*face*Central Forces Introduction: Lecture Notes*computer*Effective Potentials*computer*Mathematica Activity30 min.

##### Effective Potentials

Central Forces 2022 (2 years) 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.*groups*Air Hockey*groups*Whole Class Activity10 min.

##### Air Hockey

Central Forces 2022 (2 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.*group*Electric Field of Two Charged Plates*group*Small Group Activity30 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
- objects with like charge repel and opposite charge attract,
- object tend to move toward lower energy configurations
- The potential energy of a charged particle is related to its charge: \(U=qV\)
- The force on a charged particle is related to its charge: \(\vec{F}=q\vec{E}\)

*assignment*Find Force Law: Logarithmic Spiral Orbit*assignment*Homework##### Find Force Law: Logarithmic Spiral Orbit

Central Forces 2022 (2 years)In science fiction movies, characters often talk about a spaceship “spiralling in” right before it hits the planet. But all orbits in a \(1/r^2\) force are conic sections, not spirals. This spiralling in happens because the spaceship hits atmosphere and the drag from the atmosphere changes the shape of the orbit. But, in an alternate universe, we might have other force laws.

Find the force law for a mass \(\mu\), under the influence of a central-force field, that moves in a logarithmic spiral orbit given by \(r = ke^{\alpha \phi}\), where \(k\) and \(\alpha\) are constants.

*assignment*Find Force Law: Spiral Orbit*assignment*Homework##### Find Force Law: Spiral Orbit

Central Forces 2022 (2 years)In science fiction movies, characters often talk about a spaceship “spiralling in” right before it hits the planet. But all orbits in a \(1/r^2\) force are conic sections, not spirals. This spiralling in happens because the spaceship hits atmosphere and the drag from the atmosphere changes the shape of the orbit. But, in an alternate universe, we might have other force laws.

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.

*assignment*Undo Formulas for Reduced Mass (Geometry)*assignment*Homework##### Undo Formulas for Reduced Mass (Geometry)

Central Forces 2022 (2 years)The figure below shows the position vector \(\vec r\) and the orbit of a “fictitious” reduced mass \(\mu\).

- Suppose \(m_1=m_2\), Sketch the position vectors and orbits for \(m_1\) and \(m_2\) corresponding to \(\vec{r}\). Describe a common physics example of central force motion for which \(m_1=m_2\).
- Repeat, for \(m_2>m_1\).

*assignment*Normalization of Quantum States*assignment*Homework##### Normalization of Quantum States

Central Forces 2022 (2 years) Show that if a linear combination of ring energy eigenstates is normalized, then the coefficients must satisfy \begin{equation} \sum_{m=-\infty}^{\infty} \vert c_m\vert^2=1 \end{equation}*assignment*Ring Table*assignment*Homework##### Ring Table

Central Forces 2022 (2 years)Attached, you will find a table showing different representations of physical quantities associated with a quantum particle confined to a ring. Fill in all of the missing entries. Hint: You may look ahead. We filled out a number of the entries throughout the table to give you hints about what the forms of the other entries might be. pdf link for the Table or doc link for the Table

*group*Gravitational Force*group*Small Group Activity30 min.

##### Gravitational Force

Mechanics Gravitational Force Gravitational Potential Energy Derivatives Introductory Physics

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.-
Central Forces 2022 (2 years)
Which of the following forces can be central forces? which cannot? If the force CAN be a central force, explain the circumstances that would allow it to be a central force.

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