Съдържание

Условие Отговори Решение

USAPhO 2002, Полуфинал — Задача 4. A4

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Всички материали от USAPhO 2002

2002 Semi-Final Exam

Указания

INSTRUCTIONS

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. It may not be used in graphing mode. Calculators may not be shared. You may not use any tables, books, or collections of formulas. You may use a ruler or straight edge.
Questions with the same point value are not necessarily of the same difficulty.
Good luck!

Possibly Useful Information

Gravitational field at the Earth’s surface	$g = 9.8\ \mathrm{N/kg}$
Newton’s gravitational constant	$G = 6.67 \times 10^{-11}\ \mathrm{N\cdot m^2/kg^2}$
Coulomb’s constant	$k = 1/4\pi\varepsilon_o = 9.0 \times 10^{9}\ \mathrm{N\cdot m^2/C^2}$
Biot-Savart constant	$k_m = \mu_o/4\pi = 10^{-7}\ \mathrm{T\cdot m/A}$
Speed of light in a vacuum	$c = 3.0 \times 10^{8}\ \mathrm{m/s}$
Boltzmann’s constant	$k_B = 1.38 \times 10^{-23}\ \mathrm{J/K}$
Avogadro’s number	$N_A = 6.02 \times 10^{23}\ \mathrm{(mol)^{-1}}$
Ideal gas constant	$R = N_Ak_B = 8.31\ \mathrm{J/(mol\cdot K)}$
Stefan-Boltzmann constant	$\sigma = 5.67 \times 10^{-8}\ \mathrm{J/(s\cdot m^2\cdot K^4)}$
Elementary charge	$e = 1.6 \times 10^{-19}\ \mathrm{C}$
1 electron volt	$1\ \mathrm{eV} = 1.6 \times 10^{-19}\ \mathrm{J}$
Planck’s constant	$h = 6.63 \times 10^{-34}\ \mathrm{J\cdot s} = 4.14 \times 10^{-15}\ \mathrm{eV\cdot s}$
Electron mass	$m = 9.1 \times 10^{-31}\ \mathrm{kg} = 0.51\mathrm{MeV}/c^2$
Binomial expansion	$(1 + x)^n \approx 1 + nx$ for $\lvert x\rvert \ll 1$

Условие

A 0.75-m rod has a uniform linear mass density of $\lambda$ . A small mass $m$ with negligible volume is attached to one end of the rod. The rod with the attached mass is placed in a container of unknown fluid and after oscillating briefly comes to rest in its equilibrium position. At equilibrium it floats vertically (mass $m$ end down) with 2/3 of its length submerged. If the rod were fully submerged it would displace $7.5\ \mathrm{x}10^{-4}$ kg of fluid.

a. What is the maximum value that the mass $m$ can have? [10 т.]

b. What is the minimum value that the mass $m$ can have? [10 т.]

c. Sketch a graph that shows the values of $\lambda$ as a function of $m$ . [5 т.]

Отговори

Решение

a. Let $L = 0.75$ m, the length of the rod.

$\rho$ = the density of the unknown fluid

$V$ = the total volume of the rod

$\lambda L$ = mass of the rod

Since if the rod were fully submerged it would displaced $7.5\times10^{-4}$ kgof fluid, $\quad 7.5\times10^{-4}\ \mathrm{kg} = \rho V$

In the present case the rod is only 2/3 submerged so The buoyant force acting on the rod is

B = W_{fl\ dis} = \rho\left(\tfrac{2}{3}V\right)g = \tfrac{2}{3}\left(7.5\times10^{-4}\ \mathrm{kg}\right)g = \left(5.0\times10^{-4}\ \mathrm{kg}\right)g

Since the system is in equilibrium the buoyant force balances the total weight force.

\left(5.0\times10^{-4}\ \mathrm{kg}\right)g = mg + \lambda Lg

5.0\times10^{-4}\ \mathrm{kg} = m + \lambda L. \qquad \text{(A4-1)}

Since $\lambda$ cannot be negative the largest value m can have is $\quad 5.0\times10^{-4}\ \mathrm{kg} \ge m$ .

b. The system oscillates and comes to rest. It must be in stable equilibrium. For this to occur the center of gravity must be below the center of buoyancy. The fluid is uniform. The center of buoyancy is at the mid point of the submerged length. Measuring distances from the bottom end of the rod, the location of the center of buoyancy is

y_{cb} = \tfrac{1}{2}\left(\tfrac{2}{3}L\right) = \tfrac{1}{3}L.

Mass $m$ is located at $y = 0$ . The center of gravity of the rod is at $L/2$ . The combined center of gravity of the system $y_{cg}$ is

y_{cg}\sum m_ig = \sum y_im_ig

y_{cg}(m + \lambda L)g = 0mg + (L/2)(\lambda L)g

y_{cg} = \frac{\lambda L^2}{2(m + \lambda L)}

Requiring the center of gravity to be below the center of buoyancy

y_{cg} < y_{cb}

\frac{\lambda L^2}{2(m + \lambda L)} < \frac{1}{3}L

3\lambda L < 2(m + \lambda L)

\lambda L < 2m

Combining this with (A4-1)

5.0\times10^{-4}\ \mathrm{kg} < m + 2m

m > \frac{1}{3}\left(5.0\times10^{-4}\ \mathrm{kg}\right) = 1.7\times10^{-4}\ \mathrm{kg}

c. Solving (A4-1) for $\lambda$

\lambda = \frac{5.0\times10^{-4}\ \mathrm{kg} - m}{L} = \frac{0.5\ \mathrm{grams} - m}{0.75\ \mathrm{m}}.

At its minimum value $m_{min} = 1.7\times10^{-4}\ \mathrm{kg} = 0.17$ grams, $\lambda = 0.44$ grams / m

At its maximum value $m_{max} = 5.0\times10^{-4}\ \mathrm{kg} = 0.50$ grams, $\lambda = 0$ .

The graph is a straight line with negative slope between these two points.

Graph of λ in g/m (vertical axis) against m in g (horizontal axis): a straight line falling from λ = 0.44 at m = 0.17 to λ = 0 at m = 0.500. Axis labels: “λ in g/m”, “m in g”; tick labels 0.44, 0.17, 0.500.

Автори

The 2002 Semi-Final Examination questions were written by the coaches of the United States Physics Team. The coaches are: Academic Director, Mary Mogge – Professor of Physics at California State Polytechnic University, Pomona, CA; Senior Coach, Leaf Turner – Physicist in the Theoretical Division of Los Alamos National Laboratory, Los Alamos, NM; Warren Turner – Physics Teacher at Massachusetts Academy of Math and Science, Worcester, MA; Robert Shurtz – Physics Teacher at the Hawken School, Gates Mills, OH; Boris Zbarsky –undergraduate student at MIT, Cambridge, MA; Andrew Lin – undergraduate student at Yale University, New Haven, CT. Questions A2, A4, B1, and B2 were suggested by Leaf Turner and Questions A1 and A3 by Mary Mogge.

Оригинал в Архива: Semifinal 2002.pdf · официални решения: Semifinal 2002 Solutions.pdf