assignment Homework

Central Force Definition
Central Forces 2023 (3 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.

  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. \(q\vec v \times \vec B\)

assignment Homework

Find Force Law: Logarithmic Spiral Orbit
Central Forces 2023 (3 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 Homework

Normalization of Quantum States
Central Forces 2023 (3 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 Homework

Ring Table
Central Forces 2023 (3 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

assignment Homework

Find Force Law: Spiral Orbit
Central Forces 2023 (3 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 Homework

Undo Formulas for Reduced Mass (Geometry)
Central Forces 2023 (3 years)

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

  1. 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 \(m_1=m_2\).
  2. Repeat, for \(m_2>m_1\).

group Small Group Activity

5 min.

Spherical Harmonics
Central Forces 2023 (3 years)

computer Mathematica Activity

30 min.

Effective Potentials
Central Forces 2023 (3 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.

assignment Homework

Polar vs. Spherical Coordinates
Central Forces 2023 (3 years)

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.

face Lecture

5 min.

Central Forces Introduction: Lecture Notes
Central Forces 2023 (2 years)

assignment Homework

Reduced Mass
Central Forces 2023 (3 years)

Using your favorite graphing package, make a plot of the reduced mass \begin{equation} \mu=\frac{m_1\, m_2}{m_1+m_2} \end{equation} as a function of \(m_1\) and \(m_2\). What about the shape of this graph tells you something about the physical world that you would like to remember. You should be able to find at least three things. Hint: Think limiting cases.

groups Whole Class Activity

10 min.

Air Hockey
Central Forces 2023 (3 years)

central forces potential energy classical mechanics

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.

face Lecture

5 min.

Quantum Reference Sheet
Central Forces 2023 (6 years)

assignment Homework

Sum Shift
Central Forces 2023 (3 years)

In each of the following sums, shift the index \(n\rightarrow n+2\). Don't forget to shift the limits of the sum as well. Then write out all of the terms in the sum (if the sum has a finite number of terms) or the first five terms in the sum (if the sum has an infinite number of terms) and convince yourself that the two different expressions for each sum are the same:

  1. \begin{equation} \sum_{n=0}^3 n \end{equation}
  2. \begin{equation} \sum_{n=1}^5 e^{in\phi} \end{equation}
  3. \begin{equation} \sum_{n=0}^{\infty} a_n n(n-1)z^{n-2} \end{equation}

assignment Homework

Lines in Polar Coordinates
Central Forces 2023 (3 years)

The general equation for a straight line in polar coordinates is given by: \begin{equation} r(\phi)=\frac{r_0}{\cos(\phi-\delta)} \end{equation} where \(r_0\) and \(\delta\) are constant parameters. Find the polar equation for the straight lines below. You do NOT need to evaluate any complicated trig or inverse trig functions. You may want to try plotting the general polar equation to figure out the roles of the parameters.

  1. \(y=3\)
  2. \(x=3\)
  3. \(y=-3x+2\)

assignment Homework

Center of Mass for Two Uncoupled Particles
Central Forces 2023 (3 years)

Consider two particles of equal mass \(m\). The forces on the particles are \(\vec F_1=0\) and \(\vec F_2=F_0\hat{x}\). If the particles are initially at rest at the origin, find the position, velocity, and acceleration of the center of mass as functions of time. Solve this problem in two ways, with or without theorems about the center of mass motion. Write a short description comparing the two solutions.

face Lecture

10 min.

Systems of Particles Lecture Notes
Central Forces 2023 (3 years)

group Small Group Activity

60 min.

Raising and Lowering Operators for Spin
Central Forces 2023 (2 years)

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.

face Lecture

10 min.

Angular Momentum Commutation Relations: Lecture
Central Forces 2023 (3 years)