Students should be able to:


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Homework for Central Forces

  1. (Scattering) What happens if you change from an attractive to a repulsive potential? This problem has many parts. Don't forget your lessons from introductory physics!

    Consider a very light particle of mass $\mu$ scattering from a very heavy, stationary particle of mass $M$. The force between the two particles is a {\bf repulsive} Coulomb force ${k\over r^2}$. The impact parameter $b$ in a scattering problem is defined to be the distance which would be the closest approach if there were no interaction (See Figure). The initial velocity (far from the scattering event) of the mass $\mu$ is $\Vec v_0$. Answer the following questions about this situation in terms of $k$, $M$, $\mu$, $\Vec v_0$, and $b$. It is not necessarily wise to answer these questions in order.)

    \medskip \centerline{\includegraphics[height=1.0truein]{\TOP Figures/cfimpact}} \medskip

    1. What is the initial angular momentum of the system?

    2. What is the initial total energy of the system?

    3. What is the distance of closest approach $r_{\rm{min}}$ with the interaction)?

    4. Sketch the effective potential.

    5. What is the angular momentum at $r_{\rm{min}}$?

    6. What is the total energy of the system at $r_{\rm{min}}$?

    7. What is the radial component of the velocity at $r_{\rm{min}}$?

    8. What is the tangential component of the velocity at $r_{\rm{min}}$?

  2. (HardConeScattering)

    Consider the case of “hard cone scattering,” where the target particles are all shaped like identical cones, with their points oriented directly into the beam of incident particles. Describe the angular distribution of the scattered particles.% (Taylor 14.7)

  3. (SolidAngle)

    Calculate the solid angle subtended by the moon and by the sun, both as seen from the earth.

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