(Quick) Purpose: Recognize the definition of a central force. Build experience about which common physical situations represent central forces and which don't.

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

  • Found in: Central Forces course(s)

(Use the equation for orbit shape.) Gain experience with unusual force laws.

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.

  • Found in: Central Forces course(s)

assignment_ind Small White Board Question

5 min.

Newton's 2nd Law SWBQ
  • Found in: Central Forces course(s)

face Lecture

5 min.

Spherical Harmonics
  • Found in: Central Forces course(s)
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}
  • Found in: Central Forces course(s)

None

Ring Table

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

  • Found in: Central Forces course(s)

computer Mathematica Activity

30 min.

Effective Potentials
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.

***The solution to this problem is incorrect. It should use the orbit shape equation r(phi). See the solution to the logarithmic spiral orbit (548) and mimic that.

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.

  • Found in: Central Forces course(s)

(Sketch limiting cases) Purpose: For two central force systems that share the same reduced mass system, discover how the motions of the original systems are the same and different.

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 \(m_2>m_1\).

  • Found in: Central Forces course(s)
This handout lists Motivating Questions, Key Activities/Problems, Unit Learning Outcomes, and an Equation Sheet for a Unit on Classical Mechanics Orbits. It can be used both to introduce the unit and, even better, for review.
  • Found in: Central Forces course(s)

face Lecture

10 min.

Introduction to Central Forces
  • Found in: Central Forces course(s)

groups Whole Class Activity

10 min.

Air Hockey
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.
  • Found in: Central Forces course(s)
Show that the plane polar coordinates are equivalent to spherical coordinates if we make the choices:
  1. The direction of \(\theta=0\) in spherical coordinates is the same as the direction of out of the plane in plane polar coordinates.
  2. Given the correspondance above, then if we choose the \(\theta\) of spherical coordinates is to be \(\pi/2\), we restrict to the equatorial plane of spherical coordinates.
  • Found in: Central Forces course(s)

(Simple graphing) Purpose: Discover some of the properties of reduced mass by exploring the graph.

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.

  • Found in: Central Forces course(s)

face Lecture

5 min.

Quantum Reference Sheet
  • Found in: Central Forces course(s)

assignment_ind Small White Board Question

5 min.

Angular Momentum SWBQ
  • Found in: Central Forces course(s)

computer Mathematica Activity

30 min.

Visualizing Combinations of Spherical Harmonics
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.

(Algebra involving trigonometric functions) Purpose: Practice with polar equations.

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

  • Found in: Central Forces course(s)