Newton's Laws of Motion | 牛顿运动定律

📚 Newton’s Laws of Motion | 牛顿运动定律

Newton’s laws of motion form the foundation of classical mechanics and are essential for AQA A-Level Physics. They describe how forces affect the motion of objects, linking acceleration, mass, and momentum.

牛顿运动定律是经典力学的基础，也是 AQA A-Level 物理的核心内容。它们描述了力如何影响物体的运动，将加速度、质量和动量联系起来。

1. Newton’s First Law | 牛顿第一定律

Newton’s first law states that an object remains at rest or moves with constant velocity unless acted upon by a resultant force. This is also known as the law of inertia.

牛顿第一定律指出：若物体不受合力作用，则保持静止或匀速直线运动状态。这也被称为惯性定律。

Inertia is a measure of an object’s resistance to changes in its motion. It depends directly on mass: the greater the mass, the greater the inertia.

惯性是物体抵抗运动状态变化的量度，它直接取决于质量：质量越大，惯性越大。

ΣF = 0 ⇔ v = constant

2. Newton’s Second Law | 牛顿第二定律

Newton’s second law relates the resultant force acting on an object to its mass and acceleration. In equation form, F = ma, where F is the resultant force in newtons, m is the mass in kilograms, and a is the acceleration in metres per second squared.

牛顿第二定律将物体所受合力与其质量和加速度联系起来。公式为 F = ma，其中 F 是合力（单位：N），m 是质量（单位：kg），a 是加速度（单位：m s⁻²）。

F = ma

It is important to note that F represents the vector sum of all external forces. The acceleration is always in the same direction as the resultant force.

需要注意，F 表示所有外力的矢量之和。加速度方向始终与合力方向相同。

For a variable force, the instantaneous acceleration is given by a = F/m at that instant. This is useful when analysing oscillating or non-uniform motion.

对于变力，瞬时加速度由 a = F/m 给出。这在分析振动或非匀变速运动时非常有用。

3. Momentum and Impulse | 动量与冲量

Momentum is defined as the product of mass and velocity. It is a vector quantity with units of kg m s⁻¹. The momentum p of an object is given by p = mv.

动量定义为质量与速度的乘积，是矢量，单位为 kg m s⁻¹。物体动量 p 由 p = mv 给出。

p = mv

Newton’s second law can be expressed in terms of momentum: the resultant force is equal to the rate of change of momentum. This is the original form stated by Newton.

牛顿第二定律可用动量表达：合力等于动量变化率。这是牛顿最初表述的形式。

F = Δp / Δt

Impulse is the product of force and the time interval over which it acts. Impulse equals the change in momentum, which is the impulse-momentum theorem.

冲量是力与其作用时间间隔的乘积。冲量等于动量变化，即冲量-动量定理。

Impulse = F Δt = Δp

4. Conservation of Linear Momentum | 动量守恒定律

In a closed system where no external resultant force acts, the total linear momentum remains constant. This is the principle of conservation of linear momentum.

在不受外力影响的封闭系统中，总动量保持不变。这就是动量守恒定律。

For two objects A and B colliding, the total momentum before collision equals the total momentum after collision:

对于两个物体 A 和 B 碰撞，碰撞前总动量等于碰撞后总动量：

m_A u_A + m_B u_B = m_A v_A + m_B v_B

This principle is particularly useful for analysing collisions and explosions. In an explosion, the initial momentum is zero, so the final momenta of the fragments must sum to zero.

该定律特别适用于分析碰撞和爆炸。在爆炸中，初始动量为零，因此碎片末动量之和必为零。

5. Elastic and Inelastic Collisions | 弹性碰撞与非弹性碰撞

Collisions are classified as elastic or inelastic based on whether kinetic energy is conserved.

碰撞根据动能是否守恒分为弹性碰撞和非弹性碰撞。

In an elastic collision, both momentum and kinetic energy are conserved. No energy is dissipated as heat, sound, or deformation.

在弹性碰撞中，动量和动能均守恒，没有能量以热、声或形变的形式耗散。

In an inelastic collision, momentum is conserved but kinetic energy is not. Some initial kinetic energy is transformed into other forms.

在非弹性碰撞中，动量守恒但动能不守恒，部分初始动能转化为其他形式。

A perfectly inelastic collision is one where the objects stick together after impact. This results in the maximum possible loss of kinetic energy consistent with momentum conservation.

完全非弹性碰撞是指物体碰撞后粘在一起，这会在动量守恒的前提下损失最大的动能。

When calculating the kinetic energy before and after, always use the speeds, not velocities, because kinetic energy is a scalar.

计算碰撞前后动能时，一定要用速率而非速度，因为动能是标量。

6. Newton’s Third Law | 牛顿第三定律

Newton’s third law states that if object A exerts a force on object B, then object B exerts an equal and opposite force on object A. These forces are equal in magnitude, opposite in direction, and act on different objects.

牛顿第三定律指出：如果物体 A 对物体 B 施力，则物体 B 同时对物体 A 施以等大反向的力。这对力大小相等、方向相反，且作用在不同物体上。

F_AB = −F_BA

It is essential to remember that the action-reaction pair acts on different bodies, so they do not cancel each other out when considering the motion of a single object.

必须记住，作用力与反作用力作用在不同物体上，因此在分析单个物体运动时它们不能相互抵消。

Common examples include a rocket pushing gases backward while the gases push the rocket forward, and a person walking pushing the ground backward while the ground pushes the person forward.

常见实例包括：火箭向后喷射气体，气体同时向前推动火箭；人走路时向后蹬地，地同时向前推人。

7. Applications of Newton’s Laws | 牛顿定律的应用

Newton’s laws are applied in many real-world situations and exam questions. One key area is the analysis of systems involving connected bodies, such as two masses joined by a string over a pulley.

牛顿定律在许多真实情境和考试题目中都有应用。一个关键领域是连接体系统分析，例如通过定滑轮用绳子连接的两个物体。

For a system of connected objects moving together, treat the whole system as a single mass to find the acceleration, then analyse individual objects to find internal forces.

对于一起运动的连接体系统，先把整个系统视为一个整体求加速度，再分析单个物体求内力。

Another important application is the concept of apparent weight in a lift. When a lift accelerates upward, the normal reaction increases; when it accelerates downward, the reaction decreases.

另一个重要应用是电梯中的视重。当电梯向上加速时，支持力增大；向下加速时，支持力减小。

N = m(g ± a)

Here, N is the normal force, g is gravitational field strength (9.81 N kg⁻¹), and a is the lift’s acceleration. Take the plus sign for upward acceleration and the minus sign for downward acceleration.

其中 N 是支持力，g 是重力场强度（9.81 N kg⁻¹），a 是电梯加速度。向上加速取加号，向下加速取减号。

8. Free-Body Diagrams and Resolving Forces | 受力分析图与力的分解

Solving any Newton’s laws problem begins with a clear free-body diagram showing all forces acting on the object. Forces are then resolved into components along chosen axes.

解决任何牛顿定律问题，都要先画出清晰的受力分析图，标出物体所受的所有力，然后沿选定的坐标轴分解力。

For an object on a slope, the weight mg can be resolved into components parallel and perpendicular to the slope:

对于斜面上的物体，重力 mg 可分解为平行于斜面和垂直于斜面的两个分量：

Parallel component: mg sin θ

Perpendicular component: mg cos θ

If the surface is smooth, the perpendicular component is balanced by the normal reaction. The parallel component, if unbalanced, causes acceleration down the slope.

若表面光滑，垂直分量被支持力平衡；平行分量若不平衡，则使物体沿斜面加速下滑。

Always choose axes that simplify the calculations, often along the direction of motion and perpendicular to it.

要选择能简化计算的坐标轴，通常沿运动方向和垂直运动方向。

9. Terminal Velocity and Drag | 收尾速度与阻力

When an object moves through a fluid, it experiences a drag force that opposes motion. The drag force generally increases with speed.

物体在流体中运动时会受到阻碍运动的阻力，阻力通常随速度增大而增大。

For a falling object, two forces act: weight downward and drag upward. As speed increases, drag increases, so the resultant downward force decreases.

对于下落物体，受两个力：向下的重力和向上的阻力。随着速度增大，阻力增大，因此向下合力减小。

Eventually, the drag force equals the weight, and the resultant force becomes zero. The object then falls with constant velocity, known as terminal velocity.

最终阻力等于重力，合力为零，物体以恒定速度下落，该速度称为收尾速度。

mg = D(v) → v = v_terminal

At terminal velocity, acceleration is zero, and the object continues moving at constant velocity. The value of terminal velocity depends on the object’s shape, mass, and the fluid’s viscosity.

在收尾速度时，加速度为零，物体保持匀速运动。收尾速度的大小取决于物体的形状、质量以及流体的黏度。

10. Centre of Mass and Stability | 质心与稳定性

The centre of mass of an object is the point where the entire mass of the object can be considered to act. For a uniform symmetric object, it lies at the geometric centre.

物体的质心是可将整个物体质量视为集中作用的点。对于均匀对称的物体，质心位于几何中心。

The position of the centre of mass is important for analysing equilibrium. An object topples when the line of action of its weight falls outside its base of support.

质心位置对分析平衡非常重要。当重力作用线落在支撑面之外时，物体就会倾倒。

In AQA exams, you may need to calculate the centre of mass of a system of particles using the principle of moments:

在 AQA 考试中，你可能需要用力矩原理计算质点系的质心：

x̄ = Σ(m_i x_i) / Σm_i

For continuous shapes, integration may be required, but in A-Level, two-dimensional laminas are often treated using symmetry.

对于连续形状可能需用积分，但在 A-Level 中，二维薄板常利用对称性处理。

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Newton’s Laws of Motion | 牛顿运动定律