Physics 303 Final

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Questions (3 pts. apiece) Answer questions 1-6 in complete, well-written sentences WITHIN the spaces provided. For multiple-choice questions 7-8 circle the correct answer.

1. What is the evidence for the existence of extrasolar planets?

2. What is the evidence for dark matter halos around galaxies?

3. Recall our study of the force of gravity from a spherical shell of density $\rho(r^\prime)$. What is the mass of a shell of radius $r^\prime$ and thickness $dr^\prime$?

4. The plot below shows a model of the velocity of Vesna Vulovic when she fell out of a Yugoslavian passenger plane in 1972. Why does her velocity decrease linearly in the early part of her fall? Why does her velocity become constant later on?

   

5. For angular momentum to be conserved in some system, what feature must the Lagrangian of that system display?

6. When we studied Rutherford scattering we used the conservation laws (energy, momentum, etc) instead of Newton's Laws. Why?

7. It is possible that the Newtonian theory of gravitation may need to be modified at short range. Suppose that the potential energy between two masses $m$ and $m^\prime$ is given by
   
   $$V(r) = - \frac{G m m^\prime}{r} (1- a e^{-r/\lambda}) \quad .$$
   
   
   For short distances $r<<\lambda$ calculate the force between $m$ and $m^\prime$.

   A.	$F=-Gmm^\prime/r^2$	B.	$F=-Gmm^\prime a/\lambda$
   B.	$F=-Gmm^\prime(1-a)/r^2$	D.	$F=Gmm^\prime(1-a)r^2$
   C.	$F=-Gmm^\prime(1+a)/r^2$

8. A 10 g bullet is fired into a 2 kg ballistic pendulum as shown in the figure. The bullet remains in the block after the collision and the system rises to a maximum height of 20 cm. Find the initial speed of the bullet.

   
   

   A.	28.0 m/s	B.	23.8 m/s
   C.	3.98 m/s	D.	719 m/s
   E.	398 m/s

   
   

   

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

1.

12 pts.

A particle moves subject to the potential energy $$V(x) = V_0 \left ( \frac{a}{x} + \frac{x}{a} \right )$$ where $V_0$ and $a$ are positive. Locate any equilibrium points, determine which are stable, and obtain the frequency of small oscillations about those points.

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

2.	12 pts.	Two masses $m_1$ and $m_2$ with coordinates $x_1$ and $x_2$ in one dimension are connected by a spring of spring constant $k$. Use Lagrangian methods to find the equations of motion. What is the angular frequency of simple harmonic motion for relative displacement $x_2 - x_1$ of the two masses?
3.	12 pts.	Two particles on a line are mutually attracted by a force $$F = -fr$$ where $f$ is a constant and $r$ is the distance of separation. At time $t=0$, particle $A$ of mass $M$ is located at $x=6~cm$, and particle $B$ of mass $\frac{M}{4}$ is located at $x=11~cm$. If the particles are at rest at $t=0$, at what value of $x$ do they collide?
4.	12 pts.	Show the drag force on a satellite moving with velocity $v$ in the Earth's upper atmosphere is approximately $f_D = \rho A v^2$ where $\rho$ is the atmospheric density and $A$ is the cross-sectional area perpendicular to the direction of motion. Assume the air molecules are moving slowly compared with $v$ and their collisions with the satellite are completely inelastic (i.e., they stick together).
5.	14 pts.	In August 2004, observations of the star $\mu$ Arae revealed an oscillatory structure with a period $T=9.5~days$ shown in the figure. From its spectral type the mass of $\mu$ Arae is 1.10 solar masses. What is the minimum mass of this planet and its distance from $\mu$ Arae? How does this mass compare with planets in our solar system?

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

6.

14 pts.

Recall the way we used the conservation of energy in analyzing Rutherford scattering. We started with the following form of the energy equation $$E = \frac{1}{2} \mu {\dot r}^2 + \frac{l^2}{2\mu r^2} + V(r)$$ where $l$ is the angular momentum (and constant), $\mu$ is the reduced mass, and $V(r)$ is the potential energy. We then obtained the orbit equation which relates the distance between the two masses and the angular position $\theta$ of the projectile. Now consider a different potential energy function than the Coulomb or gravitational ones we used before. The dipole-dipole interaction is common is atomic and sub-atomic physics and is of the form $$V(r) = -\frac{\alpha}{r^2}$$ where $\alpha$ is some constant. a. Obtain an equation for $\dot r$ using the energy equation above. b. Use the chain rule and and the fact that $l = \mu r^2 \ddot \theta$ to change variable from time $t$ to angle $\theta$. c. Now separate the $\theta$ and $r$ portions of the result of the previous part and integrate the equation you get to obtain a relationship between $\theta$ and $r$. This is the new orbit equation for the dipole-dipole interaction.

Equations

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

$$\vec F_g = - m g \hat y \quad \vec F_s = - k r \hat r \quad ... ..._f = - c v^2 \hat v \quad F_c = m a_c = m {v_\theta^2 \over r}$$

$$\int {df \over dx } dx = \int df \quad \ddot y + A \dot y + ... ... A \sin(\omega_0 t + \phi) \quad \omega_0 = \sqrt{\frac{k}{m}}$$

$$V = - \int_{x_s}^x \vec F(\vec r^{\ \prime}) d\vec r^{\ \pri... ...\over 2} \quad V_g = mgy \quad V_G = -{Gm_1 m_2 \over r} \quad$$

$$V_C = - {k q_1 q_2 \over r} \quad V(x) = V(x)\vert _{x=x_e}... ...\over 2 \mu r^2} + V(r) \quad \sin \theta \approx \theta \quad$$

$$L = K - V \quad {d\ \over dt}\left ( {\partial L \over \dot ... ... p_0 = \vec p_1 \quad l = \mu r^2 \dot \theta = \mu v_i b\quad$$

$$m_1 \vec {v_1}^{\ \prime} + m_2 \vec {v_2}^{\ \prime} = 0 \q... ...r^2 + r^2 \dot \theta^2) \quad \mu = {m_1 m_2 \over m_1 + m_2}$$

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

$${ d\sigma \over d \Omega} = \left ( {\alpha \over 4 E_{cm}} ... ...} \right )^{1/3} \quad \alpha = Gm_1m_2 \ \ {\rm or}\ -kq_1q_2$$

$$\Phi = {V \over m} \quad \Phi = -{GM \over R} \quad (R>a) \q... ...r a} \quad (R<a) \quad \vec F (r) = -{GmM(r) \over r^2} \hat r$$

Constants and conversion factors

$\rho$ (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	$6.37\times 10^6~m$

More constants and conversion factors

Earth-Moon distance	$3.84 \times 10^{8}~m$	Moon's mass	$7.36\times 10^{22}~kg$
Earth-Sun distance	$1.50 \times 10^{11}~m$	Earth's mass	$5.97\times 10^{24}~kg$
Coulomb constant ($k_e$)	$8.99\times 10^{9} {N-m^2 \over C^2}$	Jupiter's mass	$1.90\times 10^{27}~kg$
Elementary charge ($e$)	$1.60\times 10^{-19}~C$	Proton/Neutron mass	$1.67\times 10^{-27}~kg$
Planck's constant ($h$)	$6.626\times 10^{-34}~J-s$	Planck's constant ($h$)	$4.1357\times 10^{-15}~eV-s$
Planck constant ($\hbar$)	$1.0546\times 10^{-34}~J-s$	Planck's constant ($\hbar$)	$6.5821\times 10^{-16}~eV-s$
Planck's constant ($\hbar c$)	$197~MeV-fm$	Planck's constant ($\hbar c$)	$1970~eV-{\rm\AA}$
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	$3.2\times 10^{-19}~C$
$\rm 1 ~MeV$	$10^6 ~ eV$	$\rm 1.0~eV$	$1.6\times 10^{-19}~J$
$\rm 1.0~ \AA$	$10^{-10} ~m$	$\rm 1~fm$	$10^{-15}~m$