Cone Surface

• assignment Icecream Mass

  assignment Homework

  Icecream Mass

  Static Fields 2023 (6 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).

• assignment Find Area/Volume from $d\vec{r}$

  assignment Homework

  Find Area/Volume from \(d\vec{r}\)

  Static Fields 2023 (5 years)

  Start with \(d\vec{r}\) in rectangular, cylindrical, and spherical coordinates. Use these expressions to write the scalar area elements \(dA\) (for different coordinate equals constant surfaces) and the volume element \(d\tau\). It might help you to think of the following surfaces: The various sides of a rectangular box, a finite cylinder with a top and a bottom, a half cylinder, and a hemisphere with both a curved and a flat side, and a cone.

  1. Rectangular: \begin{align} dA&=\\ d\tau&= \end{align}
  2. Cylindrical: \begin{align} dA&=\\ d\tau&= \end{align}
  3. Spherical: \begin{align} dA&=\\ d\tau&= \end{align}

• group Flux through a Cone

  group Small Group Activity

  30 min.

  Flux through a Cone

  Static Fields 2021 (4 years)

  Integration Sequence

  Students calculate the flux from the vector field \(\vec{F} = C\, z\, \hat{z}\) through a right cone of height \(H\) and radius \(R\) .
• keyboard Electric field for a waffle cone of charge

  keyboard Computational Activity

  120 min.

  Electric field for a waffle cone of charge

  Computational Physics Lab II 2022

  electric field cone

  Students integrate numerically to find the electric field due to a cone of surface charge, and then visualize the result. This integral can be done in either spherical or cylindrical coordinates, giving students a chance to reason about which coordinate system would be more convenient.
• assignment Sphere in Cylindrical Coordinates

  assignment Homework

  Sphere in Cylindrical Coordinates

  Static Fields 2023 (4 years) Find the surface area of a sphere using cylindrical coordinates. Note: The fact that you can describe spheres nicely in cylindrical coordinates underlies the equal area cylindrical map project that allows you to draw maps of the earth where everything has the correct area, even if the shapes seem distorted. If you want to plot something like population density, you need an area preserving map projection.
• group Applying the equipartition theorem

  group Small Group Activity

  30 min.

  Applying the equipartition theorem

  Contemporary Challenges 2021 (4 years)

  equipartition theorem

  Students count the quadratic degrees of freedom of a few toy molecules to predict their internal energy at temperature \(T\).
• assignment_ind Curvilinear Coordinates Introduction

  assignment_ind Small White Board Question

  10 min.

  Curvilinear Coordinates Introduction

  Static Fields 2023 (10 years)

  Cylindrical coordinates spherical coordinates curvilinear coordinates

  Curvilinear Coordinate Sequence

  First, students are shown diagrams of cylindrical and spherical coordinates. Common notation systems are discussed, especially that physicists and mathematicians use opposite conventions for the angles \(\theta\) and \(\phi\). Then students are asked to check their understanding by sketching several coordinate equals constant surfaces on their small whiteboards.
• group Calculating Coefficients for a Power Series

  group Small Group Activity

  30 min.

  Calculating Coefficients for a Power Series

  Theoretical Mechanics (8 years)

  Taylor series power series approximation

  Power Series Sequence (E&M)

  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.

• groups Pineapples and Pumpkins

  groups Whole Class Activity

  10 min.

  Pineapples and Pumpkins

  Static Fields 2023 (6 years)

  Integration Sequence

  There are two versions of this activity:

  As a whole class activity, the instructor cuts a pumpkin in order to produce a small volume element \(d\tau\), interspersing their work with a sequence of small whiteboard questions. This version of the activity is described here.

  As a small group activity, students are given pineapple rounds and pumpkin wedges to explore area volume elements in cylindrical and spherical coordinate systems. In this version of the activity, the fruit is distribued to the students with appropriate children's pumpkin cutting equipment, as part of activities Vector Differential--Curvilinear, Scalar Surface and Volume Elements, or Vector Surface and Volume Elements.

• face Statistical Analysis of Stern-Gerlach Experiments

  face Lecture

  30 min.

  Statistical Analysis of Stern-Gerlach Experiments

• Static Fields 2023 (6 years)
  • Find \(dA\) on the surface of an (open) cone in both cylindrical and spherical coordinates. Hint: Be smart about how you coordinatize the cone.
  • Using integration, find the surface area of an (open) cone with height \(H\) and radius \(R\). Do this problem in both cylindrical and spherical coordinates.