Activity: Systems of Particles Lecture Notes

Central Forces 2023 (3 years)

Consider a system of \(n\) different masses \(m_i\), interacting with each other and being acted on by external forces. We can write Newton's second law for the positions \(\vec{r}_i\) of each of these masses with respect to a fixed origin \(\cal{O}\), thereby obtaining a system of equations governing the motion of the masses. \begin{align} m_1 \frac{d^2\, \vec{r}_1}{dt^2} &=\vec{F}_1+\;\; 0\;\, +\vec{f}_{12}+\vec{f}_{13}+\;\,\dots\;\, +\vec{f}_{1n}\nonumber\\ m_2 \frac{d^2\, \vec{r}_2}{dt^2} &=\vec{F}_2+\vec{f}_{21}+\;\; 0\;\, +\vec{f}_{23}+\;\,\dots\;\, +\vec{f}_{2n} \label{NewtonSystem}\\ \vdots\nonumber\\ m_n \frac{d^2\, \vec{r}_n}{dt^2} &=\vec{F}_n+\vec{f}_{n1}+\vec{f}_{n2}+\dots+\vec{f}_{n(n-1)}+0\quad\nonumber \end{align} Here, we have chosen the notation \(\vec{F}_i\) for the net external forces acting on mass \(m_i\) and \(\vec{f}_{ij}\) for the internal force of mass \(m_j\) acting on \(m_i\).

In general, each internal force \(\vec{f}_{ij}\) will depend on the positions of the particles \(\vec{r}_i\) and \(\vec{r}_j\) in some complicated way, making \((\ref{NewtonSystem})\), a set of coupled differential equations. To solve \((\ref{NewtonSystem})\), we first need to decouple the differential equations, i.e. find an equivalent set of differential equations in which each equation contains only one variable.

The weak form of Newton's third law states that the force \(\vec{f}_{12}\) of \(m_2\) on \(m_1\) is equal and opposite to the force \(\vec{f}_{21}\) of \(m_1\) on \(m_2\). We see that each internal force appears twice in the system of equations \((\ref{NewtonSystem})\), once with a positive sign and once with a negative sign. Therefore, if we add all of the equations together, the internal forces will all cancel, leaving: \begin{equation} \sum_{i=1}^n m_i \frac{d^2 \vec{r}_i}{dt^2} =\sum_{i=1}^n\vec{F}_i\label{NewtonCOM} \end{equation}

Notice what a surprising equation \((\ref{NewtonCOM})\) is. The right-hand side directs us to add up all of the external forces, each of which acts on a different mass; something you were taught never to do in introductory physics.

The left-hand side of \((\ref{NewtonCOM})\) directs us to add up (the second derivatives of) \(n\) “weighted" position vectors pointing from the origin to different masses. We can simplify the left-hand side of \((\ref{NewtonCOM})\) if we multiply and divide by the total mass \(M=m_1+m_2+\dots+m_n\) and use the linearity of differentiation to “factor out” the derivative operator: \begin{align} \sum_{i=1}^n m_i \frac{d^2 \vec{r}_i}{dt^2} &=M\frac{d^2}{dt^2} \left(\sum_{i=1}^n \frac{m_i}{M}\, \vec{r}_i\right)\label{CenterOfMass1}\\ &=M\frac{d^2 \vec{R}_{cm}}{dt^2}\label{CenterOfMass2} \end{align} We recognize (or define) the quantity in the parentheses on the right-hand side of \((\ref{CenterOfMass1})\) as the position vector \(\vec{R}_{cm}\) from the origin to the “center of mass” of the system of particles, i.e. \begin{equation} \vec{R}_{cm}=\sum_{i=1}^n\frac{m_i}{M}\, \vec{r}_i\label{CenterOfMass3} \end{equation} With these simplifications, equation (\ref{NewtonCOM}) becomes: \begin{equation} M \frac{d^2 \vec{R}_{cm}}{dt^2} =\sum_{i=1}^n\vec{F}_i\label{NewtonCOM2} \end{equation} which has the form of Newton's 2nd Law for a fictitious particle with mass \(M\) sitting at the center of mass of the system of particles and acted on by all of the external forces from the original system.

We can define the momentum of the center of mass as the total mass times the time derivative of the position of the center of mass: \begin{equation} \vec{P}_{cm}=M\frac{d\vec{R}_{cm}}{dt} \end{equation} If there are no external forces acting, then the acceleration of the center of mass is zero and the momentum of the center of mass is constant in time (conserved). \begin{equation} M\frac{d^2 \vec{R}_{cm}}{dt^2}=\frac{d\vec{P}_{cm}}{dt}=0 \label{MomentumConservation} \end{equation}

Notice that the entire discussion above applies even if all of the internal forces are zero \(\vec{f}_{ij}=0\), i.e. none of the particles have any way of knowing that the others are even present. Such particles are called non-interacting. The position of the center of mass of the system will still move according to equation \((\ref{NewtonCOM2})\).

  • assignment Undo Formulas for Reduced Mass (Algebra)

    assignment Homework

    Undo Formulas for Reduced Mass (Algebra)
    Central Forces 2023 (2 years) For systems of particles, we used the formulas \begin{align} \vec{R}_{cm}&=\frac{1}{M}\left(m_1\vec{r}_1+m_2\vec{r}_2\right) \nonumber\\ \vec{r}&=\vec{r}_2-\vec{r}_1 \label{cm} \end{align} to switch from a rectangular coordinate system that is unrelated to the system to coordinates adapted to the center-of-mass. After you have solved the equations of motion in the center-of-mass coordinates, you may want to transform back to the original coordinate system. Find the inverse transformation, i.e. solve for: \begin{align} \vec{r}_1&=\\ \vec{r}_2&= \end{align} Hint: The system of equations (\ref{cm}) is linear, i.e. each variable is to the first power, even though the variables are vectors. In this case, you can use all of the methods you learned for solving systems of equations while keeping the variables vector valued, i.e. you can safely ignore the fact that the \(\vec{r}\)s are vectors while you are doing the algebra.
  • assignment Ring Function

    assignment Homework

    Ring Function
    Central Forces 2023 (3 years) Consider the normalized wavefunction \(\Phi\left(\phi\right)\) for a quantum mechanical particle of mass \(\mu\) constrained to move on a circle of radius \(r_0\), given by: \begin{equation} \Phi\left(\phi\right)= \frac{N}{2+\cos(3\phi)} \end{equation} where \(N\) is the normalization constant.
    1. Find \(N\).

    2. Plot this wave function.
    3. Plot the probability density.
    4. Find the probability that if you measured \(L_z\) you would get \(3\hbar\).
    5. What is the expectation value of \(L_z\) in this state?
  • assignment Find Force Law: Spiral Orbit

    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 Diatomic hydrogen

    assignment Homework

    Diatomic hydrogen
    rigid rotor hamiltonian angular momentum ground state hydrogen diatomic probability Energy and Entropy 2021 (2 years)

    At low temperatures, a diatomic molecule can be well described as a rigid rotor. The Hamiltonian of such a system is simply proportional to the square of the angular momentum \begin{align} H &= \frac{1}{2I}L^2 \end{align} and the energy eigenvalues are \begin{align} E_{\ell m} &= \hbar^2 \frac{\ell(\ell+1)}{2I} \end{align}

    1. What is the energy of the ground state and the first and second excited states of the \(H_2\) molecule? i.e. the lowest three distinct energy eigenvalues.

    2. At room temperature, what is the relative probability of finding a hydrogen molecule in the \(\ell=0\) state versus finding it in any one of the \(\ell=1\) states?
      i.e. what is \(P_{\ell=0,m=0}/\left(P_{\ell=1,m=-1} + P_{\ell=1,m=0} + P_{\ell=1,m=1}\right)\)

    3. At what temperature is the value of this ratio 1?

    4. At room temperature, what is the probability of finding a hydrogen molecule in any one of the \(\ell=2\) states versus that of finding it in the ground state?
      i.e. what is \(P_{\ell=0,m=0}/\left(P_{\ell=2,m=-2} + P_{\ell=2,m=-1} + \cdots + P_{\ell=2,m=2}\right)\)

  • group Electrostatic Potential Due to a Pair of Charges (without Series)

    group Small Group Activity

    30 min.

    Electrostatic Potential Due to a Pair of Charges (without Series)
    Static Fields 2022 (3 years) Students work in small groups to use the superposition principle \[V(\vec{r}) = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{\vert\vec{r}-\vec{r}_i\vert}\] to find the electrostatic potential \(V\) everywhere in space due to a pair of charges (either identical charges or a dipole). This activity can be paired with activity 29 to find the limiting cases of the potential on the axes of symmetry.
  • assignment Isolength and Isoforce Stretchability

    assignment Homework

    Isolength and Isoforce Stretchability
    Energy and Entropy 2021 (2 years)

    In class, you measured the isolength stretchability and the isoforce stretchability of your systems in the PDM. We found that for some systems these were very different, while for others they were identical.

    Show with algebra (NOT experiment) that the ratio of isolength stretchability to isoforce stretchability is the same for both the left-hand side of the system and the right-hand side of the system. i.e.: \begin{align} \frac{\left(\frac{\partial {x_L}}{\partial {F_L}}\right)_{x_R}}{\left(\frac{\partial {x_L}}{\partial {F_L}}\right)_{F_R}} &= \frac{\left(\frac{\partial {x_R}}{\partial {F_R}}\right)_{x_L}}{\left(\frac{\partial {x_R}}{\partial {F_R}}\right)_{F_L}} \label{eq:ratios} \end{align}

    You will need to make use of the cyclic chain rule: \begin{align} \left(\frac{\partial {A}}{\partial {B}}\right)_{C} = -\left(\frac{\partial {A}}{\partial {C}}\right)_{B}\left(\frac{\partial {C}}{\partial {B}}\right)_{A} \end{align}
    You will also need the ordinary chain rule: \begin{align} \left(\frac{\partial {A}}{\partial {B}}\right)_{D} = \left(\frac{\partial {A}}{\partial {C}}\right)_{D}\left(\frac{\partial {C}}{\partial {B}}\right)_{D} \end{align}

  • assignment Reduced Mass

    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.

  • group Mass is not Conserved

    group Small Group Activity

    30 min.

    Mass is not Conserved
    Theoretical Mechanics (4 years)

    energy conservation mass conservation collision

    Groups are asked to analyze the following standard problem:

    Two identical lumps of clay of (rest) mass m collide head on, with each moving at 3/5 the speed of light. What is the mass of the resulting lump of clay?

  • assignment Mass-radius relationship for white dwarfs

    assignment Homework

    Mass-radius relationship for white dwarfs
    White dwarf Mass Density Energy Thermal and Statistical Physics 2020

    Consider a white dwarf of mass \(M\) and radius \(R\). The dwarf consists of ionized hydrogen, thus a bunch of free electrons and protons, each of which are fermions. Let the electrons be degenerate but nonrelativistic; the protons are nondegenerate.

    1. Show that the order of magnitude of the gravitational self-energy is \(-\frac{GM^2}{R}\), where \(G\) is the gravitational constant. (If the mass density is constant within the sphere of radius \(R\), the exact potential energy is \(-\frac53\frac{GM^2}{R}\)).

    2. Show that the order of magnitude of the kinetic energy of the electrons in the ground state is \begin{align} \frac{\hbar^2N^{\frac53}}{mR^2} \approx \frac{\hbar^2M^{\frac53}}{mM_H^{\frac53}R^2} \end{align} where \(m\) is the mass of an electron and \(M_H\) is the mas of a proton.

    3. Show that if the gravitational and kinetic energies are of the same order of magnitude (as required by the virial theorem of mechanics), \(M^{\frac13}R \approx 10^{20} \text{g}^{\frac13}\text{cm}\).

    4. If the mass is equal to that of the Sun (\(2\times 10^{33}g\)), what is the density of the white dwarf?

    5. It is believed that pulsars are stars composed of a cold degenerate gas of neutrons (i.e. neutron stars). Show that for a neutron star \(M^{\frac13}R \approx 10^{17}\text{g}^{\frac13}\text{cm}\). What is the value of the radius for a neutron star with a mass equal to that of the Sun? Express the result in \(\text{km}\).

  • assignment Center of Mass for Two Uncoupled Particles

    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.

Learning Outcomes