题型概述：万有引力、轨道与圆周运动Question type: gravitation, orbits & circular motion
Top Gold这是 BPhO Round 1 最稳定的考点之一：Section 1 每年 1–3 道小题（每题 2–5 分），Section 2 还常有整道大题，合计约 15 分/年。物理本身只有两条主线——"引力提供向心力"和"合力指向圆心"。失分几乎都出在三处：单位换算（km/h、光年、天 → 秒）、半径从哪里量（从球心 / 从转轴），以及角度从哪条线量（竖直还是水平，sin 还是 tan）。One of the most reliable topics in BPhO Round 1: Section 1 has 1–3 short items every year (2–5 marks each) and Section 2 often has a full question, about 15 marks a year in total. The physics is just two ideas: "gravity supplies the centripetal force" and "the net force points to the centre". Marks are lost in three places: unit conversions (km/h, light years, days → seconds), where the radius is measured from (the centre of the planet / the axis), and which line the angle is measured from (vertical or horizontal, sin or tan).
1万有引力与表面重力加速度 gNewton's law of gravitation and surface g
两个质点（或均匀球体，按球心算）之间的引力与质量乘积成正比、与距离平方成反比。放在星球表面的物体所受引力除以质量就是 g。把质量写成"密度 × 体积"后，表面 g 只和 ρR 成正比——这正是"比值题"最常用的一步。Two point masses (or uniform spheres, measured centre to centre) attract with a force proportional to the product of the masses and inversely proportional to the square of the distance. Divide the force on a surface object by its mass to get g. Writing the mass as "density × volume" shows that surface g is proportional to ρR alone — the key step in most ratio questions.
F = GMm/r², g(r) = GM/r² = g0(R/r)², gsurface = G·(4/3)πR³ρ/R² = (4/3)πGρ Rr 永远从星球球心量起：高度 h 处 r = R + h。比值题里 G、π 全部约掉，不必代数值。r is always measured from the centre of the planet: at height h, r = R + h. In ratio questions G and π cancel, so no numbers are needed for them.
2圆轨道：v、T 与开普勒第三定律Circular orbits: v, T and Kepler's third law
卫星做匀速圆周运动，引力就是向心力。令 GMm/r² = mv²/r = mω²r，卫星质量 m 约掉，得到轨道速度和周期只由中心天体质量 M 和轨道半径 r 决定。越远越慢、周期越长；T² ∝ r³ 就是开普勒第三定律（等价写法 ω²r³ = GM = 常量）。A satellite moves in uniform circular motion and gravity is the centripetal force. Setting GMm/r² = mv²/r = mω²r, the satellite mass m cancels: the orbital speed and period depend only on the central mass M and the radius r. Further out means slower and longer; T² ∝ r³ is Kepler's third law (equivalently ω²r³ = GM = constant).
v = 2πr/T = √(GM/r), T² = (4π²/GM) r³, a = v²/r = ω²r = GM/r²地球同步轨道：T = 1 恒星日（23.93 h），代入得 r ≈ 42 200 km ≈ 6.6 R⊕，只能在赤道上空。拖动下面的滑块比较近地轨道和同步轨道。Geostationary orbit: T = 1 sidereal day (23.93 h) gives r ≈ 42 200 km ≈ 6.6 R⊕, and it must lie above the equator. Use the slider below to compare a low orbit with the geostationary one.
动画 1 s = 2 h。地球上的红点随地球自转；选"地球同步轨道"时卫星始终停在红点正上方。Animation: 1 s = 2 h. The red dot turns with the Earth; in the geostationary orbit the satellite stays above it.
3圆周运动的受力：合力指向圆心Forces in circular motion: the net force points to the centre
"向心力"不是一个新的力，它是真实受力（重力、拉力、支持力、摩擦力……）的合力在指向圆心方向上的分量。受力图里只画真实的力，再令"指向圆心的合力 = mv²/r"。最常见的一族是斜支撑：圆锥摆的绳、倾斜轨道的支持力、转弯时自行车的倾斜——竖直分量托住重力，水平分量提供向心力，两式相除得到同一个公式。竖直圆周（过山车、绳球）则在最高点取临界条件，再用能量守恒连接最高点和最低点。"Centripetal force" is not a new force: it is the component, towards the centre, of the resultant of the real forces (weight, tension, normal reaction, friction …). Draw only real forces, then set "net force towards the centre = mv²/r". The most common family is the tilted support: a conical-pendulum string, the normal force on a banked track, a leaning cyclist — the vertical component holds up the weight, the horizontal component supplies the centripetal force, and dividing the two gives one formula. For vertical circles (loops, a ball on a string) use the critical condition at the top, then link top and bottom with energy conservation.
F cos θ = mg, F sin θ = mv²/r ⇒ tan θ = v²/(rg) = ω²r/g 竖直圆最高点top of a vertical circle: vtop² ≥ grθ 是支撑力（绳、法向力）与竖直方向的夹角。地球自转也属于这一类：赤道上"视重" = mg − mω²R，所以赤道的 g 比两极小约 0.34%。θ is the angle between the support force (string, normal reaction) and the vertical. The Earth's spin belongs here too: apparent weight at the equator = mg − mω²R, so g is about 0.34% smaller at the equator than at the poles.
| 子题型Sub-type | 识别特征How to spot it | 核心步骤Core step | 真题Past paper |
|---|---|---|---|
| 轨道速度 v = 2πr/TOrbital speed v = 2πr/T | 给出轨道半径和周期，单位是光年、秒差距、年Radius and period given in light years, parsecs, years | 每个轨道 v = 2πr/T；比值中 2π 约掉，统一"光距离/年"v = 2πr/T for each orbit; 2π cancels in the ratio; use light-distance per year | 2024 Q1(b) |
| 开普勒第三定律比值Kepler III ratios | 同一中心天体的两颗卫星，给周期比Two satellites of the same body, period ratio given | T² ∝ r³ 求半径比，再用 a = ω²r 或 GM/r²T² ∝ r³ for the radius ratio, then a = ω²r or GM/r² | 2022 Q1(d) |
| 其他星球 / 高空的 gg on other planets / at altitude | "density … radius … times that of the Earth""density … radius … times that of the Earth" | g ∝ ρR（表面）或 g ∝ M/r²g ∝ ρR (surface) or g ∝ M/r² | 2018 Q1(k) |
| 圆锥摆Conical pendulum | 绳与竖直成 θ，物体在水平圆上转String at θ to the vertical, mass on a horizontal circle | 竖直平衡 + 水平向心，tanθ = rω²/gVertical balance + horizontal centripetal: tanθ = rω²/g | 2024 Q1(d) |
| 倾斜弯道 / 骑车倾斜Banked curve / leaning cyclist | "banked"、"only a normal reaction"、"leans over""banked", "only a normal reaction", "leans over" | tanθ = v²/(rg)，再处理几何（h = d sinθ）tanθ = v²/(rg), then geometry (h = d sinθ) | 2021 Q1(l)；同型 2022(c); same type 2022(c) |
| 竖直圆周 / 地球自转视重Vertical circles / apparent weight on a spinning Earth | "top of the loop"、"tension is zero"、"equator""top of the loop", "tension is zero", "equator" | 最高点 mg ≤ mv²/r，能量守恒连两端；视重 = mg − mv²/RAt the top mg ≤ mv²/r, energy links top and bottom; apparent weight = mg − mv²/R | 2017(k)、2016(q)、2019(m)、2015(k)（本模块未单列）2017(k), 2016(q), 2019(m), 2015(k) (not included here) |