## Central Force

• computer Effective Potentials

computer Mathematica Activity

30 min.

##### Effective Potentials
Central Forces Spring 2021 Students use a pre-written Mathematica notebook or a Geogebra applet to explore how the shape of the effective potential function changes as the various parameters (angular momentum, force constant, reduced mass) are varied.
• 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: $\vec{F}=q\vec{E}$
• groups Air Hockey

groups Whole Class Activity

10 min.

##### Air Hockey
Central Forces Spring 2021

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.
• group Gravitational Force

group Small Group Activity

30 min.

##### Gravitational Force

Students examine a plastic "surface" graph of the gravitational potential energy of a Earth-satellite system to make connections between gravitational force and gravitational potential energy.
• assignment Spiral Orbit

assignment Homework

##### Spiral Orbit
Central Forces Spring 2021 A mass $\mu$, under the influence of a central-force field, moves in a logarithmic spiral orbit given by $r = ke^{\alpha \phi}$, where $k$ and $\alpha$ are constants. Determine the force law of this central-force field.
• assignment Find Force Law

assignment Homework

##### Find Force Law
Central Forces Spring 2021

Find the force law for a central-force field that allows a particle to move in a spiral orbit given by $r=k\phi^2$, where $k$ is a constant.

• 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 Spring 2021 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 Statistical Analysis of Stern-Gerlach Experiments

face Lecture

30 min.

##### Statistical Analysis of Stern-Gerlach Experiments
• assignment Polar vs. Spherical Coordinates

assignment Homework

##### Polar vs. Spherical Coordinates
Central Forces Spring 2021

Show that the plane polar coordinates we have chosen are equivalent to spherical coordinates if we make the choices:

1. The direction of $z$ in spherical coordinates is the same as the direction of $\vec L$.
2. The $\theta$ of spherical coordinates is chosen to be $\pi/2$, so that the orbit is in the equatorial plane of spherical coordinates.

• assignment Potential vs. Potential Energy

assignment Homework

##### Potential vs. Potential Energy
AIMS Maxwell AIMS 21 Static Fields Winter 2021

In this course, two of the primary examples we will be using are the potential due to gravity and the potential due to an electric charge. Both of these forces vary like $\frac{1}{r}$, so they will have many, many similarities. Most of the calculations we do for the one case will be true for the other. But there are some extremely important differences:

1. Find the value of the electrostatic potential energy of a system consisting of a hydrogen nucleus and an electron separated by the Bohr radius. Find the value of the gravitational potential energy of the same two particles at the same radius. Use the same system of units in both cases. Compare and the contrast the two answers.
2. Find the value of the electrostatic potential due to the nucleus of a hydrogen atom at the Bohr radius. Find the gravitational potential due to the nucleus at the same radius. Use the same system of units in both cases. Compare and contrast the two answers.
3. Briefly discuss at least one other fundamental difference between electromagnetic and gravitational systems. Hint: Why are we bound to the earth gravitationally, but not electromagnetically?

• Central Forces Spring 2021

Which of the following forces can be central forces? which cannot?

1. The force on a test mass $m$ in a gravitational field $\vec{g~}$, i.e. $m\vec g$
2. The force on a test charge $q$ in an electric field $\vec E$, i.e. $q\vec E$
3. The force on a test charge $q$ moving at velocity $\vec{v~}$ in a magnetic field $\vec B$, i.e. $q\vec v \times \vec B$