Classical mechanics - Kinematics and Newtonian Dynamics

Description of Objects and Their Motion in Newtonian Mechanics Introduction Classical mechanics describes how objects move and change due to forces acting on them. To do this rigorously, we need precise definitions of motion (kinematics) and the relationship between forces and motion (dynamics). This chapter establishes the foundational concepts and frameworks that allow us to predict and analyze the behavior of physical systems, from falling objects to orbiting planets. What We Mean by "Objects": Point Particles and Extended Bodies In classical mechanics, we simplify real objects in a clever way. Rather than tracking every atom in a complex object, we often treat objects as point particles—objects with mass but negligible physical size, concentrated at a single location in space. This seems unrealistic at first. A baseball clearly has size! However, this approximation works remarkably well when the object's size is small compared to the distances involved in the problem. For instance, when analyzing Earth's orbit around the Sun, treating Earth as a point particle is perfectly reasonable because Earth's radius is tiny compared to the Sun-Earth distance. When we cannot ignore an object's size and shape, we treat it as an extended object—a collection of many point particles rigidly connected together. A key insight is that even for extended objects, we can predict the motion of the system's center of mass using the same laws that apply to point particles. The center of mass moves as though all the object's mass were concentrated there, and all external forces acted at that point. This principle allows us to extend Newton's laws to spinning tops, thrown hammers, and other complex systems. Kinematics: Describing Motion Before we discuss what causes motion (forces), we must establish how to describe motion mathematically. This is kinematics. Position specifies an object's location in space. We define it relative to a chosen origin point $O$ using the position vector $\mathbf{r}$, which points from the origin to the particle's current location. Velocity describes how quickly position changes. It is defined as the time derivative of position: $$\mathbf{v} = \frac{d\mathbf{r}}{dt}$$ Velocity is a vector, meaning it has both magnitude (speed) and direction. The magnitude of velocity is the speed, which is always non-negative. Acceleration describes how quickly velocity changes. It is the time derivative of velocity: $$\mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{r}}{dt^2}$$ Here's a potentially confusing point: acceleration doesn't necessarily mean "speeding up." Acceleration means any change in velocity—this includes slowing down (when acceleration and velocity point in opposite directions) or changing direction at constant speed (like circular motion). When working with multiple objects or observers, velocities add according to vector rules. If object A has velocity $\mathbf{v}A$ relative to the ground and object B has velocity $\mathbf{v}B$ relative to the ground, then A's velocity relative to B is $\mathbf{v}A - \mathbf{v}B$. Reference Frames and Inertial Frames All motion is relative—we always measure position, velocity, and acceleration relative to some reference frame. Different observers in different reference frames will disagree about velocities and accelerations, but physics should describe the same underlying reality. An inertial frame is a special reference frame with a crucial property: objects on which no net force acts move with constant velocity (including remaining at rest). In other words, inertial frames do not accelerate. Newton's second law, $\mathbf{F} = m\mathbf{a}$, holds only in inertial frames. This is not arbitrary; it's part of how inertial frames are defined. An inertial frame is precisely a frame in which Newton's laws work as we expect. Non-inertial frames accelerate relative to inertial frames. The critical consequence: in non-inertial frames, Newton's second law does not hold without modification. Instead, fictitious forces (also called pseudo-forces) appear—forces with no physical source. A passenger in an accelerating car feels pushed backward against the seat; this is a centrifugal force, which only exists in the car's non-inertial reference frame. In the inertial frame of the ground, the car accelerates forward while the passenger's body, obeying Newton's first law, tends to stay at constant velocity until the seat pushes forward. For most of classical mechanics, we work in inertial frames to keep the mathematics clean. Earth itself is not a perfect inertial frame (it rotates and orbits), but for most everyday problems, treating it as inertial introduces negligible error. Galilean Transformations: Relating Different Inertial Frames Different inertial frames move at constant velocities relative to each other. How do measurements in one frame relate to measurements in another? Consider two inertial frames: frame $S$ and frame $S'$, where $S'$ moves at constant velocity $\mathbf{u}$ relative to $S$ (say, along the $x$-axis). For simplicity, assume both origins coincided at time $t = 0$. The Galilean transformation gives the relationship: $$x' = x - ut$$ $$y' = y$$ $$z' = z$$ $$t' = t$$ Time is the same in both frames—this is a crucial assumption in classical mechanics (unlike relativistic mechanics, which we will not cover here). An important consequence: velocities transform as $v'x = vx - u$, meaning velocities are subtracted. However, accelerations do not change: $a'x = ax$. And neither do forces: $F'x = Fx$. This explains why Newton's laws have the same form in all inertial frames. The acceleration is unchanged under a Galilean transformation, so if $\mathbf{F} = m\mathbf{a}$ is true in one inertial frame, it is true in all inertial frames. Newtonian Mechanics: Forces and Motion Forces and Newton's Laws A force is any interaction that changes an object's velocity. Pushing, pulling, gravity, magnetic attraction—all are forces. Mathematically, a force produces acceleration. Newton's second law is the central equation of mechanics: $$\mathbf{F} = \frac{d\mathbf{p}}{dt}$$ where $\mathbf{p} = m\mathbf{v}$ is the momentum. This form emphasizes that force is the time rate of change of momentum. When mass is constant (the usual case in classical mechanics), this becomes the familiar: $$\mathbf{F} = m\mathbf{a}$$ Newton's second law is not derived; it is a fundamental statement about how the universe works. It tells us that applying more force produces greater acceleration, and that more massive objects accelerate less under the same force. Newton's Third Law: Action and Reaction Newton's third law states: If object A exerts a force $\mathbf{F}$ on object B, then object B exerts a force $-\mathbf{F}$ on object A. These are called action and reaction forces, and they are always equal in magnitude and opposite in direction. A subtle but important point: the strong form of Newton's third law requires these forces to act along the line connecting the two objects. This is true for many common forces like gravity and spring forces (called central forces). However, some forces—notably the Lorentz force that acts on moving electric charges in magnetic fields—do not satisfy the strong form. For these forces, we can rely on a weak form: while individual action-reaction pairs may not obey the strong form, the total momentum of an isolated system is still conserved. This turns out to be the deeper principle. A common misconception: Why don't the action and reaction forces cancel out? They don't because they act on different objects. If you push a wall, the wall pushes back on you with equal force—these forces act on different bodies, so they produce different accelerations (the wall accelerates less because it is much more massive). Work, Energy, and Conservation Work and the Work-Energy Theorem Work measures the energy transferred when a force moves an object. For a constant force $\mathbf{F}$ acting on an object that moves from position $\mathbf{r}{\text{initial}}$ to $\mathbf{r}{\text{final}}$: $$W = \mathbf{F} \cdot \Delta\mathbf{r}$$ where $\Delta\mathbf{r} = \mathbf{r}{\text{final}} - \mathbf{r}{\text{initial}}$ and the dot denotes the scalar (dot) product. Only the component of force in the direction of motion does work. For variable forces (when force changes along the path), work is computed as a line integral: $$W = \intC \mathbf{F} \cdot d\mathbf{r}$$ The integral is taken along the path $C$ the object follows. Kinetic energy is the energy an object possesses due to its motion: $$K = \frac{1}{2}mv^2$$ The work-energy theorem states: $$W{\text{total}} = \Delta K$$ The total work done on an object equals its change in kinetic energy. This is a powerful consequence of Newton's second law and is often easier to use than $\mathbf{F} = m\mathbf{a}$ directly. Conservative Forces and Potential Energy A force is conservative if the work it performs is independent of the path taken between two points—it depends only on the starting and ending positions. Gravity and ideal spring forces are conservative; friction is not. For conservative forces, we can define a potential energy $U$ such that: $$\mathbf{F}{\text{conservative}} = -\nabla U$$ (The gradient $\nabla U$ is the vector of partial derivatives: $\nabla U = \left(\frac{\partial U}{\partial x}, \frac{\partial U}{\partial y}, \frac{\partial U}{\partial z}\right)$.) This definition connects force and potential energy: the force points in the direction of decreasing potential energy. Conservation of Mechanical Energy When only conservative forces act on a system, we have: $$K + U = \text{constant}$$ This is the conservation of mechanical energy. Kinetic and potential energy can convert into each other, but their sum remains constant. This principle often simplifies problems enormously—instead of solving differential equations for motion, we can use energy conservation to relate velocities at different positions. <extrainfo> Example: Velocity-Dependent Friction As an illustration of how forces produce motion, consider velocity-dependent friction, where the friction force is proportional to velocity with some constant $\lambda > 0$: $$\mathbf{F}f = -\lambda \mathbf{v}$$ Applying Newton's second law: $$m\frac{d\mathbf{v}}{dt} = -\lambda \mathbf{v}$$ This is a first-order differential equation whose solution shows that velocity decays exponentially over time. Objects subject to velocity-dependent friction slow down gradually and asymptotically approach rest. Real fluid drag (air resistance at moderate speeds) behaves roughly this way, making this a useful model for many practical situations. </extrainfo> Summary You now have the foundational language and concepts of classical mechanics: Objects are modeled as point particles or aggregates of rigidly connected particles. Kinematics (position, velocity, acceleration) describes motion without referring to causes. Inertial frames are reference frames where Newton's laws hold without modification. Forces cause accelerations according to Newton's second law: $\mathbf{F} = m\mathbf{a}$. Energy provides an alternative and often simpler way to analyze motion. These tools allow you to predict how any system will move under applied forces. In subsequent chapters, we apply these principles to specific scenarios—orbital motion, oscillations, collisions, and rotation—to develop a complete picture of classical mechanics.