sun”.
Students hold rulers and meter sticks to represent a vector field. The instructor holds a hula hoop to represent a small area element. Students are asked to describe the flux of the vector field through the area element.
Students are shown a topographic map of an oval hill and imagine that the classroom is on the hill. They are asked to point in the direction of the gradient vector appropriate to the point on the hill where they are "standing".
Students learn about the geometric meaning of the amplitude and period parameters in the sine function. They also practice sketching the sum of two functions by hand.
Many students do not know what it means to add two functions graphically. Students are shown graphs of two simple functions and asked to sketch the sum.
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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.
Students learn how to express Angular Momentum as a vector quantity in polar coordinates, and then in Cylindrical and Spherical Coordinates
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In this activity, students apply the Stefan-Boltzmann equation and the principle of energy balance in steady state to find the steady state temperature of a black object in near-Earth orbit.
Problem
These notes, from the third week of https://paradigms.oregonstate.edu/courses/ph441 cover the canonical ensemble and Helmholtz free energy. They include a number of small group activities.
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Students solve for the equations of motion of a box sliding down (frictionlessly) a wedge, which itself slides on a horizontal surface, in order to answer the question "how much time does it take for the box to slide a distance \(d\) down the wedge?". This activities highlights finding kinetic energies when the coordinate system is not orthonormal and checking special cases, functional behavior, and dimensions.
Students are placed into small groups and asked to calculate the total differential of a function of two variables, each of which is in turn expressed in terms of two other variables.
This activity starts with a brief lecture introduction to power series and a short derivation of the formula for calculating the power series coefficients.
\[c_n={1\over n!}\, f^{(n)}(z_0)\]
Students use this formula to compute the power series coefficients for a \(\sin\theta\) (around both the origin and (if time allows) \(\frac{\pi}{6}\)). The meaning of these coefficients and the convergence behavior for each approximation is discussed in the whole-class wrap-up and in the follow-up activity: Visualization of Power Series Approximations.
Problem
This small group activity is designed to provide practice with the chain rule and to develop familiarity with polar coordinates. Students work in small groups to relate partial derivatives in rectangular and polar coordinates. The whole class wrap-up discussion emphasizes the importance of specifying what quantities are being held constant.
This small group activity using surfaces combines practice with the multivariable chain rule while emphasizing numerical representations of derivatives. Students work in small groups to measure partial derivatives in both rectangular and polar coordinates, then verify their results using the chain rule. The whole class wrap-up discussion emphasizes the relationship between a directional derivative in the \(r\)-direction and derivatives in \(x\)- and \(y\)-directions using the chain rule.
Students consider the change in internal energy during three different processes involving a container of water vapor on a stove. Using the 1st Law of Thermodynamics, students reason about how the internal energy would change and then compare this prediction with data from NIST presented as a contour plot.
Students work in small groups to use completeness relations to change the basis of quantum states.
Students use a plastic surface representing the potential due to a charged sphere to explore the electrostatic potential, equipotential lines, and the relationship between potential and electric field.
These notes from the fifth week of https://paradigms.oregonstate.edu/courses/ph441 cover the grand canonical ensemble. They include several small group activities.