Fundamental Equations of Motion

Why This Matters

Every dynamics problem you'll encounter reduces to one of these fundamental equations. Spacecraft docking, car crash analysis, spinning turbines: they all come back to the same core principles. Your job is 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. They aren't isolated formulas to memorize; they're interconnected principles describing 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. 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

$\vec{F} = m\vec{a}$ is the foundational equation: net force equals mass times acceleration. It applies to any particle or to the center of mass of any system.

Because this is a vector equation, you must apply it independently in each coordinate direction. Choosing your coordinate system wisely (e.g., aligning one axis along an incline) can dramatically simplify the math. If forces are known and constant, this gives you acceleration directly, which you then feed into kinematic equations.

Equations of Motion for Particles

The kinematic equations relate displacement $s$, velocity $v$, acceleration $a$, and time $t$ when acceleration is constant:

• $v = v_0 + at$
• $s = s_0 + v_0 t + \frac{1}{2}at^2$
• $v^2 = v_0^2 + 2a(s - s_0)$

These are derived from Newton's Second Law by integrating acceleration with respect to time or position. That derivation matters because it tells you the key limitation: they're only valid for constant acceleration. If forces vary with position or time, 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

$U_{1 \to 2} = T_2 - T_1$ states that the work done by all forces equals the change in kinetic energy, where $T = \frac{1}{2}mv^2$.

This is ideal for variable forces because work is calculated as $U = \int \vec{F} \cdot d\vec{r}$, which naturally handles force changes along the path. And since it's a scalar equation, there are no coordinate system headaches. You just track energy in and energy out.

Conservation of Energy

$T_1 + V_1 = T_2 + V_2$ says total mechanical energy stays constant when only conservative forces (gravity, springs) do work.

Conservative forces have associated potential energy functions:

• Gravitational: $V_g = mgh$
• Elastic: $V_e = \frac{1}{2}kx^2$

This is the fastest solution method when friction and other non-conservative forces are absent. One equation relates initial and final states directly, with no need to track the path between them.

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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

$\int \vec{F} \, dt = m\vec{v}_2 - m\vec{v}_1$ connects the impulse (force applied over time) to the change in momentum.

This is essential for impact problems where forces are large but act over very short time intervals. You often know the impulse even when the force magnitude itself is unknown. For problems that give you a collision duration, rearrange to find average force: $\vec{F}_{avg} = \frac{\Delta(m\vec{v})}{\Delta t}$.

Conservation of Linear Momentum

$m_A\vec{v}_{A1} + m_B\vec{v}_{B1} = m_A\vec{v}_{A2} + m_B\vec{v}_{B2}$ holds when no net external force (or impulse) acts on the system.

This is the cornerstone of collision analysis. It applies to all collision types:

• Elastic collisions: both momentum and kinetic energy are conserved
• Inelastic collisions: momentum is conserved, kinetic energy is not
• Perfectly inelastic collisions: objects stick together, maximum kinetic energy loss

System selection is critical. Draw your system boundary to exclude external impulses. Internal forces between colliding objects always cancel in pairs (Newton's Third Law), so they don't affect the system's total momentum.

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Angular Momentum and Rotation

Rotational motion has 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

$\vec{H} = I\vec{\omega}$ gives the angular momentum of a rigid body rotating about a fixed axis, where $I$ is the moment of inertia and $\omega$ is angular velocity.

The rotational analog of Newton's Second Law is $\sum \vec{M} = \dot{\vec{H}}$, stating that net torque equals the rate of change of angular momentum. For a rigid body about a fixed axis, this simplifies to $\sum M = I\alpha$, where $\alpha$ is angular acceleration.

Moment of inertia depends on mass distribution, not just total mass. The same mass arranged differently (solid cylinder vs. hollow cylinder, for example) gives a different $I$, directly affecting how the body responds to torques.

Conservation of Angular Momentum

$I_1\omega_1 = I_2\omega_2$ holds when no external torques act on the system.

This explains many spinning phenomena: a figure skater spins faster when pulling arms in because $I$ decreases, so $\omega$ must increase to keep $H$ constant. It also applies in orbital mechanics, where planets move faster at closer approach to the sun because angular momentum about the sun is conserved (this is Kepler's Second Law).

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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

Two equations govern rigid body motion:

• $\sum \vec{F} = m\vec{a}_G$ 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.

These are coupled equations. For rolling, sliding, or constrained motion, you need a kinematic relationship to link them. The most common one: for rolling without slip, $a_G = r\alpha$, which ties the translational acceleration of the center to the angular acceleration.

Principle of Virtual Work

$\delta U = \sum \vec{F} \cdot \delta \vec{r} = 0$ for systems in equilibrium: the virtual work done during any compatible virtual displacement is zero.

This is powerful for constrained systems because it eliminates unknown constraint forces automatically, reducing complex multi-body problems to single equations. It connects to D'Alembert's Principle for dynamics, where you treat inertial terms ($-m\vec{a}$) as fictitious forces and then apply virtual work to derive 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 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.