Rolling · 滚动
A race down a ramp: which wheel wins?
- Roll a solid disk and a hollow ring down the same ramp. They start together — but the disk wins every time.
- A rolling object is doing two things at once: moving forward and spinning.
- Its energy is shared between the two, and the split depends on the shape.
- Understanding that split is the key to rolling motion.
斜坡上的比赛:哪个轮子赢?
- 让实心圆盘和空心圆环滚下同一个斜坡。它们一起出发——但圆盘每次都赢。
- 滚动的物体同时做两件事:向前移动和旋转。
- 它的能量在两者之间分配,分配比例取决于形状。
- 理解这种分配是滚动运动的关键。
Two kinds of kinetic energy at once
- A rolling body has translational KE $\tfrac12 mv^2$ and rotational KE $\tfrac12 I\omega^2$.
- Its total kinetic energy is the sum: $E_k = \tfrac12 mv^2 + \tfrac12 I\omega^2$.
- Rolling without slipping links the two through $v = r\omega$.
- Energy released (say from a ramp) must fill both accounts.
同时有两种动能
- 滚动的物体有平动动能 $\tfrac12 mv^2$ 和****转动动能 $\tfrac12 I\omega^2$。
- 它的总动能是两者之和:$E_k = \tfrac12 mv^2 + \tfrac12 I\omega^2$。
- 无滑滚动通过 $v = r\omega$ 把两者联系起来。
- 释放的能量(比如从斜坡)必须填满两个账户。
The total kinetic energy of a rolling object is: · 滚动物体的总动能为：
A rolling body moves and spins, so its KE is the sum $\tfrac12 mv^2 + \tfrac12 I\omega^2$. · 滚动体既移动又旋转，因此其动能是$\tfrac12 mv^2 + \tfrac12 I\omega^2$之和。
Rolling without slipping links linear and angular speed by $v = r\_\_$ (fill in the symbol). · 纯滚动通过$v = r\_\_$（填入符号）将线速度和角速度联系起来。
$v = r\omega$ ties how fast it travels to how fast it spins. · $v = r\omega$ 将其移动速度与旋转速度联系起来。
Why the solid disk wins
- The ring has more rotational inertia, so more of its energy goes into spinning.
- That leaves less energy for moving, so the ring accelerates down the ramp more slowly.
- The solid disk keeps more energy for translation and reaches the bottom first.
- Shape decides the race, not weight — a heavy and light disk of the same shape tie.
为什么实心圆盘赢
- 圆环的转动惯量更大,所以它更多的能量进入旋转。
- 这样留给移动的能量更少,所以圆环沿斜坡加速更慢。
- 实心圆盘为平动保留更多能量,率先到达底部。
- 决定比赛的是形状,而非重量——同形状的重盘和轻盘打平。
A solid disk and a hollow ring of the same mass and radius roll down a ramp. Which reaches the bottom first? · 一个实心圆盘和一个空心圆环具有相同的质量和半径，沿斜面滚下。哪一个先到达底部？
The ring has more rotational inertia, so more energy goes to spinning, leaving less for moving — the disk wins. · 圆环具有更大的转动惯量，因此更多能量用于旋转，留给平移动的较少——圆盘获胜。
Select all · 所有 true statements about a body rolling without slipping. · 选择所有关于纯滚动体的正确陈述。
Rolling combines moving and spinning, obeys $v = r\omega$, and has a momentarily still contact point — but not all · 所有 rotational. · 滚动结合了移动和旋转，遵循$v = r\omega$，并且有一个瞬时静止的接触点——但不是全部转动动能。
The contact point is momentarily still
- In rolling without slipping, the point touching the ground is instantaneously at rest.
- The centre moves at $v$, and the top of the wheel moves at $2v$.
- Because the contact point doesn't slide, static friction acts — and does no work.
- This is why a rolling wheel loses so little energy compared with a sliding block.
接触点在那一瞬间静止
- 在无滑滚动中,接触地面的点在那一瞬间静止。
- 中心以 $v$ 运动,轮子顶部以 $2v$ 运动。
- 因为接触点不滑动,静摩擦力作用——而且不做功。
- 这就是为什么滚动的轮子比滑动的方块损失的能量少得多。
Rolling without slipping · 纯滚动（无滑动滚动）
For a wheel that rolls without slipping, v = r omega. Sort each case. · 对于纯滚动的轮子，v = r omega。对每种情况进行排序。
For a wheel rolling without slipping, the point touching the ground is momentarily at rest. · 对于纯滚动的轮子，接触地面的点瞬时静止。
The contact point has zero velocity at that instant; the top moves at $2v$. · 接触点在那一瞬间速度为零；顶部以 $2v$ 移动。
A rolling object's energy is not just $\tfrac12 mv^2$. You must add the rotational part $\tfrac12 I\omega^2$. Forgetting the spin term gives too large a speed at the bottom of a ramp — a very common mistake.
滚动物体的能量不只是 $\tfrac12 mv^2$。你必须加上转动部分 $\tfrac12 I\omega^2$。忘掉旋转项会给出斜坡底部过大的速度——一个非常常见的错误。
A solid disk ($I = \tfrac12 mR^2$) rolls without slipping. What fraction of its kinetic energy is rotational? (Give a decimal.) · 一个实心圆盘（$I = \tfrac12 mR^2$）做纯滚动。其动能中有多少比例是转动动能？（给出小数。）
Rotational $= \tfrac14 mv^2$, translational $= \tfrac12 mv^2$; fraction $= \tfrac{1/4}{3/4} = \tfrac13 \approx 0.33$. · 转动$= \tfrac14 mv^2$，平动$= \tfrac12 mv^2$；分数$= \tfrac{1/4}{3/4} = \tfrac13 \approx 0.33$。
A solid disk ($I = \tfrac12 mR^2$) rolls without slipping at speed $v$. What fraction of its kinetic energy is rotational?
- Rotational: $\tfrac12 I\omega^2 = \tfrac12(\tfrac12 mR^2)(v/R)^2 = \tfrac14 mv^2$.
- Translational: $\tfrac12 mv^2$. So rotational is $\tfrac{1/4}{1/4 + 1/2} = \tfrac13$ of the total.
一个实心圆盘($I = \tfrac12 mR^2$)以速率 $v$ 无滑滚动。它动能的多少比例是转动的?
- 转动:$\tfrac12 I\omega^2 = \tfrac12(\tfrac12 mR^2)(v/R)^2 = \tfrac14 mv^2$。
- 平动:$\tfrac12 mv^2$。所以转动占总量的 $\tfrac{1/4}{1/4 + 1/2} = \tfrac13$。
A rolling body has both translational KE $\tfrac12 mv^2$ and rotational KE $\tfrac12 I\omega^2$, linked by $v = r\omega$. Objects with more rotational inertia put more energy into spinning, so they roll down a ramp slower. The contact point is momentarily at rest, so rolling friction does no work.
滚动的物体同时有平动动能 $\tfrac12 mv^2$ 和转动动能 $\tfrac12 I\omega^2$,由 $v = r\omega$ 联系。转动惯量更大的物体把更多能量放进旋转,所以滚下斜坡更慢。接触点在那一瞬间静止,所以滚动摩擦不做功。