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## Homework for Reduced Mass

- (CentralForce)
*Determine whether several common forces in nature are central forces.*Which of the following forces can be central forces? which cannot?

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. $q\Vec v \times \Vec B$

- (FreeCentralForce)
*A simple check on your understanding of center-of-mass motion.*If a central force is the only force acting on a system of two masses (i.e. no external forces), what will the motion of the center of mass be?

- (PlanarOrbit)
*A simple check on your understanding of classical angular momentum.*}Show that the plane of the orbit is perpendicular to the angular momentum vector $\Vec L$.

- (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.

- (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.

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

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

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

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

Repeat the calculation for the Sun–Jupiter system.

- (CMLandT)
*Explicitly show how the kinetic energy and angular momentum of a two particle system is related to the energy and angular momentum of the center of mass and reduced mass system.*Consider a system of two particles.

Show that the total kinetic energy of the system is the same as that of two “fictitious” particles: one of mass $M=m_1+m_2$ moving with the speed of the CM (center of mass) and one of mass $\mu$ (the reduced mass) moving with the speed of the relative position $\vec{r}=\vec{r}_2-\vec{r}_1$.

Show that the total angular momentum of the system can be similarly decomposed into the angular momenta of these two fictitious particles.