Physics 303 Test 2

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Questions (5 pts. apiece) Answer questions 1-3 in complete, well-written sentences WITHIN the spaces provided. For multiple-choice questions 4-5 circle the correct answer.

1. Recall our study of the $\rm CO_2$ molecule. What are normal modes of vibration?

2. Suppose an asteroid struck a glancing, inelastic blow on the Earth and went off into space. Would this situation be better or worse than a perfectly inelastic collision? Explain.

3. In calculating the Rutherford scattering cross section we made some assumptions about the impact parameters for the target-projectile collisions. What were those those assumptions and why are they valid?

4. The potential energy of a particle moving in one dimension is given by $U = 1/2 kx^2 + 1/4 bx^4$. Determine the force.

   A.	$-kx - bx^3$	B.	$kx + bx^3$
   C.	$1/6 kx^3 + 1/20 bx^5$	D.	$-1/6 kx^3 - 1/20 bx^5$
   E.	$-kx-bx^2$

5. Find the distance of closest approach for the following elastic nuclear reaction.
   
   $$\rm ^7_3Li ~+~ ^{208}_{82}Pb$$
   
   
   Assume that only the Coulomb force is important. The Li nucleus is accelerated to a kinetic energy of 50.0 MeV.

   A.	1.12 fm	B.	2.24 fm
   C.	3.54 fm	D.	7.08 fm
   E.	8.20 fm

   
   

   
   
   
   

Problems. Clearly show all reasoning for full credit. Use a separate sheet to show your work.

1.	20 pts.	What is the distance of the center of mass of the earth-moon system from the center of the earth?
2.	25 pts.	A proton of mass $m_1=1~u$ and energy $E=6~MeV$ scatters off another proton and comes off at an angle $\theta_1=25^\circ$ in the lab system. What is its final speed?
3.	30 pts.	In our study of Rutherford scattering we ignored the effect of the atomic electrons outside the nucleus. We now want to test that assumption. Consider a gold target with $Z_1=79$ and atomic radius $R_g=1.4 \rm\AA$. For simplicity, treat the negative charge of the atom as a uniform spherical shell of charge $-Z_1$. What is the maximum deflection due solely to this negative charge of the gold atom for an incident $\rm ^4_2He$ nucleus with charge $Z_2=+2$ and energy $E=5.407~MeV$ if it does not penetrate the shell? In other words, ignore the effect of the positive charge in the gold nucleus and find the biggest angle the $\rm ^4_2He$ nucleus will be scattered into. Does your result validate the assumption we made? Extra Credit: What is the Coulomb force on the incident nucleus if it penetrates the spherical shell? How would this change the trajectory?

Equations, Conversions, and Constants

$$\vec F = m \vec a = \dot {\vec p} = - \nabla V \qquad \vec F... ...ver r^2} \hat r \qquad \vec F_C = {k q_1 q_2 \over r^2} \hat r$$

$$\vec F_g = - m g \hat y \qquad \vec F_s = - k r \hat r \qquad \vec F_f = -b v \hat v \qquad \vec F_f = - c v^2 \hat v$$

$$\int {df \over dx } dx = \int df \qquad \ddot y + A \dot y +... ...y + \omega_0^2 y = 0 \Rightarrow y = A \sin(\omega_0 t + \phi)$$

$$V = - \int_{x_s}^x \vec F(\vec r^{\ \prime}) \cdot d\vec r^{... ...uad V_G = -{Gm_1 m_2 \over r} \qquad V_C = {k q_1 q_2 \over r}$$

$$K = {1 \over 2} mv^2 \quad L = K - V \qquad {d\ \over dt}\le... ...0 \qquad l = \mu r^2 \dot \theta \quad \vec p = m \vec v \quad$$

$${1 \over r} = {\mu \alpha \over l^2} \left (1 + \epsilon ... ...\sin \left ( {\theta_s \over 2} \right ) = { 1 \over \epsilon}$$

$${ d\sigma \over d \Omega} = \left ( {\alpha \over 4 E_{cm}} ... ...ke^2 Z_1 Z_2 \qquad ke^2 = {\hbar c \over 137} = 1.44 ~ MeV-fm$$

$$V_{eff}(r) = {l^2 \over 2 \mu r^2} + V(r) \quad (m_1 + m_2) ... ... \vec r_1 + m_2 \vec r_2 \quad \mu = \frac{m_1 m_2}{m_1 + m_2}$$

$\rho$ (water)	$1.0\times 10^3 kg/m^3$	$P_{atm}$	$1.05\times 10^5 ~N/m^2$
$k_B$	$1.38\times 10^{-23}~J/K$	Speed of light ($c$)	$3.0\times 10^{8}~m/s$
$R$	$8.31J/K-mole$	$g$	$9.8~m/s^2$
$0~K$	$\rm -273^\circ~C$	$1 ~ u$	$1.67\times 10^{-27}~kg$
Gravitation constant ($G$)	$6.67 \times 10^{-11}~N-m^2/kg^2$	Earth's radius	$6.37\times 10^6~m$
Earth-Moon distance	$3.84 \times 10^{8}~m$	Moon's mass	$7.36\times 10^{22}~kg$
Coulomb constant ($k_e$)	$8.99\times 10^{9} {N-m^2 \over C^2}$	Earth's mass	$5.97\times 10^{24}~kg$
Elementary charge ($e$)	$1.60\times 10^{-19}~C$	Proton/Neutron mass	$1.67\times 10^{-27}~kg$
Planck's constant ($h$)	$6.626\times 10^{-34}~J-s$	Proton/Neutron mass	$932\times 10^{6}~eV/c^2$
Permittivity constant ($\epsilon_0$)	$8.85\times 10^{-12} {kg^2\over N-m^2}$	Electron mass	$9.11\times 10^{-31}~kg$
Permeability constant ($\mu_0$)	$4\pi\times 10^{-7} N/A^2$	Electron mass	$0.55\times 10^{6}~MeV/c^2$
$\rm ^4 He$ mass	$6.68 \times 10^{-27}~kg$	$\rm ^4 He$ charge	$3.2\times 10^{-19}~C$
$\rm 1 ~MeV$	$10^6 ~ eV$	$\rm 1.0~eV$	$1.6\times 10^{-19}~J$
$\rm 1.0~ \AA$	$10^{-10} ~m$	$\rm 1~fm$	$10^{-15}~m$