Motion of Orbiting Satellites · 轨道卫星的运动
The Moon is falling — and always missing
- The Moon is pulled toward Earth by gravity, so it is constantly "falling".
- Yet it never lands — because it also moves sideways fast enough to keep missing.
- An orbit is exactly this: falling toward a planet while sailing past it forever.
- Gravity is not fighting the orbit; gravity is what bends the path into a circle.
月球在坠落——却永远擦身而过
- 月球被引力拉向地球,所以它在不断"坠落"。
- 可它从不落地——因为它同时横向运动得足够快,能一直擦过去。
- 轨道正是如此:朝行星坠落,同时永远从它旁边掠过。
- 引力不是在对抗轨道;引力正是把路径弯成圆的原因。
Gravity is the centripetal force
- For a circular orbit, gravity supplies the whole centripetal force.
- Set them equal: $\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$.
- The satellite's mass $m$ cancels — orbits don't care how heavy the satellite is.
- This single equation controls every circular orbit.
引力就是向心力
- 对圆轨道,引力提供全部向心力。
- 令它们相等:$\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$。
- 卫星的质量 $m$ 消掉了——轨道不在乎卫星有多重。
- 这一个方程控制着每一个圆轨道。
Trace the orbit · 追踪轨道
Change the orbit radius and see how the geometry of the circular path changes. · 改变轨道半径并观察圆形路径几何形状的变化。
What provides the centripetal force that keeps a satellite in a circular orbit? · 什么提供了使卫星保持在圆形轨道中的向心力？
Gravity is the centripetal force: $\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$. · 重力是向心力：$\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$。
In $\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$, the satellite's ____ cancels from both sides. · 在$\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$中，卫星的____在两边抵消。
The satellite mass $m$ cancels, so orbital speed does not depend on it. · 卫星质量$m$抵消，因此轨道速度不依赖于它。
Orbital speed and radius
- Solving for speed: $v = \sqrt{\dfrac{GM}{r}}$.
- A higher orbit (larger $r$) means a slower speed — counter-intuitive but true.
- Low satellites (like the ISS) whip around in about $90$ minutes; the far-off Moon takes a month.
- Every orbit radius has exactly one speed that keeps it circular.
轨道速率与半径
- 解出速率:$v = \sqrt{\dfrac{GM}{r}}$。
- 更高的轨道(更大的 $r$)意味着更慢的速率——违反直觉但正确。
- 低轨卫星(如国际空间站)约 $90$ 分钟绕一圈;遥远的月球要一个月。
- 每个轨道半径都恰好对应一个使其保持圆形的速率。
At an orbit where $GM/r = 25$ (scaled units), what is the orbital speed $v = \sqrt{GM/r}$? · 在$GM/r = 25$（缩放单位）的轨道处，轨道速度$v = \sqrt{GM/r}$是多少？
$v = \sqrt{GM/r} = \sqrt{25} = 5$.
A satellite is moved to a higher · 更高 circular orbit. Its orbital speed: · 卫星被移至更高的圆形轨道。其轨道速度：
$v = \sqrt{GM/r}$: a larger $r$ gives a smaller $v$ — higher orbits are slower. · $v = \sqrt{GM/r}$：更大的 $r$ 导致更小的 $v$ ——更高的轨道速度更慢。
Falling forever
- A satellite is in constant free fall — its only force is gravity, pulling it inward.
- It feels "weightless" not because gravity is gone, but because it is falling with everything around it.
- Give it more sideways speed than the orbital value and it climbs to a higher orbit.
- Too little, and it spirals back down — orbits are a delicate balance of speed and pull.
永远地坠落
- 卫星处于持续的自由落体中——它唯一的力是引力,把它向内拉。
- 它感到"失重"不是因为引力消失了,而是因为它正与周围的一切一起坠落。
- 给它比轨道值更大的横向速率,它就爬升到更高的轨道。
- 太小了,它就螺旋落回——轨道是速率与拉力之间的微妙平衡。
A satellite in orbit is in continuous free fall, which is why astronauts feel weightless. · 轨道上的卫星处于持续自由落体状态，这就是宇航员感觉失重的原因。
They fall together with everything around them, so they float — gravity is still present. · 他们与周围的一切一起下落，所以漂浮——重力依然存在。
Select all · 所有 true statements about a satellite in a circular orbit. · 选择所有关于圆形轨道卫星的正确陈述。
Gravity is the centripetal force, the satellite free-falls, and higher orbits are slower. Gravity is very much present. · 重力是向心力，卫星自由落体，且高轨道速度较慢。重力确实存在。
A satellite in orbit is not beyond gravity — gravity is the very force holding it in orbit. Astronauts float because they are in continuous free fall, not because there is "no gravity" up there. At the ISS, gravity is still nearly as strong as on the ground.
在轨的卫星并非超出了引力——引力正是把它维持在轨道上的那个力。宇航员漂浮是因为他们处于持续的自由落体,而不是因为那里"没有引力"。在国际空间站,引力仍然几乎与地面上一样强。
At a certain orbit, $\dfrac{GM}{r} = 25\ \tfrac{\text{m}^2}{\text{s}^2}$ (in scaled units).
- Orbital speed $v = \sqrt{\dfrac{GM}{r}} = \sqrt{25} = 5$ (speed units).
Move to a higher orbit where $GM/r = 16$, and the speed drops to $\sqrt{16} = 4$ — farther out is slower.
在某个轨道上,$\dfrac{GM}{r} = 25\ \tfrac{\text{m}^2}{\text{s}^2}$(用缩放单位)。
- 轨道速率 $v = \sqrt{\dfrac{GM}{r}} = \sqrt{25} = 5$(速率单位)。
移到 $GM/r = 16$ 的更高轨道,速率降到 $\sqrt{16} = 4$——越远越慢。
For a circular orbit, gravity is the centripetal force: $\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$, giving orbital speed $v = \sqrt{GM/r}$. Higher orbits are slower. A satellite is in constant free fall — "weightlessness" is falling, not the absence of gravity.
对圆轨道,引力就是向心力:$\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$,给出轨道速率 $v = \sqrt{GM/r}$。更高的轨道更慢。卫星处于持续的自由落体——"失重"是坠落,而非没有引力。