Work, Energy and Power | 功、能量与功率

📚 Work, Energy and Power | 功、能量与功率

Work, energy and power are central ideas in mechanics. Work is done when a force moves its point of application, energy is the capacity to do work, and power is the rate at which energy is transferred or work is done. In CIE A-Level Physics, you must be able to calculate work done by constant and variable forces, apply the work-energy principle, use conservation of energy, and solve problems involving power and efficiency.

功、能量和功率是力学的核心概念。当力使其作用点发生位移时，力就做了功；能量是做功的本领；功率是能量转移或做功的速率。在 CIE A-Level 物理中，你必须能够计算恒力和变力所做的功，应用功能原理，使用能量守恒，并解决涉及功率和效率的问题。

1. Work Done by a Constant Force | 恒力做的功

In mechanics, work is done when a force moves its point of application in the direction of the force. For a constant force F acting at an angle θ to a displacement s, the work done W is given by:

在力学中，当力使其作用点沿力的方向发生位移时，力就做了功。若恒力 F 与位移 s 之间的夹角为 θ，则所做的功 W 由下式给出：

W = F s cos θ

Work is a scalar quantity: it has magnitude but no direction. If the force and displacement are in the same direction, θ = 0 and W = F s. If the force is perpendicular to the displacement, θ = 90° and cos θ = 0, so no work is done by that force. For example, the normal contact force on a box sliding along a horizontal floor does no work because its line of action is perpendicular to the motion.

功是标量：只有大小，没有方向。若力与位移同向，θ = 0，则 W = F s。若力垂直于位移，θ = 90°，cos θ = 0，因此该力不做功。例如，木箱沿水平地面滑动时，法向接触力不做功，因为它的作用线垂直于运动方向。

2. Work and the Angle Between Force and Displacement | 功与力和位移的夹角

The sign of work depends on the angle θ. When 0° ≤ θ < 90°, cos θ is positive, so the work done is positive. This means the force transfers energy to the object. When θ = 90°, no work is done. When 90° < θ ≤ 180°, cos θ is negative, so the work done is negative. The force removes energy from the object, often because it opposes the motion.

功的正负取决于夹角 θ。当 0° ≤ θ < 90° 时，cos θ 为正，因此所做的功为正。这意味着该力向物体传递能量。当 θ = 90° 时，不做功。当 90° < θ ≤ 180° 时，cos θ 为负，因此所做的功为负。该力从物体中移走能量，通常是因为它阻碍运动。

Friction acting on a sliding block is a common example of negative work. The friction force is opposite to the displacement, so θ = 180° and cos θ = −1. The work done by friction is therefore −F_f s, where F_f is the magnitude of the friction force and s is the distance moved.

作用在滑动木块上的摩擦力是负功的常见例子。摩擦力与位移方向相反，因此 θ = 180°，cos θ = −1。因此摩擦力所做的功为 −F_f s，其中 F_f 是摩擦力的大小，s 是移动的距离。

3. Units of Work and Energy | 功和能量的单位

The SI unit of work and energy is the joule, symbol J. One joule is defined as the work done when a force of one newton moves its point of application through one metre in the direction of the force.

功和能量的国际单位是焦耳，符号为 J。1 焦耳的定义是：1 牛顿的力使力的作用点沿力的方向移动 1 米时所做的功。

1 J = 1 N m = 1 kg m² s⁻²

Because the joule is made from base units, Work and energy are equivalent quantities. Kinetic energy, gravitational potential energy, elastic potential energy and internal energy are all measured in joules.

由于焦耳由基本单位组成，功和能量是等价的物理量。动能、重力势能、弹性势能和内能都以焦耳为单位。

4. Kinetic Energy | 动能

The kinetic energy of an object of mass m moving with speed v is given by:

质量为 m、速度为 v 的物体的动能由下式给出：

KE = ½ m v²

Kinetic energy depends on mass and on the square of the speed. Doubling the speed while keeping the mass constant increases the kinetic energy by a factor of four. This quadratic relationship is important in road safety: at higher speeds, a vehicle has much more kinetic energy, so braking distances increase dramatically.

动能取决于质量和速度的平方。在质量不变的情况下，速度加倍会使动能增大到原来的四倍。这种平方关系在道路安全中非常重要：在较高速度下，车辆具有大得多的动能，因此制动距离会急剧增加。

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

功能原理指出，作用在物体上的净功等于其动能的变化：

Net work = ΔKE = ½ m v² − ½ m u²

Here u is the initial speed and v is the final speed. This principle is particularly useful when a force accelerates or decelerates an object along a straight line.

其中 u 是初速度，v 是末速度。当力沿直线使物体加速或减速时，这一原理特别有用。

5. Gravitational Potential Energy | 重力势能

Near the Earth’s surface, the change in gravitational potential energy of an object of mass m raised through a vertical height Δh is:

在地球表面附近，质量为 m 的物体升高竖直高度 Δh 时，重力势能的变化为：

ΔEₚ = m g Δh

The value of gravitational potential energy depends on the chosen reference level, but changes in gravitational potential energy do not. The symbol g is the gravitational field strength, which is approximately 9.81 N kg⁻¹ on Earth. If an object moves downwards, Δh is negative, so the change in gravitational potential energy is negative and kinetic energy may increase.

重力势能的值取决于所选参考面，但重力势能的变化与参考面无关。符号 g 是重力场强度，在地球上约为 9.81 N kg⁻¹。如果物体向下运动，Δh 为负，因此重力势能的变化为负，动能可能增加。

6. Conservation of Mechanical Energy | 机械能守恒

If the only forces doing work on an object are conservative forces such as gravity, the total mechanical energy remains constant. Mechanical energy is the sum of kinetic energy and gravitational potential energy. Energy may transfer between kinetic energy and gravitational potential energy, but the sum does not change.

如果对物体做功的力只有重力等保守力，则总机械能保持不变。机械能是动能和重力势能的总和。能量可以在动能和重力势能之间转化，但总和不变。

KE₁ + Eₚ₁ = KE₂ + Eₚ₂

For example, a pendulum bob exchanges kinetic energy and gravitational potential energy during each swing. At the lowest point, speed is greatest and gravitational potential energy is least. At the highest point, speed is momentarily zero and gravitational potential energy is greatest. In the absence of air resistance, the bob would return to the same height on the opposite side.

例如，单摆摆锤在每次摆动中交换动能和重力势能。在最低点，速度最大，重力势能最小。在最高点，速度瞬时为零，重力势能最大。如果没有空气阻力，摆锤将回到另一侧相同的高度。

7. Work Done Against Friction and Energy Dissipation | 克服摩擦做功与能量耗散

When friction or air resistance acts, mechanical energy is no longer conserved. Work done against friction transfers energy from the mechanical store to the internal energy store of the surfaces and the surroundings, raising their temperature. The work done against friction is the product of the friction force and the distance moved along the surface:

当存在摩擦或空气阻力时，机械能不再守恒。克服摩擦做功会将能量从机械能储存转移到表面和周围环境的内能储存中，使温度升高。克服摩擦所做的功等于摩擦力与沿表面移动距离的乘积：

W_friction = F_f s

Total energy is always conserved, but the useful mechanical energy decreases. This is why sliding objects slow down and eventually stop. The dissipated energy is not lost; it is spread out as internal energy in the system and surroundings.

总能量始终守恒，但有用的机械能会减少。这就是滑动物体逐渐减速并最终停下来的原因。耗散的能量并没有消失；它以系统及周围环境的内能形式散开。

8. Power | 功率

Power is the rate of doing work or transferring energy. Average power is given by:

功率是做功或转移能量的速率。平均功率由下式给出：

P = W / t = ΔE / t

The SI unit of power is the watt, symbol W. One watt is equal to one joule per second: 1 W = 1 J s⁻¹. A powerful machine transfers the same amount of energy in a shorter time, or transfers more energy in the same time.

功率的国际单位是瓦特，符号为 W。1 瓦特等于 1 焦耳每秒：1 W = 1 J s⁻¹。功率大的机器能在更短时间内转移相同的能量，或在相同时间内转移更多的能量。

9. Power as Force × Velocity | 功率与力和速度的关系

For a constant force F acting on an object moving with constant velocity v in the direction of the force, the instantaneous power output is:

若恒力 F 作用在沿力的方向以恒定速度 v 运动的物体上，则瞬时功率输出为：

P = F v

If the force and velocity are at an angle θ, the power is P = F v cos θ. This relation is especially useful for cars, trains and aircraft, where an engine provides thrust against resistive forces at a given speed. At constant power, a vehicle engine produces less driving force at higher speed, which is why acceleration falls as speed increases.

如果力与速度之间的夹角为 θ，则功率为 P = F v cos θ。这一关系对汽车、火车和飞机尤其有用，因为发动机在给定速度下提供推力以克服阻力。在功率恒定时，车辆发动机在较高速度下产生的驱动力较小，这就是为什么随着速度增大，加速度会减小。

10. Efficiency | 效率

Efficiency measures how much of the input energy or power is converted into useful output. It is usually expressed as a percentage:

效率衡量输入能量或功率有多少转化为有用的输出。它通常用百分比表示：

Efficiency = (useful output energy / input energy) × 100% = (useful output power / input power) × 100%

Real machines always have an efficiency less than 100% because some energy is transferred to internal energy through friction, sound or heat. For example, an electric motor with an input power of 500 W and a useful mechanical output of 400 W has an efficiency of 80%.

真实机器的效率总是小于 100%，因为有些能量通过摩擦、声音或热传递转化为内能。例如，一台输入功率为 500 W、有用机械输出为 400 W 的电动机，其效率为 80%。

11. Work Done by a Variable Force | 变力做的功

When the force is not constant, the work done cannot be found directly from W = F s. Instead, the work done is the area under a force–distance graph. If the force varies with displacement x, then:

当力不是恒力时，不能直接用 W = F s 计算功。此时，功等于力—距离图像下方的面积。如果力随位移 x 变化，则：

W = area under F–x graph

For a spring obeying Hooke’s law, the force is F = kx, where k is the spring constant. The work done in stretching the spring from extension 0 to extension e is the area of a triangle under the F–x graph:

对于满足胡克定律的弹簧，力为 F = kx，其中 k 是弹簧劲度系数。将弹簧从伸长量 0 拉伸到伸长量 e 所做的功等于 F–x 图像下方三角形的面积：

W = ½ k e²

This amount of work is stored as elastic potential energy in the spring. It can be released when the spring returns to its original length.

这部分功以弹性势能的形式储存在弹簧中。当弹簧恢复原

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Work, Energy and Power | 功、能量与功率