4.2 Kinematics and dynamics of particles and rigid bodies

Kinematics and dynamics of particles and rigid bodies are foundational concepts in engineering physics. They explain how objects move and what causes that motion. Whether you're analyzing a car's wheels spinning or a satellite maintaining its orbit, these principles apply directly.

This section covers motion description (kinematics), force-based analysis (dynamics using Newton's laws), energy and momentum methods, and the distinction between linear and angular momentum.

Particle and Rigid Body Motion

Kinematics Fundamentals

Kinematics describes how objects move without worrying about why they move. You're tracking position, velocity, and acceleration across one, two, or three dimensions, but you're not yet thinking about forces.

For motion with constant acceleration, four kinematic equations relate displacement, velocity, acceleration, and time:

• $v = u + at$
• $s = ut + \frac{1}{2}at^2$
• $v^2 = u^2 + 2as$
• $s = \frac{u+v}{2}t$

Where $s$ is displacement, $u$ is initial velocity, $v$ is final velocity, $a$ is acceleration, and $t$ is time. These only work when acceleration is constant, so don't try to apply them to a car that's accelerating unevenly.

Relative motion describes how one object moves from the perspective of another moving object. You handle this with vector addition of velocities. For example, if you're on a train moving at 30 m/s and you walk forward at 1 m/s, a person standing on the platform sees you moving at 31 m/s.

Curvilinear motion covers paths that aren't straight lines, like projectile motion (a ball thrown at an angle) and circular motion (a car rounding a curve). These require breaking motion into components and often use vector or parametric descriptions.

Rigid Body Kinematics

A rigid body isn't just a point; it has size and shape, so it can rotate. That means you need rotational kinematic variables alongside the linear ones:

• Angular position ($\theta$): where the body is in its rotation
• Angular velocity ($\omega$): how fast it's rotating
• Angular acceleration ($\alpha$): how quickly the rotation rate is changing

The connection between linear and angular variables depends on the radius $r$ from the axis of rotation:

• $v = r\omega$
• $a_t = r\alpha$ (tangential acceleration, not centripetal)

So a point on the rim of a spinning wheel moves faster than a point closer to the hub, even though both have the same angular velocity. A car wheel with radius 0.3 m spinning at $\omega = 50$ rad/s has a rim speed of $v = 0.3 \times 50 = 15$ m/s.

Applying Newton's Laws

Newton's Laws of Motion

Newton's three laws are the foundation of dynamics:

1. First Law (Inertia): An object stays at rest or moves at constant velocity unless a net external force acts on it. A book sitting on a table stays put because the normal force balances gravity.
2. Second Law: $F = ma$. The net force on an object equals its mass times its acceleration. This is the workhorse equation for most dynamics problems.
3. Third Law: For every force one object exerts on another, the second object exerts an equal and opposite force back. A rocket pushes exhaust gases downward; the gases push the rocket upward.

Free body diagrams are your most important tool here. Before solving any dynamics problem, draw the object isolated from its surroundings and sketch every force acting on it with arrows showing direction and magnitude. Skipping this step is the most common source of errors.

Extended Applications

For rigid bodies that rotate, Newton's Second Law has a rotational counterpart:

$\tau = I\alpha$

where $\tau$ is the net torque, $I$ is the moment of inertia (a measure of how hard the body is to spin, depending on mass distribution), and $\alpha$ is angular acceleration. Think of torque as the rotational equivalent of force, and moment of inertia as the rotational equivalent of mass.

A practical example: when you use a torque wrench to tighten a bolt, applying force farther from the bolt increases the torque, making it easier to turn.

Two additional principles worth knowing at this level:

• The principle of transmissibility says you can slide a force along its line of action on a rigid body without changing its effect. This simplifies many force diagrams.
• D'Alembert's principle lets you convert a dynamics problem into a statics-like problem by adding a fictitious "inertial force" ($-ma$) to the free body diagram. Some textbooks use this approach; others don't.

Problem Solving in Dynamics

Force Analysis

Real problems involve specific types of forces. Here are the most common ones you'll encounter:

• Friction: Opposes relative motion between surfaces. Static friction keeps objects from sliding (up to a maximum of $f_s = \mu_s N$), while kinetic friction acts on sliding objects ($f_k = \mu_k N$). The kinetic coefficient is typically smaller than the static one.
• Gravity near Earth's surface: Treated as constant, $F = mg$, with $g \approx 9.81$ m/s².
• Universal gravitation (for larger scales like orbits): $F = G\frac{m_1 m_2}{r^2}$
• Spring forces follow Hooke's law: $F = -kx$, where $k$ is the spring constant and $x$ is displacement from the equilibrium position. The negative sign means the force always pulls back toward equilibrium.
• Tension in ropes or cables, usually analyzed by assuming the rope is massless and doesn't stretch.
• Normal forces and other constraint forces, determined by the requirement that objects follow their constrained paths (e.g., a block can't fall through a surface).

A classic problem: a block sliding down an inclined plane. You draw the free body diagram, resolve gravity into components parallel and perpendicular to the surface, set the perpendicular component equal to the normal force, and use the parallel component minus friction to find acceleration via $F = ma$.

Energy and Momentum Methods

Force analysis isn't always the easiest approach. Sometimes energy or momentum methods get you to the answer faster.

Work-energy methods:

• The work-energy theorem says the net work done on an object equals its change in kinetic energy: $W_{net} = \Delta KE$
• Conservation of energy applies when only conservative forces (gravity, springs) do work: total mechanical energy (kinetic + potential) stays constant. This is how you'd analyze a roller coaster: at the top of a hill, energy is mostly potential; at the bottom, it's mostly kinetic.

Impulse-momentum methods:

• Impulse equals the change in momentum: $J = F \cdot \Delta t = \Delta p$
• Conservation of momentum applies when no external forces act on a system. This is essential for collision problems, where forces between colliding objects are huge but internal to the system.

Use energy methods when you care about speeds and heights but not time. Use momentum methods when you're dealing with collisions or impacts.

Linear vs Angular Momentum

Linear Momentum

Linear momentum is defined as:

$p = mv$

where $m$ is mass and $v$ is velocity. It's a vector quantity pointing in the direction of motion.

Conservation of linear momentum: In a closed system with no net external force, total momentum stays constant. This is why in a billiard ball collision, the total momentum of all balls before the hit equals the total momentum after.

Impulse ($J = F \Delta t$) equals the change in momentum. This explains why car airbags work: they increase the time over which your momentum changes during a crash, which reduces the force on your body.

Angular Momentum

Angular momentum is the rotational analog of linear momentum.

For a particle at distance $r$ from an axis:

$L = r \times p$

For a rigid body spinning about a fixed axis:

$L = I\omega$

where $I$ is the moment of inertia and $\omega$ is angular velocity.

Conservation of angular momentum: Without external torques, total angular momentum stays constant. The classic example is a figure skater pulling their arms in during a spin. Pulling arms in decreases $I$, so $\omega$ must increase to keep $L$ constant. That's why they spin faster.

Two useful theorems for calculating moments of inertia of complex shapes:

• Parallel axis theorem: Lets you find $I$ about any axis if you know $I$ about a parallel axis through the center of mass: $I = I_{cm} + md^2$
• Perpendicular axis theorem: For flat (planar) objects, the moment of inertia about an axis perpendicular to the plane equals the sum of moments about two perpendicular axes in the plane.

Finally, the center of mass is the point where you can treat all the mass as concentrated for translational motion. For a system of particles, the center of mass moves as if all external forces act on a single particle of total mass located there. This simplifies many rigid body problems considerably.