Systems of Particles

Students should be able to:

• Write Newton's second law as a differential equation.
• - reduced mass

1. (FreeCoM) Determine whether several common forces in nature are central forces.

   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.

2. (FreeCoMLG) Determine whether several common forces in nature are central forces.

   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.

3. (ReducedMassLG) How does the reduced mass depend on the two original masses.?

   Using your favorite graphing package, make a plot of the reduced mass $\mu$ 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.

4. (UndoReduced) Once you have solved for the motion of the reduced mass, you must “undo” the substitutions that you made to find the motions of your original two masses.

   The figure below shows the position vector $\bf r$ and the orbit of a “fictitious” reduced mass.

   1. Assuming that $m_2=m_1$, draw on the figure the position vectors for $m_1$ and $m_2$ corresponding to $\bf r$. Also draw the orbits for $m_1$ and $m_2$. Describe a common physics example of central force motion for which $m_1=m_2$.

      \bigskip \centerline{\includegraphics[height=2.5truein]{\TOP Figures/cfellipse2}} \medskip

   2. Repeat the previous problem for $m_2=3 m_1$.

      \bigskip \centerline{\includegraphics[height=2.7truein]{\TOP Figures/cfellipse2}} \medskip

5. (SunJupiter) Get a sense for the position of the center of mass for planets in our solar system.

   1. Find ${\bf r}_{\rm sun}-{\bf r}_{\rm cm}$ and $\mu$ for the Sun–Earth system. Compare ${\bf r}_{\rm sun}-{\bf r}_{\rm cm}$ to the radius of the Sun and to the distance from the Sun to the Earth. Compare $\mu$ to the mass of the Sun and the mass of the Earth.

   2. Repeat the calculation for the Sun–Jupiter system.