Conservation of Energy · 能量守恒
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| mechanical energy/mɪˈkænɪkl ˈenədʒi/ | 机械能 | jī xiè néng |
| nonconservative forces/ˌnɒŋkənˈsɜːvətɪv ˈfɔːsɪz/ | 非保守力 | fēi bǎo shǒu lì |
| thermal energy/ˈθɜːml ˈenədʒi/ | 热能 | rè néng |
A pendulum trades height for speed
- A swinging pendulum is fastest at the bottom and momentarily stops at the top of each swing.
- It is constantly trading height for speed and back again.
- Add up its motion energy and its height energy, and the total never changes.
- That unshakable total is one of physics' great bookkeeping laws.
钟摆用高度换速度
- 摆动的钟摆在底部最快,在每次摆动的顶端瞬间停下。
- 它不断地用高度换速度,再换回来。
- 把它的运动能量和高度能量加起来,总和永远不变。
- 那个不可动摇的总和,是物理学伟大的记账定律之一。
Conservation of mechanical energy
- With only conservative forces, mechanical energy 机械能 (kinetic + potential) is conserved:
- Energy shifts between kinetic and potential, but the sum stays fixed.
- Solve motion problems without ever finding the force.
机械能守恒
- 在只有保守力时,机械能(动能 + 势能)守恒:
- 能量在动能和势能之间转移,但总和保持不变。
- 无需求出力就能解运动问题。
A ball is dropped from $2\ \text{m}$ (no air resistance, $g = 10$). Its speed at the bottom (in m/s)? · 一个球从高度 $2\ \text{m}$ 处落下（无空气阻力，$g = 10$）。到达底部时的速度是多少（单位：m/s）？
$mgh = \tfrac{1}{2}mv^2$, so $v = \sqrt{2gh} = \sqrt{2 \times 10 \times 2} = \sqrt{40} \approx 6.32\ \tfrac{\text{m}}{\text{s}}$. · $mgh = \tfrac{1}{2}mv^2$，所以 $v = \sqrt{2gh} = \sqrt{2 \times 10 \times 2} = \sqrt{40} \approx 6.32\ \tfrac{\text{m}}{\text{s}}$。
With only conservative forces, the sum of kinetic and potential energy (the ____ energy) is conserved. · 仅受保守力作用时，动能和势能之和（即 ____ 能量）是守恒的。
$K + U$ is the mechanical energy, conserved when no friction or drag acts. · $K + U$ 是机械能，在无摩擦或阻力作用时守恒。
A frictionless pendulum is at its lowest point. There, its energy is... · 一个无摩擦的摆锤处于最低点。在那里，它的能量是...
At the bottom, height (and $U$) is lowest and speed is highest -- the energy is all kinetic. · 在底部，高度（和 $U$）最低且速度最高——能量全部为动能。
Conservative versus not
- Conservative forces (gravity, springs) conserve mechanical energy.
- Nonconservative forces 非保守力 (friction, drag) drain it away.
- The difference decides whether $K + U$ stays constant.
保守力与非保守力
- 保守力(重力、弹簧)使机械能守恒。
- 非保守力(摩擦力、拽力)把它耗散掉。
- 这个区别决定了 $K + U$ 是否保持不变。
When friction acts, the mechanical energy $K + U$ stays constant. · 当摩擦力作用时，机械能 $K + U$ 保持不变。
Friction is nonconservative -- it drains mechanical energy into heat, so $K + U$ decreases. · 摩擦力是非保守力——它将机械能耗散为热量，因此 $K + U$ 减小。
Select all · 所有 forces that conserve mechanical energy. · 选择所有能维持机械能守恒的力。
Gravity and springs are conservative. Kinetic friction is nonconservative -- it dissipates energy. · 重力和弹簧力是保守力。动摩擦力是非保守力——它会耗散能量。
Where the energy goes
- Friction converts mechanical energy into thermal energy 热能 (heat).
- Energy is still conserved overall -- it just leaves the mechanical account.
- The work done by friction equals the mechanical energy lost.
能量去哪了
- 摩擦力把机械能转化为热能(热)。
- 能量总体上仍然守恒——它只是离开了机械能账户。
- 摩擦力所做的功等于损失的机械能。
Conservation of energy · 能量守恒
As an object rises and falls, energy shifts between kinetic and potential while the total stays constant. · 当物体上升和下降时，能量在动能和势能之间转换，而总能量保持不变。
A block slides to a stop on a rough floor. Where did its kinetic energy go? · 一个滑块在粗糙地板上滑行直至停止。它的动能去哪了？
Friction converted the kinetic energy to heat. Total energy is still conserved; it just left the mechanical account. · 摩擦力将动能转化为热量。总能量仍然守恒；只是离开了机械账户。
Energy accounting
- Track energy with bar charts or the equation $K_i + U_i = K_f + U_f\ (+\ \text{heat})$.
- List every form before and after; they must balance.
- This makes even messy multi-step motions solvable.
能量记账
- 用条形图或方程 $K_i + U_i = K_f + U_f$(有摩擦时再加一个热项)来追踪能量。
- 列出前后的每一种形式;它们必须平衡。
- 这让即使是乱糟糟的多步运动也能求解。
A $1\ \text{kg}$ ball is dropped from $5\ \text{m}$ (no air resistance).
- All its $U_g = mgh = 1 \times 9.8 \times 5 = 49\ \text{J}$ becomes kinetic energy at the bottom.
- So $\tfrac{1}{2}mv^2 = 49\ \text{J}$ gives $v = \sqrt{98} \approx 9.9\ \tfrac{\text{m}}{\text{s}}$.
一个 $1\ \text{kg}$ 的球从 $5\ \text{m}$ 处落下(无空气阻力)。
- 它全部的 $U_g = mgh = 1 \times 9.8 \times 5 = 49\ \text{J}$ 在底部变成动能。
- 所以 $\tfrac{1}{2}mv^2 = 49\ \text{J}$ 给出 $v = \sqrt{98} \approx 9.9\ \tfrac{\text{m}}{\text{s}}$。
"Energy is conserved" always holds -- but mechanical energy is conserved only without friction or drag. When friction acts, do not set $K_i + U_i = K_f + U_f$; you must add the heat term, or you will over-count the final speed.
"能量守恒"永远成立——但只有在没有摩擦或拽力时机械能才守恒。当有摩擦时,不要令 $K_i + U_i = K_f + U_f$;你必须加上热项,否则会把末速度算多。
With only conservative forces, mechanical energy is conserved: $K_i + U_i = K_f + U_f$. Nonconservative forces like friction convert it to thermal energy -- total energy is still conserved, but you must track the heat. Energy bookkeeping solves motion without forces.
在只有保守力时,机械能守恒:$K_i + U_i = K_f + U_f$。像摩擦这样的非保守力会把它转化为热能——总能量仍然守恒,但你必须追踪热。能量记账无需力就能解运动。