Work-Energy Theorem and Its Applications | 动能定理及其应用

📚 Work-Energy Theorem and Its Applications | 动能定理及其应用

The work-energy theorem is one of the most powerful tools in physics. It connects the net work done on an object to its change in kinetic energy, providing a direct bridge between force, displacement, and motion.

动能定理是物理学中最强大的工具之一。它将作用在物体上的合外力所做的功与物体动能的变化直接联系起来，在力、位移和运动之间架起了一座桥梁。

1. Definition of Kinetic Energy | 动能的定义

Kinetic energy is the energy an object possesses due to its motion. For a particle of mass \( m \) moving with speed \( v \), the kinetic energy is given by:

动能是物体由于运动而具有的能量。对于质量为 \( m \)、速度为 \( v \) 的质点，动能表达式为：

Eₖ = ½ m v²

Kinetic energy is a scalar quantity and is always positive or zero. It depends only on the speed, not on the direction of motion.

动能是标量，总为非负值。它只取决于速度的大小，与运动方向无关。

2. Work Done by a Constant Force | 恒力做功

When a constant force \( F \) acts on an object while it undergoes a displacement \( s \), the work done is defined as \( W = F s \cos θ \), where \( θ \) is the angle between the force and the displacement.

当恒力 \( F \) 作用在物体上，且物体发生位移 \( s \) 时，功的定义为 \( W = F s \cos θ \)，其中 \( θ \) 是力与位移之间的夹角。

If the force is in the same direction as the displacement, \( θ = 0° \), so \( W = F s \). If the force is perpendicular to the displacement, \( θ = 90° \), and no work is done.

如果力与位移同向，则 \( θ = 0° \)，此时 \( W = F s \)。如果力与位移垂直，则 \( θ = 90° \)，此时不做功。

3. The Work-Energy Theorem | 动能定理的表述

The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy:

动能定理指出：合外力对物体所做的功等于物体动能的变化量：

W_net = ΔEₖ = Eₖ_final – Eₖ_initial = ½ m v_f² – ½ m v_i²

This theorem holds for both constant and variable forces, as long as we consider the net work done by all forces acting on the object.

该定理对恒力和变力均适用，只要考虑作用在物体上所有力的合功即可。

4. Derivation from Newton’s Second Law | 从牛顿第二定律推导

For a constant net force \( F \) acting on an object of mass \( m \), Newton’s second law gives \( F = m a \). If the object starts from rest at \( s = 0 \) and accelerates over a distance \( s \), the kinematic equation \( v² = u² + 2 a s \) can be rearranged as \( a s = (v² – u²)/2 \).

对于质量 \( m \) 的物体，设合外力 \( F \) 恒定，牛顿第二定律给出 \( F = m a \)。若物体从初速度 \( u \) 经过位移 \( s \) 后速度为 \( v \)，运动学方程 \( v² = u² + 2 a s \) 可变形为 \( a s = (v² – u²)/2 \)。

Multiplying by mass \( m \), we get \( F s = ½ m v² – ½ m u² \), which is exactly the work-energy theorem.

两边同乘质量 \( m \)，得到 \( F s = ½ m v² – ½ m u² \)，这正是动能定理。

5. Work Done by Multiple Forces | 多个力的做功

When several forces act on an object, the net work is the algebraic sum of the work done by each individual force. The work-energy theorem then uses this net work.

当多个力同时作用在物体上时，合功等于各力做功的代数和。此时动能定理中的功应为合功。

Work can be positive, negative, or zero. A force that assists motion does positive work; a force that opposes motion does negative work.

功可以为正、为负或为零。帮助运动的力做正功；阻碍运动的力做负功。
Gravity does positive work when an object falls, and negative work when it rises.

物体下落时重力做正功，上升时重力做负功。
Friction always does negative work, reducing the kinetic energy of the object.

摩擦力总是做负功，减小物体的动能。

6. Application: Braking Distance | 应用：刹车距离

Suppose a car of mass \( m \) is moving at speed \( v \) and then brakes. The friction force \( f \) does negative work over a distance \( d \) until the car stops. Using the work-energy theorem:

设质量为 \( m \) 的汽车以速度 \( v \) 行驶，刹车时摩擦力 \( f \) 在距离 \( d \) 内做负功使车停下。由动能定理：

– f d = 0 – ½ m v² ⇒ d = (m v²) / (2 f)

For a constant friction force, the stopping distance is proportional to the square of the speed. Doubling the speed makes the stopping distance four times as large.

若摩擦力恒定，刹车距离与速度的平方成正比。速度加倍时，刹车距离变为原来的四倍。

7. Application: Projectile Motion | 应用：抛体运动

In projectile motion, the only force doing work is gravity (ignoring air resistance). Since gravity acts vertically, the horizontal component of velocity does not change, and the kinetic energy changes only due to vertical motion.

在抛体运动中，忽略空气阻力时，只有重力做功。因为重力沿竖直方向作用，速度的水平分量不变，动能的变化仅由竖直方向的运动决定。

At the highest point, the vertical velocity is zero, but horizontal velocity remains. The kinetic energy is minimum but not zero. The work-energy theorem can be used to find the speed at any height directly from the change in potential energy.

在最高点，竖直分速度为零，但水平分速度仍然存在，此时动能最小但不为零。利用动能定理可以直接从势能变化求出任意高度处的速度。

8. Application: Variable Forces | 应用：变力做功

For a force that varies with position, such as a spring force \( F = -k x \), the work done cannot be calculated simply as \( F s \). However, the work-energy theorem still applies: the net work equals the change in kinetic energy, regardless of how the force varies.

对于随位置变化的力，如弹簧力 \( F = -k x \)，其做功不能简单用 \( F s \) 计算。但动能定理仍然适用：无论力如何变化，合功总等于动能的变化。

For a spring, the work done by the spring is \( W_s = -½ k x² \), which can be derived by integrating the force over displacement. This work equals the change in kinetic energy of the attached mass.

对于弹簧，弹簧做功为 \( W_s = -½ k x² \)，可以通过对力在位移上积分得到。该功等于所连接物体动能的变化。

9. Relationship with Potential Energy | 与势能的关系

When only conservative forces (like gravity or spring force) do work, the total mechanical energy (kinetic + potential) is conserved. The work-energy theorem then reduces to \( \Delta Eₖ = – \Delta Eₚ \).

当只有保守力（如重力或弹簧力）做功时，机械能（动能加势能）守恒。此时动能定理可写为 \( \Delta Eₖ = – \Delta Eₚ \)。

For non-conservative forces like friction, the work done by friction is converted into thermal energy, and the work-energy theorem becomes \( W_f = \Delta Eₖ + \Delta Eₚ \).

对于摩擦力等非保守力，摩擦力做功转化为热能，动能定理可写为 \( W_f = \Delta Eₖ + \Delta Eₚ \)。

10. Solving Problems Using the Theorem | 用动能定理解题

The theorem is especially useful when forces are not constant or when the path is complicated. It avoids dealing with acceleration and time in many cases.

当力不恒定或路径复杂时，动能定理特别有用。在许多情况下，它避免了处理加速度和时间。

Identify all forces acting on the object and compute the work done by each force.

找出作用在物体上的所有力，并计算每个力所做的功。
Write the initial and final kinetic energy of the object.

写出物体的初动能和末动能。
Set the net work equal to the change in kinetic energy, and solve for the unknown quantity.

令合功等于动能变化，解出未知量。

This approach is particularly powerful for F=ma problems involving curved paths, loops, or variable friction.

这种方法对于涉及曲线路径、环路或变化摩擦的动力学问题尤为强大。

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Work-Energy Theorem and Its Applications | 动能定理及其应用