USAPhO 2009, Полуфинал — Задача 3. Question A3

United States Physics Team 2009 — Semifinal Exam · 25 т.

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Possibly Useful Information. You may use this sheet for both parts of the exam.
g=9.8 N/kgg = 9.8\ \mathrm{N/kg}G=6.67×10−11 N⋅m2/kg2G = 6.67 \times 10^{-11}\ \mathrm{N \cdot m^2/kg^2}
k=1/4πϵ0=8.99×109 N⋅m2/C2k = 1/4\pi\epsilon_0 = 8.99 \times 10^{9}\ \mathrm{N \cdot m^2/C^2}km=μ0/4π=10−7 T⋅m/Ak_{\mathrm{m}} = \mu_0/4\pi = 10^{-7}\ \mathrm{T \cdot m/A}
c=3.00×108 m/sc = 3.00 \times 10^{8}\ \mathrm{m/s}kB=1.38×10−23 J/Kk_{\mathrm{B}} = 1.38 \times 10^{-23}\ \mathrm{J/K}
NA=6.02×1023 (mol)−1N_{\mathrm{A}} = 6.02 \times 10^{23}\ \mathrm{(mol)^{-1}}R=NAkB=8.31 J/(mol⋅K)R = N_{\mathrm{A}} k_{\mathrm{B}} = 8.31\ \mathrm{J/(mol \cdot K)}
σ=5.67×10−8 J/(s⋅m2⋅K4)\sigma = 5.67 \times 10^{-8}\ \mathrm{J/(s \cdot m^2 \cdot K^4)}e=1.602×10−19 Ce = 1.602 \times 10^{-19}\ \mathrm{C}
1eV=1.602×10−19 J1\mathrm{eV} = 1.602 \times 10^{-19}\ \mathrm{J}h=6.63×10−34 J⋅s=4.14×10−15 eV⋅sh = 6.63 \times 10^{-34}\ \mathrm{J \cdot s} = 4.14 \times 10^{-15}\ \mathrm{eV \cdot s}
me=9.109×10−31 kg=0.511 MeV/c2m_e = 9.109 \times 10^{-31}\ \mathrm{kg} = 0.511\ \mathrm{MeV/c^2}(1+x)n≈1+nx(1+x)^n \approx 1 + nx for ∣x∣≪1\lvert x \rvert \ll 1
sin⁡θ≈θ−16θ3\sin\theta \approx \theta - \frac{1}{6}\theta^3 for ∣θ∣≪1\lvert\theta\rvert \ll 1cos⁡θ≈1−12θ2\cos\theta \approx 1 - \frac{1}{2}\theta^2 for ∣θ∣≪1\lvert\theta\rvert \ll 1

Условие

Two stars, each of mass MM and separated by a distance dd, orbit about their center of mass. A planetoid of mass mm (m≪Mm \ll M) moves along the axis of this system perpendicular to the orbital plane.

Two equal stars on opposite sides of an elliptical (perspective circular) orbit joined by a double arrow labelled d; a dashed vertical line through the center is labelled 'axis perpendicular to plane of orbit', and a small dot on it above the plane is at height z, marked with a small double arrow.

Let TpT_p be the period of simple harmonic motion for the planetoid for small displacements from the center of mass along the zz-axis, and let TsT_s be the period of motion for the two stars. Determine the ratio Tp/TsT_p/T_s.

This problem was adapted from a problem by French in Newtonian Mechanics.

Отговори

Решение

Оригинал в Архива: 2009_SemiFinal.pdf · официални решения: USAPhO-2009-Solutions.pdf