Fundamental Equations of Motion

Why This Matters

Every dynamics problem you'll encounter—whether it's a spacecraft docking maneuver, a car crash analysis, or a spinning turbine—reduces to one of these fundamental equations. You're being tested on your ability to recognize which equation applies to which scenario and why that equation captures the physics at play. The concepts here—force-acceleration relationships, energy methods, momentum principles, and rigid body motion—form the analytical toolkit you'll use throughout your engineering career.

These equations aren't isolated formulas to memorize; they're interconnected principles that describe how objects move and interact. Newton's Second Law underlies everything, but knowing when to use an energy approach versus a momentum approach is what separates students who struggle from those who solve problems efficiently. Don't just memorize the math—understand what physical quantity each equation conserves or relates, and you'll know exactly which tool to reach for on exam day.

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Force-Acceleration Relationships

These equations directly connect forces to the resulting motion. When you know the forces acting on a system and need to find accelerations (or vice versa), this is your starting point.

Newton's Second Law of Motion

• $\vec{F} = m\vec{a}$—the foundational equation stating that net force equals mass times acceleration, applicable to any particle or center of mass
• Vector equation means you must apply it independently in each coordinate direction; choose your coordinate system wisely to simplify the math
• Starting point for most dynamics problems—if forces are known and constant, this gives you acceleration directly for kinematic analysis

Equations of Motion for Particles

• Kinematic equations relate $s$, $v$, $a$, and $t$ when acceleration is constant: $v = v_0 + at$, $s = s_0 + v_0t + \frac{1}{2}at^2$, $v^2 = v_0^2 + 2a(s - s_0)$
• Derived from Newton's Second Law by integrating acceleration with respect to time or position—understanding this connection helps you know when they apply
• Only valid for constant acceleration—if forces vary, you'll need energy or momentum methods instead

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Energy Methods

Energy approaches are powerful when forces vary along a path or when you care about speeds rather than accelerations. These methods use scalar quantities, eliminating the need for vector components.

Work-Energy Principle

• $U_{1 \to 2} = T_2 - T_1$—the work done by all forces equals the change in kinetic energy, where $T = \frac{1}{2}mv^2$
• Ideal for variable forces because work is calculated as $U = \int \vec{F} \cdot d\vec{r}$, naturally handling force changes along the path
• Scalar equation means no coordinate system headaches—just track energy in and energy out

Conservation of Energy

• $T_1 + V_1 = T_2 + V_2$—total mechanical energy remains constant when only conservative forces (gravity, springs) do work
• Conservative forces have associated potential energy functions: gravitational $V_g = mgh$, elastic $V_e = \frac{1}{2}kx^2$
• Fastest solution method when friction and other non-conservative forces are absent—one equation relates initial and final states directly

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Linear Momentum Methods

Momentum methods shine when forces are unknown or impulsive, especially in collisions. These approaches track how motion transfers between objects rather than analyzing forces directly.

Impulse-Momentum Theorem

• $\int \vec{F} \, dt = m\vec{v}_2 - m\vec{v}_1$—impulse equals change in momentum, connecting force applied over time to velocity change
• Essential for impact problems where forces are large but act over short time intervals—you often know impulse even when force magnitude is unknown
• Average force calculation: rearrange to $\vec{F}_{avg} = \frac{\Delta(m\vec{v})}{\Delta t}$ for problems involving collision duration

Conservation of Linear Momentum

• $m_A\vec{v}_{A1} + m_B\vec{v}_{B1} = m_A\vec{v}_{A2} + m_B\vec{v}_{B2}$—total momentum is conserved when no external forces act on the system
• Collision analysis cornerstone—applies to elastic collisions (energy conserved), inelastic collisions (energy lost), and perfectly inelastic collisions (objects stick together)
• System selection is critical—draw your boundary to exclude external impulses; internal forces between colliding objects cancel automatically

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Angular Momentum and Rotation

Rotational motion requires its own set of equations that parallel the linear case. Moment of inertia replaces mass, angular velocity replaces linear velocity, and torque replaces force.

Angular Momentum Equation

• $\vec{H} = I\vec{\omega}$ for rigid bodies rotating about a fixed axis, where $I$ is moment of inertia and $\omega$ is angular velocity
• $\sum \vec{M} = \dot{\vec{H}}$—the rotational analog of Newton's Second Law, stating that net torque equals rate of change of angular momentum
• Moment of inertia depends on mass distribution—same mass arranged differently gives different $I$, directly affecting rotational response to torques

Conservation of Angular Momentum

• $I_1\omega_1 = I_2\omega_2$—angular momentum is conserved when no external torques act on the system
• Explains spinning phenomena: figure skaters spin faster when pulling arms in ($I$ decreases, so $\omega$ increases to conserve $H$)
• Applies to orbital mechanics—planets move faster when closer to the sun because angular momentum about the sun is conserved

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Rigid Body Dynamics

Real engineering systems involve extended bodies that can translate and rotate simultaneously. These equations couple linear and angular motion for complete analysis.

Equations of Motion for Rigid Bodies

• $\sum \vec{F} = m\vec{a}_G$ governs translation of the mass center, independent of how the body rotates
• $\sum \vec{M}_G = I_G\vec{\alpha}$ governs rotation about the mass center, where $\alpha$ is angular acceleration
• Coupled equations—for rolling, sliding, or constrained motion, kinematic relationships (like $a_G = r\alpha$ for rolling without slip) connect these equations

Principle of Virtual Work

• $\delta U = \sum \vec{F} \cdot \delta \vec{r} = 0$ for systems in equilibrium—virtual work done during any compatible virtual displacement is zero
• Powerful for constrained systems—eliminates unknown constraint forces automatically, reducing complex problems to single equations
• Connects to D'Alembert's Principle for dynamics: treat inertial terms as forces and apply virtual work to solve equations of motion

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Quick Reference Table

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Self-Check Questions

1. A block slides down a rough incline. Which two equations would you combine to find its speed at the bottom, and why can't you use Conservation of Energy alone?

2. In a collision between two hockey pucks on frictionless ice, which quantity is definitely conserved? Under what additional condition would kinetic energy also be conserved?

3. Compare and contrast how you would analyze a swinging pendulum using (a) Newton's Second Law and (b) Conservation of Energy. Which approach finds tension in the string more easily?

4. A figure skater pulls her arms inward while spinning. Which conservation principle explains why her angular velocity increases, and what physical quantity remains constant?

5. FRQ-style prompt: A disk rolls without slipping down an incline. Explain why you need both $\sum F = ma_G$ and $\sum M_G = I_G\alpha$, and identify the kinematic constraint that links them.