Two carts, each with a mass of , are fixed to move on a horizontal track. · OlympiadBench #925

Physics · electromagnetism

Problem

Two carts, each with a mass of 300 g300 \mathrm{~g}, are fixed to move on a horizontal track. As shown in the figure, the first cart has a strong, tiny permanent magnet of dipole moment 0.5 A⋅m20.5 \mathrm{~A} \cdot \mathrm{m}^{2} attached to it, which is aligned along the axis of the track pointing toward the other cart. On the second cart, a copper tube of radius 7 mm7 \mathrm{~mm}, thickness 0.5 mm0.5 \mathrm{~mm}, resistivity 1.73⋅10−8Ω1.73 \cdot 10^{-8} \Omega, and length 30 cm30 \mathrm{~cm} is attached. The masses of the magnet and coil are negligible compared to the mass of the carts. At the moment its magnet enters through the right end of the copper tube, the velocity of the first cart is 0.3 m/s0.3 \mathrm{~m} / \mathrm{s} and the distance between the two ends of each cart is 50 cm50 \mathrm{~cm}, find the minimum distance achieved between the two ends of the carts in centimeters. While on the track, the carts experience an effective coefficient of static friction (i.e., what it would be as if they did not have wheels) of 0.01 . Neglect the self-inductance of the copper tube. <image_1> A picture of the two cart setup. The black rectangle represents the magnet while the gold rectangle represents the copper tube. Hint 1: The magnetic field due to a dipole of moment μ⃗\vec{\mu}, at a position r⃗\vec{r} away from the dipole can be written as B⃗=μ04π(3r^(r⃗⋅μ⃗)−μ⃗)r3r^\vec{B}=\frac{\mu_{0}}{4 \pi} \frac{(3 \hat{r}(\vec{r} \cdot \vec{\mu})-\vec{\mu})}{r^{3}} \hat{r} where r^\hat{r} is the unit vector in the direction of r⃗\vec{r}. Hint 2: The following mathematical identity may be useful: ∫−∞∞u2du(1+u2)5=5π128\int_{-\infty}^{\infty} \frac{u^{2} d u}{\left(1+u^{2}\right)^{5}}=\frac{5 \pi}{128} Given constants: - Proton mass, mp=1.67⋅10−27 kgm_{p}=1.67 \cdot 10^{-27} \mathrm{~kg} - Neutron mass, mn=1.67⋅10−27 kgm_{n}=1.67 \cdot 10^{-27} \mathrm{~kg} - Electron mass, me=9.11⋅10−31 kgm_{e}=9.11 \cdot 10^{-31} \mathrm{~kg} - Avogadro's constant, N0=6.02⋅1023 mol−1N_{0}=6.02 \cdot 10^{23} \mathrm{~mol}^{-1} - Universal gas constant, R=8.31 J/(mol⋅K)R=8.31 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K}) - Boltzmann's constant, kB=1.38⋅10−23 J/Kk_{B}=1.38 \cdot 10^{-23} \mathrm{~J} / \mathrm{K} - Electron charge magnitude, e=1.60⋅10−19Ce=1.60 \cdot 10^{-19} \mathrm{C} - 1 electron volt, 1eV=1.60⋅10−19 J1 \mathrm{eV}=1.60 \cdot 10^{-19} \mathrm{~J} - Speed of light, c=3.00⋅108 m/sc=3.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s} - Universal Gravitational constant, G=6.67⋅10−11( N⋅m2)/kg2G=6.67 \cdot 10^{-11}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{kg}^{2} - Solar Mass M⊙=1.988⋅1030 kgM_{\odot}=1.988 \cdot 10^{30} \mathrm{~kg} - Acceleration due to gravity, g=9.8 m/s2g=9.8 \mathrm{~m} / \mathrm{s}^{2} - 1 unified atomic mass unit, 1u=1.66⋅10−27 kg=931MeV/c21 \mathrm{u}=1.66 \cdot 10^{-27} \mathrm{~kg}=931 \mathrm{MeV} / \mathrm{c}^{2} - Planck's constant, h=6.63⋅10−34 J⋅s=4.41⋅10−15eV⋅sh=6.63 \cdot 10^{-34} \mathrm{~J} \cdot \mathrm{s}=4.41 \cdot 10^{-15} \mathrm{eV} \cdot \mathrm{s} - Permittivity of free space, ϵ0=8.85⋅10−12C2/(N⋅m2)\epsilon_{0}=8.85 \cdot 10^{-12} \mathrm{C}^{2} /\left(\mathrm{N} \cdot \mathrm{m}^{2}\right) - Coulomb's law constant, k=14πϵ0=8.99⋅109( N⋅m2)/C2k=\frac{1}{4 \pi \epsilon_{0}}=8.99 \cdot 10^{9}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{C}^{2} - Permeability of free space, μ0=4π⋅10−7 T⋅m/A\mu_{0}=4 \pi \cdot 10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} - Magnetic constant, μ04π=1⋅10−7( T⋅m)/A\frac{\mu_{0}}{4 \pi}=1 \cdot 10^{-7}(\mathrm{~T} \cdot \mathrm{m}) / \mathrm{A} - 1 atmospheric pressure, 1 atm=1.01⋅105 N/m2=1.01⋅105 Pa1 \mathrm{~atm}=1.01 \cdot 10^{5} \mathrm{~N} / \mathrm{m}^{2}=1.01 \cdot 10^{5} \mathrm{~Pa} - Wien's displacement constant, b=2.9b=2.9. 10−3 m⋅K10^{-3} \mathrm{~m} \cdot \mathrm{K} - Stefan-Boltzmann constant, σ=5.67⋅10−8 W/m2/K4\sigma=5.67 \cdot 10^{-8} \mathrm{~W} / \mathrm{m}^{2} / \mathrm{K}^{4}

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