转动系统的能量与动量

A spinning body has rotational kinetic energy 转动动能

the rotational twin of $\tfrac12mv^2$, with the rotational inertia 转动惯量 $I$ playing the role of mass. A body that both moves and spins carries both terms:

Worked example. A uniform cylinder ($I=\tfrac12mr^2$) rolls at speed $v$. Its kinetic energy is $K=\tfrac12mv^2+\tfrac12\big(\tfrac12mr^2\big)\big(\tfrac{v}{r}\big)^2=\tfrac34mv^2$ – one third of it is rotational. The same ball of energy bookkeeping decides every rolling problem.

A torque 力矩 acting through an angular displacement 角位移 does work, and power is torque times angular velocity:

These are the rotational forms of $W=\int F\,dx$ and $P=Fv$ – the whole translational energy toolkit carries over with $F\to\tau$, $x\to\theta$, $v\to\omega$.

Worked example. A motor applies a constant $8.0\ \text{N}\cdot\text{m}$ torque to a flywheel for $5.0$ full turns: $W=\tau\,\Delta\theta=8.0(5.0)(2\pi)=250\ \text{J}$, which appears as rotational kinetic energy if friction is negligible.

Angular momentum 角动量 for a particle is

so even a particle moving in a straight line has angular momentum about any point not on that line ($L=mv\,d$, with $d$ the perpendicular distance). For a rigid body spinning about a fixed axis, $L=I\omega$. Newton's second law in rotational form is

and a net torque acting over time delivers an angular impulse 角冲量 $\int\tau\,dt=\Delta L$ – the rotational impulse–momentum theorem.

With zero net external torque, total angular momentum is conserved 守恒:

If $I$ shrinks, $\omega$ grows – a spinning skater speeds up pulling her arms in. The rule survives collisions and shape changes, which is what makes it so useful: pick the axis so that every external force (gravity, the pivot force) exerts no torque about it.

Worked example. A skater spins at $2.0\ \text{rev/s}$ with $I_1=4.0\ \text{kg}\cdot\text{m}^2$. Pulling in her arms drops it to $I_2=1.6\ \text{kg}\cdot\text{m}^2$: $\omega_2=\dfrac{4.0}{1.6}(2.0)=5.0\ \text{rev/s}$. Her kinetic energy rises – the extra energy is the work her muscles do pulling her arms inward.

Worked example (rotational collision). A $0.020\ \text{kg}$ bullet at $300\ \text{m/s}$ strikes the tip of a uniform rod ($M=1.5\ \text{kg}$, length $l=0.60\ \text{m}$) hanging from a pivot, and embeds. About the pivot, gravity and the pivot force exert no torque during the strike, so $L$ is conserved: $L=mvl=0.020(300)(0.60)=3.6\ \text{kg}\cdot\text{m}^2/\text{s}$. Afterwards $I=\tfrac13Ml^2+ml^2=0.18+0.0072=0.187\ \text{kg}\cdot\text{m}^2$, so $\omega=\dfrac{3.6}{0.187}\approx19\ \text{rad/s}$. (Linear momentum is not conserved here – the pivot pushes on the rod – and kinetic energy certainly is not: check both before claiming them.)

Rolling without slipping 无滑滚动 ties translation to rotation: the contact point is momentarily at rest, so

Static friction supplies the torque that keeps the spin matched to the motion, but at a point that is not sliding – so for pure rolling, friction does no work, and energy conservation is safe to use. (Rolling friction is beyond the AP course.)

Racing shapes down an incline shows the energy split. With $I=\beta mr^2$, energy conservation gives

a sphere ($\beta=\tfrac25$) beats a disk 圆盘 ($\beta=\tfrac12$), which beats a hoop 圆环 ($\beta=1$) – mass and radius cancel completely. More of the hoop's energy is locked in rotation, so its center moves slower.

Worked example. A solid sphere rolls from rest down a $30^\circ$ incline: $a=\dfrac{g\sin30^\circ}{1+\tfrac25}=\dfrac{9.8(0.50)}{1.4}=3.5\ \text{m/s}^2$, versus $4.9\ \text{m/s}^2$ for a frictionless slider – rolling objects always lose the race against sliding ones.

Gravity supplies the centripetal force 向心力 for a satellite 卫星 in a circular orbit:

– larger orbits are slower. With gravitational potential energy 引力势能 $U_g=-\dfrac{GMm}{r}$, a circular orbit obeys $K=-\tfrac12U$, so the total mechanical energy 总机械能 is

negative because the satellite is bound. To escape from radius $r$, total energy must reach zero, giving the escape velocity 逃逸速度

In an elliptical orbit 椭圆轨道, $E$ and $L$ are fixed: gravity points at the focus, so it exerts no torque about it. Conserved $L$ means the satellite sweeps equal areas in equal times – Kepler's second law 开普勒第二定律 – and therefore moves fastest at closest approach, slowest at the far point. Energy conservation connects speeds at the two ends.

Worked example. For a satellite at $r=7.0\times10^{6}\ \text{m}$ around Earth ($GM=4.0\times10^{14}\ \text{m}^3/\text{s}^2$): orbital speed $v=\sqrt{GM/r}=7.6\times10^{3}\ \text{m/s}$, while escaping from that radius needs $v_{\text{esc}}=\sqrt{2GM/r}=1.1\times10^{4}\ \text{m/s}$ – exactly $\sqrt2$ times the circular speed.

Exam skill. Orbit FRQs are energy-and-angular-momentum problems in disguise: write $E=\tfrac12mv^2-\dfrac{GMm}{r}$ and $L=mvr\sin\theta$ at the two points of interest and solve the pair – never assume the orbit formulae for circles apply to an ellipse.

• Compute the moment of inertia $I=\int r^2\,dm$ and shift axes with the parallel-axis theorem $I=I_{cm}+Md^2$.
• Rotational kinetic energy is $\tfrac12 I\omega^2$; a rolling body has both translational and rotational KE.
• Conserve angular momentum $L=I\omega$ when net external torque is zero (a spinning skater pulling in).
• Use energy conservation for rolling-without-slipping problems ($v=r\omega$ ties the two motions).
• Know standard $I$ values (hoop, disk, rod, sphere) and where the axis is.

Common mistakes

• Conserving linear momentum when a rod is hit while hanging from a pivot. The pivot pushes during the collision; conserve angular momentum about the pivot instead.
• Leaving out rotational kinetic energy for a rolling object. $K = \tfrac12 Mv^2 + \tfrac12 I\omega^2$ with $\omega = v/R$.
• Saying static friction on a rolling wheel wastes energy. With no slipping, the contact point is at rest, so static friction does no work.
• Using circular-orbit formulas for an ellipse. Use conservation of energy and of angular momentum between the two points.
• Forgetting that a skater who pulls in her arms gains kinetic energy: $L$ is conserved, not $K$.