Homework for Reduced Mass

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

Which of the following forces can be central forces? which cannot?

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

2. (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?

3. (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 $\Vec L$.

4. (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.

5. (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

6. (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.

7. (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.

1. 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 $\vec{r}=\vec{r}_2-\vec{r}_1$.

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

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