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Условие Решение

USAPhO 2005, Полуфинал — A4

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Всички материали от USAPhO 2005

2005 Semi-Final Exam · 25 т.

2005 Semi-Final Exam — Instructions

AAPT / AIP — United States Physics Team 2005.

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

Work Part A first. You have 90 minutes to complete all four problems.
After you have completed Part A, you may take a break.
Then work Part B. You have 90 minutes to complete both problems.
Show all your work. Partial credit will be given.
Start each question on a new sheet of paper. Be sure to put your name in the upper right-hand corner of each page, along with the question number and the page number/total pages for this problem. For example: Doe, Jamie; A1 – 1/3.
A hand-held calculator may be used. Its memory must be cleared of data and programs. You may use only the basic functions found on a simple scientific calculator. Calculators may not be shared. Cell phones, PDA’s, or cameras may not be used during the exam or while the exam papers are present. You may not use any tables, books, or collections of formulas.
Questions with the same point value are not necessarily of the same difficulty.
Do not discuss the contents of this exam with anyone until after the submission deadline.
Good luck!

Possibly Useful Information

Quantity	Value
Gravitational field at the Earth’s surface	$g=9.8$ N/kg
Newton’s gravitational constant	$G=6.67\times10^{-11}$ N·m $^2$ /kg $^2$
Coulomb’s constant	$k=1/4\pi\epsilon_0=8.99\times10^9$ N·m $^2$ /C $^2$
Biot-Savart constant	$k_m=\mu_0/4\pi=10^{-7}$ T·m/A
Speed of light in a vacuum	$c=3.00\times10^8$ m/s
Boltzmann’s constant	$k_B=1.38\times10^{-23}$ J/K
Avogadro’s number	$N_A=6.02\times10^{23}$ (mol) $^{-1}$
Ideal gas constant	$R=N_Ak_B=8.31$ J/(mol·K)
Stefan-Boltzmann constant	$\sigma=5.67\times10^{-8}$ J/(s·m $^2$ ·K $^4$ )
Elementary charge	$e=1.602\times10^{-19}$ C
1 electron volt	$1\text{ eV}=1.602\times10^{-19}$ J
Planck’s constant	$h=6.63\times10^{-34}\text{ J}\cdot\text{s}=4.14\times10^{-15}\text{ eV}\cdot\text{s}$
Electron mass	$m=9.109\times10^{-31}\text{ kg}=0.511\text{ MeV}/c^2$
Binomial expansion	$(1+x)^n\approx1+nx$ for $\lvert x\rvert\ll1$
Small angle approximations	$\sin\theta\approx\theta$ , $\cos\theta\approx1-\tfrac12\theta^2$

Условие

Just like electrostatic fields, gravitational fields obey Gauss’s Law. One of the arguments for dark matter in the universe is the observation that the velocity of stars orbiting far from the center of the galaxy is approximately constant. This requires a spherically symmetric mass distribution which the visible stars do not have. Let the mass density of the galaxy be $\rho(r)=Ar^n$ where $A$ is a constant, $r$ is the distance from the center of galaxy, and the power $n$ is as yet undetermined.

a. Show that the total mass $M$ enclosed by a sphere of radius $R$ is $M(R)=CR^{n+3}$ and find the constant $C$ in terms of given variables and known constants. [5 т.]

b. You may use $M(R)=CR^{n+3}$ , with $C$ unspecified, for the remaining parts of this problem.

Find the magnitude of the gravitational force on a star of mass $m$ which has a circular orbit of radius $R$ about the center of this galaxy. [5 т.]

c. Find the orbital velocity $v$ of this star. [5 т.]

d. What power $n$ is required by the observation that $v$ is constant? [5 т.]

e. If the mass density were $\rho(r)\approx0$ for $r>R$ , a reasonable assumption based on the observed visible star distribution, what would be the orbital velocity of a star with $r>R$ ? Let $M_0=$ the total mass of the galaxy. [5 т.]

Решение

a. (5 т.)

The mass of a spherical shell of radius $r$ and thickness $dr$ is

\rho(r)4\pi r^2dr=Ar^n4\pi r^2dr=Ar^{n+2}4\pi dr.\qquad (A4-1)

To find the total mass enclosed by a sphere of radius $R$ , we add up the mass of all of the spherical shells:

M(R)=\int_0^RAr^{n+2}4\pi dr=\frac{4\pi AR^{n+3}}{n+3}=CR^{n+3}\qquad (A4-2)

where $C=\frac{4\pi A}{n+3}$ . (A4-3)

b. (5 т.)

Gauss’s Law applied to a spherical shell of radius $R$ gives us

\oint\vec g\cdot d\vec A=-4\pi GM_{\rm enclosed}.\qquad (A4-4)

By symmetry, $g$ is constant over the surface of the spherical shell. $g$ is radially inward, so

\oint\vec g\cdot d\vec A=-g4\pi R^2.\qquad (A4-5)

When we substitute (A4-5) into (A4-4) and substitute (A4-2) for $M_{\rm enclosed}$ , we obtain

-g4\pi R^2=-4\pi GCR^{n+3}.\qquad (A4-6)

Solving for $g$ ,

g=GCR^{n+1}.\qquad (A4-7)

The gravitational force on a star of mass $m$ is then

F_{\rm grav}=mg=mGCR^{n+1}.\qquad (A4-8)

c. (5 т.)

Applying Newton’s second law to the circular motion of the star, we find that

F_{\rm grav}=\frac{mv^2}{R}.\qquad (A4-9)

Substituting (A4-8) into (A4-9) yields

mGCR^{n+1}=\frac{mv^2}{R}.\qquad (A4-10)

v=\sqrt{GCR^{n+2}}.\qquad (A4-11)

d. (5 т.)

If $v$ is constant, then $R^{n+2}=R^0$ , so $n=-2$ . (A4-12)

e. (5 т.)

Apply Gauss’s Law to a spherical shell of radius $r$ , where $r>R$ . Here, the mass enclosed is $M_0$ .

-g4\pi r^2=-4\pi GM_0.\qquad (A4-13)

g=\frac{GM_0}{r^2}.\qquad (A4-14)

Substituting into (A4-9)

\frac{mGM_0}{r^2}=\frac{mv^2}{r}.\qquad (A4-15)

v=\sqrt{\frac{GM_0}{r}}.\qquad (A4-16)

Оригинал в Архива: Semifinal 2005.pdf · официални решения: Semifinal 2005 Solutions.pdf