Statics and Dynamics

Introduction

Statics and Dynamics are two central branches of engineering mechanics. Statics studies systems in equilibrium, while dynamics studies motion and the forces and moments associated with changes in motion. Together they provide a modeling language for structures, machines, vehicles, robots, biomechanical systems, and many other engineered systems.

This university-level aiMOOC is designed for students in engineering, applied physics, and related fields. You will move from physical idealization and free-body diagrams to equilibrium equations, kinematics, Newton–Euler equations, energy and momentum methods, rigid-body motion, and introductory vibration. The emphasis is not only on obtaining numerical answers but also on choosing useful models, communicating assumptions, checking results, and transferring methods to unfamiliar problems.

After completing the course, you should be able to:

distinguish between a particle, a rigid body, and a deformable-body model;
construct correct free-body and kinematic diagrams;
apply equilibrium equations to particles, rigid bodies, beams, trusses, and simple machines;
describe particle and rigid-body motion using position, velocity, and acceleration;
apply Newton's laws, work–energy, and impulse–momentum principles;
connect translational and rotational dynamics through mass, center of mass, torque, and mass moment of inertia;
explain how static equilibrium appears as a special case within a broader dynamical model;
evaluate assumptions, dimensions, signs, limiting cases, and engineering plausibility.

A free-body diagram is one of the most important representations in both statics and dynamics. The diagram isolates the selected body or subsystem and replaces its surroundings by the external forces and moments that the surroundings exert on it.

The MIT OpenCourseWare lecture above reviews Newton's laws and particle kinematics. Use it to connect the mathematical descriptions in this course with a systematic engineering interpretation of motion.

Foundations of Engineering Mechanics

Models, Systems, and Reference Frames

Engineering mechanics begins with a deliberate simplification of reality. A particle model represents an object by a point mass when size and orientation are irrelevant to the question. A rigid-body model allows translation and rotation but neglects deformation. A deformable-body model is required when stress, strain, deflection, or shape change matters.

You should always identify the system boundary before writing equations. Forces between parts inside the boundary are internal to the selected system; forces exerted by objects outside the boundary are external. The same physical interaction may be internal or external depending on the chosen system.

Dynamics also requires attention to the frame of reference. Newton's second law in its standard form, $\sum 𝐅 = m 𝐚$ , is directly valid in an inertial reference frame. When you work in accelerating or rotating frames, additional inertial terms may be required. Choosing an appropriate frame can turn a difficult problem into a simple one, but an inappropriate frame can hide or misrepresent the physics.

Units, Dimensions, Scalars, and Vectors

Mechanics uses both scalar and vector quantities. Mass, time, energy, and temperature are scalars. Force, position, velocity, acceleration, and momentum are vectors. A vector has magnitude and direction, and its components depend on the chosen coordinate basis.

In SI units, force is measured in newtons, with $1 N = 1 k g m / s^{2}$ . Moments and torques have units of $N m$ . Work and energy are measured in joules, and $1 J = 1 N m$ . Although torque and energy share the same dimensional units, they represent different physical quantities and should not be treated as interchangeable.

Dimensional checking is a powerful error-detection method. If the left and right sides of an equation do not have the same physical dimensions, the equation cannot be correct.

Forces, Moments, and Couples

A force $𝐅$ acting at a point can produce both translation and rotation. Its moment about a point $O$ is

$𝐌_{O} = 𝐫 \times 𝐅$ ,

where $𝐫$ points from $O$ to any point on the force's line of action. In planar problems, the moment can be handled as a signed scalar perpendicular to the plane.

A couple consists of two equal, opposite, non-collinear forces. The resultant force is zero, but the pair produces a pure moment. A couple moment is a free vector: for a rigid body, its external effect does not depend on the point about which moments are summed.

A general force system acting on a rigid body can often be reduced at a chosen reference point to one resultant force and one resultant moment. This reduction is useful in both equilibrium and equations of motion.

Statics

Equilibrium of Particles and Rigid Bodies

A body is in mechanical equilibrium when its linear and angular accelerations are zero. In a standard statics problem the body is at rest in the chosen inertial frame. The equilibrium conditions are

$\sum 𝐅 = 𝟎$

and

$\sum 𝐌_{O} = 𝟎$ .

For a two-dimensional rigid body, these conditions normally become three independent scalar equations:

$\sum F_{x} = 0, \sum F_{y} = 0, \sum M_{O} = 0.$

For a three-dimensional rigid body, up to six independent scalar equilibrium equations are available: three force equations and three moment equations.

Equilibrium equations are statements about the entire isolated body or subsystem. They do not justify omitting a force simply because it seems small or inconvenient. Every external interaction that belongs to the model must appear on the free-body diagram.

Free-Body Diagrams and Support Reactions

A reliable free-body diagram separates modeling from algebra. First choose the body. Then remove its supports and contacts and replace them by the forces and moments they can exert. Add applied loads, body forces such as weight, and any known couples.

Common idealized supports in planar statics include:

a smooth surface or roller, which provides one reaction normal to the surface;
a pin or hinge, which can provide two force components but no ideal reaction couple;
a fixed support, which can provide two force components and a reaction moment;
a cable, which carries tension along its axis and pulls away from the body;
a two-force member, which carries equal and opposite end forces along the line joining its end connections.

Support models are idealizations. Real connections have compliance, friction, finite dimensions, and strength limits. The correct idealization is the simplest one that preserves the behavior relevant to the engineering question.

This MIT OpenCourseWare recitation focuses on free-body diagrams. As you watch, compare each diagram with the rule that only external interactions on the chosen body belong on its free-body diagram.

Trusses, Frames, and Machines

An ideal planar truss is assembled from slender two-force members connected by pins, with external loads and reactions applied at joints. Under these assumptions, each member carries only axial tension or compression. Two common solution methods are the method of joints, which enforces force equilibrium at selected joints, and the method of sections, which cuts through the truss and applies rigid-body equilibrium to one part.

A frame contains at least one member that is not a two-force member. A machine is an assembly designed to transmit or modify forces and motion. For frames and machines, you often need to isolate multiple members separately. Internal pin forces that disappear when the entire assembly is isolated become external forces when an individual member is isolated.

Counting unknowns and equations can help identify whether a structure may be statically determinate, but counting alone is not a complete stability test. Geometry, redundant constraints, and special alignments can change whether the equilibrium equations yield a unique physically meaningful solution.

Distributed Loads, Resultants, and Internal Loading

A distributed load $w (x)$ acting on a beam can be replaced, for overall equilibrium, by a resultant equal to the area under the loading curve:

$R = \int w (x) d x .$

The resultant acts through the centroid of that load distribution. For a constant load on a length $L$ , $R = w L$ and acts at the midpoint. For a triangular load, the resultant is the triangle's area and acts one-third of the base length from the larger-intensity end.

When a beam is cut, the removed part is replaced by internal resultants. In a planar beam these are commonly axial force, shear force, and bending moment. Their signs depend on the adopted convention, so you should define a convention and use it consistently.

The relationships between distributed load, shear, and bending moment are especially useful for continuous beam loading. With a consistent sign convention, the differential relations connect the slope of the shear diagram to loading and the slope of the moment diagram to shear.

Friction and Impending Motion

Dry friction is commonly modeled with Coulomb friction. Before slip begins, the static friction magnitude adjusts as needed up to a limiting value:

$| F_{s} | \leq μ_{s} N .$

At impending slip, $| F_{s} | = μ_{s} N$ . During sliding, a common model is $| F_{k} | = μ_{k} N$ , directed opposite the relative sliding velocity.

A frequent mistake is to set static friction equal to $μ_{s} N$ in every equilibrium problem. That equality is justified only when the contact is at impending slip. Otherwise, friction is an unknown force whose magnitude is determined by equilibrium, subject to the inequality limit.

Dynamics

Kinematics of Particles

Kinematics describes motion without first asking what forces cause it. For a particle with position vector $𝐫 (t)$ ,

$𝐯 = \frac{d 𝐫}{d t}, 𝐚 = \frac{d 𝐯}{d t} .$

In Cartesian coordinates with fixed unit vectors, differentiation acts directly on the scalar components. For curvilinear motion, the directions of the basis vectors may also change.

In tangential-normal coordinates,

$𝐚 = \dot{v} 𝐞_{t} + \frac{v^{2}}{ρ} 𝐞_{n},$

where $ρ$ is the local radius of curvature. The normal acceleration points toward the center of curvature.

In plane polar coordinates,

$𝐯 = \dot{r} 𝐞_{r} + r \dot{θ} 𝐞_{θ}$

and

$𝐚 = (\ddot{r} - r {\dot{θ}}^{2}) 𝐞_{r} + (r \ddot{θ} + 2 \dot{r} \dot{θ}) 𝐞_{θ} .$

The term $2 \dot{r} \dot{θ}$ is associated with the changing radial distance in a rotating basis and is closely related to Coriolis acceleration in moving-frame descriptions.

Projectile and Circular Motion

Ideal projectile motion near Earth's surface is often modeled with constant downward gravitational acceleration and negligible air resistance. Horizontal acceleration is then zero while vertical acceleration is $- g$ . The result is a parabolic trajectory in a fixed Cartesian frame.

For uniform circular motion, speed is constant but velocity is not, because the velocity direction changes continuously. The resulting acceleration has magnitude $v^{2} / r$ and points toward the center of the circle. This is a kinematic fact; the inward net force required to create that acceleration depends on the physical system.

Kinetics and the Newton–Euler Equations

Kinetics connects forces and moments to changes in motion. For a particle of constant mass in an inertial frame,

$\sum 𝐅 = m 𝐚 .$

For a system of particles or a rigid body, the translational equation can be written using the center of mass $G$ :

$\sum 𝐅_{e x t} = m 𝐚_{G} .$

For planar rigid-body motion, the moment equation about the center of mass is

$\sum M_{G} = I_{G} α,$

where $I_{G}$ is the mass moment of inertia about the axis perpendicular to the plane through $G$ , and $α$ is angular acceleration.

Do not confuse mass moment of inertia, which measures resistance to angular acceleration, with the second moment of area, which appears in beam bending and depends on geometry rather than mass distribution.

Relative Motion of Rigid Bodies

For two points $A$ and $B$ fixed in the same rigid body,

$𝐯_{B} = 𝐯_{A} + 𝝎 \times 𝐫_{B / A} .$

The corresponding acceleration relation is

$𝐚_{B} = 𝐚_{A} + 𝜶 \times 𝐫_{B / A} + 𝝎 \times (𝝎 \times 𝐫_{B / A}) .$

These equations separate the translation of a reference point from the rotational contribution. They are central to mechanisms, rolling bodies, robot links, and machine components.

For rolling without slipping on a fixed surface, the point of contact has zero instantaneous velocity relative to the surface, but its acceleration is not generally zero. Treating zero velocity as zero acceleration is a common conceptual error.

Mass Moment of Inertia and Rotational Dynamics

The mass moment of inertia about an axis is

$I = \int r_{⊥}^{2} d m,$

where $r_{⊥}$ is the perpendicular distance from the mass element to the axis. Mass farther from the axis contributes more strongly because the distance is squared.

For parallel axes separated by a distance $d$ , the parallel-axis theorem gives

$I_{O} = I_{G} + m d^{2} .$

The theorem is valid when one of the parallel axes passes through the center of mass. It should not be used as a generic relation between arbitrary parallel axes.

The MIT OpenCourseWare lecture above develops mass moment of inertia for rigid bodies and discusses principal axes and the parallel-axis theorem.

Work and Energy

The work done by a force along a path is

$U_{1 \to 2} = \int_{1}^{2} 𝐅 \cdot d 𝐫 .$

For a particle, the work–energy principle is

$T_{1} + U_{1 \to 2} = T_{2},$

with kinetic energy $T = \frac{1}{2} m v^{2}$ . A rigid body in planar motion has kinetic energy

$T = \frac{1}{2} m v_{G}^{2} + \frac{1}{2} I_{G} ω^{2} .$

When all relevant forces are conservative, it is often useful to introduce potential energy and write conservation of mechanical energy as $T_{1} + V_{1} = T_{2} + V_{2}$ . Non-conservative work, such as work by sliding friction or some actuators, must be accounted for separately.

Energy methods are especially effective when you need speeds or displacements but not the time history or detailed constraint forces.

Impulse and Momentum

Linear impulse changes linear momentum:

$\int_{t_{1}}^{t_{2}} \sum 𝐅 d t = m 𝐯_{2} - m 𝐯_{1} .$

Angular impulse similarly changes angular momentum. These formulations are useful for impacts, short-duration forces, and situations in which force varies strongly with time but its integral is known or can be estimated.

Momentum conservation applies only in directions or about points for which the relevant external impulse is zero or negligible over the time interval. It is not a universal shortcut; the system boundary and external interactions must justify it.

Mechanical Vibration

Vibration is a major application of dynamics. A linear single-degree-of-freedom mass–spring–damper model obeys

$m \ddot{x} + c \dot{x} + k x = F (t) .$

For the undamped free system, the natural circular frequency is

$ω_{n} = \sqrt{\frac{k}{m}} .$

The damping ratio is

$ζ = \frac{c}{2 \sqrt{k m}} .$

Harmonic forcing can produce large steady-state amplitudes when the forcing frequency is near the system's natural frequency, particularly when damping is small. Engineers therefore analyze resonance in structures, machinery, vehicles, instruments, and control systems.

A pendulum is a useful example because it is nonlinear in its exact form. For sufficiently small angular displacements, the approximation $\sin θ \approx θ$ produces a linear equation and reveals a simple natural-frequency relation.

This MIT OpenCourseWare lecture introduces mechanical vibration, static equilibrium coordinates, natural frequency, damping, and single-degree-of-freedom response.

Connecting Statics and Dynamics

Statics and dynamics are not isolated subjects. They use the same force and moment concepts but differ in the motion terms that appear in the governing equations. For a planar rigid body, the dynamic equations

$\sum F_{x} = m a_{G x}, \sum F_{y} = m a_{G y}, \sum M_{G} = I_{G} α$

reduce to the familiar statics equations when $𝐚_{G} = 𝟎$ and $α = 0$ .

This connection is important in engineering judgment. A loading can be treated as quasi-static only when inertia effects are negligible for the accuracy required. Slowly applied loads on a stiff structure may be approximated statically, while impact, rapid actuation, vibration, earthquake excitation, and high-speed machinery generally require dynamics.

Static equilibrium can also define the reference configuration for a dynamic problem. For example, a spring–mass system under gravity has a static deflection. If the dynamic coordinate is measured from that equilibrium position, the constant gravity term can disappear from the equation of motion, simplifying the analysis without eliminating gravity from the physical model.

Choosing a Solution Method

Different methods answer different questions. Use direct equilibrium when accelerations are zero. Use Newton–Euler equations when forces, accelerations, and constraint reactions are central. Use work–energy when a relation between configuration and speed is more important than time. Use impulse–momentum when forces act over a short interval or when momentum exchange is central. Use vibration methods when repeated or oscillatory response matters.

A good engineer does not begin by selecting a favorite formula. Begin with the physical question, system boundary, assumptions, and desired unknowns. Then choose the governing principle that exposes those unknowns with the fewest unnecessary intermediate quantities.

A Systematic Problem-Solving Workflow

Stage	Questions you should ask	Typical output
Define the system	What body or collection of bodies am I analyzing, and what is outside the boundary?	A clear system boundary and list of relevant interactions
Idealize	Is a particle, rigid body, beam, truss, spring, damper, or other model appropriate?	Explicit assumptions
Draw	What free-body and kinematic diagrams make the physics visible?	Diagrams with forces, moments, coordinates, dimensions, and motion variables
Choose coordinates	Which axes, origin, and positive directions simplify components and constraints?	A consistent sign convention
Apply principles	Is this equilibrium, Newton–Euler, work–energy, impulse–momentum, or vibration?	Governing equations
Add constraints	Are there geometric, rolling, cable-length, joint, or constitutive relations?	Sufficient independent equations
Solve	Are the equations algebraic, differential, linear, or nonlinear?	Symbolic or numerical result
Check	Are units, signs, magnitudes, limiting cases, and physical interpretations sensible?	A defensible engineering conclusion

Worked Examples

Example: Reactions on a Simply Supported Beam

A horizontal beam of length $4 m$ is supported by a pin at the left end $A$ and a roller at the right end $B$ . A downward point load of $10 k N$ acts $1 m$ from $A$ . Neglect the beam's weight.

The free-body diagram contains $A_{x}$ , $A_{y}$ , $B_{y}$ , and the applied load. There is no horizontal applied load, so $A_{x} = 0$ . Taking moments about $A$ removes both reactions at $A$ :

$4 B_{y} - 10 (1) = 0.$

Thus $B_{y} = 2.5 k N$ . Vertical force equilibrium then gives

$A_{y} + B_{y} - 10 = 0$ ,

so $A_{y} = 7.5 k N$ . The reactions sum to the applied vertical load, and the larger reaction occurs nearer the load, which is physically reasonable.

Example: Projectile Motion

A projectile is launched from level ground at $20 m / s$ and $3 0^{\circ}$ above the horizontal. Neglect air resistance and use $g = 9.81 m / s^{2}$ .

The initial components are $v_{0 x} = 20 \cos 3 0^{\circ} \approx 17.32 m / s$ and $v_{0 y} = 20 \sin 3 0^{\circ} = 10.0 m / s$ . The time to return to launch height is

$T = \frac{2 v_{0 y}}{g} \approx 2.04 s .$

The horizontal range is

$R = v_{0 x} T \approx 35.3 m .$

The maximum height above the launch point is

$H = \frac{v_{0 y}^{2}}{2 g} \approx 5.10 m .$

These values belong to the ideal constant-gravity model. Aerodynamic drag, spin, wind, elevation differences, and variation of gravity would require a more detailed model.

Example: Constant Torque on a Rigid Rotor

A rotor has a mass moment of inertia $I = 2.4 k g m^{2}$ about its fixed axis. A constant net torque of $12 N m$ acts about that axis. Neglect bearing friction.

The rotational equation is $\sum M = I α$ , so

$α = \frac{12}{2.4} = 5.0 r a d / s^{2} .$

If the rotor starts from rest, after $3 s$ its angular speed is $ω = α t = 15 r a d / s$ . This example shows the rotational analogue of constant-force particle motion.

Example: Static Deflection and Dynamic Model of a Spring–Mass System

A mass $m$ hangs vertically from a linear spring of stiffness $k$ . In static equilibrium, the spring extension from its unloaded length is determined by $k δ = m g$ , so $δ = m g / k$ .

Now define $x$ as the additional displacement measured from that static equilibrium position. If damping is neglected and there is no extra applied force, the dynamic equation becomes

$m \ddot{x} + k x = 0.$

The constant weight term does not appear because the coordinate origin was placed at the gravity-loaded equilibrium configuration. The natural frequency is $ω_{n} = \sqrt{k / m}$ . This is a direct example of statics providing the reference state for dynamics.

Common Errors and Engineering Checks

Common mistakes are often modeling errors rather than algebra errors. Watch for the following:

drawing action and reaction forces from the same interaction on one free-body diagram even though they act on different bodies;
including internal forces when analyzing a complete assembly but forgetting them when isolating a component;
assuming a support reaction direction that the ideal support cannot provide;
setting static friction to its limiting value without evidence of impending slip;
using degrees in a calculation that assumes radians for angular derivatives;
treating constant speed as zero acceleration during curved motion;
applying $\sum M = I α$ about an arbitrary moving point without the additional terms required by the general angular-momentum relation;
confusing mass moment of inertia with second moment of area;
conserving energy or momentum without checking external work or impulse;
reporting a numerical answer without dimensions, sign interpretation, or plausibility checks.

A result should be challenged before it is trusted. Check units, equilibrium of the solved system, known limiting cases, symmetry, expected direction of motion, and whether the magnitude is realistic for the modeled device or structure.

Interactive Tasks

Quiz: Test Your Knowledge

Which condition characterizes a rigid body in static equilibrium? (The net external force and net external moment are both zero) (!The velocity must increase linearly with time) (!The kinetic energy must be at its maximum) (!The mass moment of inertia must be zero)

Which statement best describes a free-body diagram? (It isolates a selected body and shows the external forces and moments acting on it) (!It shows every internal force inside every material particle) (!It is a graph of acceleration against time) (!It replaces the need to define a system boundary)

What reaction can an ideal smooth roller support provide in a planar problem? (One force normal to the supporting surface) (!Two independent force components and a moment) (!A pure couple with no force) (!A force only parallel to the supporting surface)

When is the relation between limiting static friction and normal force applied? (When the contact is at impending slip) (!Whenever the body is at rest) (!Whenever the normal force is zero) (!Only after sliding has already begun)

What does particle kinematics describe? (The position velocity and acceleration of motion) (!The material strength of a moving body) (!Only the forces that cause motion) (!Only systems that are in equilibrium)

For uniform circular motion what is true about acceleration? (It points toward the center even when speed is constant) (!It is zero because the speed is constant) (!It always points tangentially in the direction of travel) (!It points away from the center)

Which equation governs translation of the center of mass in an inertial frame? (The net external force equals mass times center of mass acceleration) (!The net external force always equals zero) (!The net external moment equals mass times velocity) (!The kinetic energy equals the linear momentum)

What does mass moment of inertia measure in rigid-body dynamics? (The dependence of rotational inertia on how mass is distributed about an axis) (!The total external force on the body) (!The elastic stiffness of the body) (!The area enclosed by the body's path)

When is a work-energy method especially useful? (When relating configuration and speed without needing the detailed time history) (!When every support reaction must be found directly) (!When no displacement occurs anywhere in the system) (!When only geometric dimensions are known)

How are statics and dynamics related for a rigid body? (Statics is obtained when the translational and angular accelerations are zero) (!Dynamics excludes forces and moments) (!Statics requires nonzero angular acceleration) (!They use unrelated physical laws)

Memory Game

Equilibrium	State in which net external force and net external moment vanish
Kinematics	Description of motion using position velocity and acceleration
Kinetics	Connection between forces moments and changes in motion
Couple	Pair of equal opposite non-collinear forces producing a pure moment
Impulse	Time integral of force associated with change in linear momentum
Truss	Pin-jointed ideal structure whose members primarily carry axial force
Inertia	Resistance of mass to changes in translational or rotational motion
Resonance	Large response that can occur when forcing frequency approaches a natural frequency

Drag and Drop

Match the correct terms.	Topic
Free-body diagram	Shows external forces and moments on an isolated system
Method of joints	Uses force equilibrium at truss connections
Work-energy principle	Relates mechanical work to change in kinetic energy
Impulse-momentum principle	Relates force integrated over time to momentum change
Parallel-axis theorem	Transfers mass moment of inertia from a center-of-mass axis to a parallel axis

...

Crossword Puzzle

Equilibrium	What condition requires zero resultant force and zero resultant moment?
Momentum	What quantity equals mass multiplied by velocity for a particle?
Friction	What contact force opposes impending or actual relative sliding?
Kinematics	What branch describes motion without first considering its causes?
Impulse	What quantity is the time integral of force?
Inertia	What property describes resistance to changes in motion?

LearningApps

Cloze Text

Statics begins by isolating a system and enforcing

. A free-body diagram should show only the relevant

interactions on the selected body. In planar rigid-body statics the independent balance equations include two force equations and one

equation. Kinematics describes position velocity and

without first specifying the forces that cause the motion. Newton's second law connects net force to mass and

. For planar rigid-body rotation the angular equation involves the mass moment of

. The work-energy principle connects work with a change in kinetic

. Linear impulse changes linear

. A rolling body can have a contact point with zero instantaneous velocity but nonzero

. A linear undamped spring-mass system has a natural frequency controlled by stiffness and

. Statics appears within dynamics when translational and angular accelerations are

. Good engineering solutions end with unit sign magnitude and plausibility

.

Open-Ended Tasks

Easy

Free-body diagram audit: Choose a familiar object such as a wall shelf, desk lamp, bicycle pedal, or suspended sign; photograph or sketch it, define one system boundary, and create a labeled free-body diagram that explains every external force and moment.
Support model comparison: Find four real supports or joints on campus and classify each using an idealized mechanics model; explain where the idealization is useful and where it could fail.
Motion data sketch: Record a short video of an object moving approximately in one dimension, extract several position observations, and sketch consistent position-time, velocity-time, and acceleration-time graphs.
Unit consistency check: Collect five equations from your mechanics notes and produce a short annotated sheet showing how dimensional analysis can confirm or reject each equation.

Standard

Beam reaction investigation: Model a bench, shelf, or simple beam with realistic loads, calculate its support reactions, and compare the predicted load distribution with an intuitive estimate before seeing the calculation.
Friction experiment: Use a block and an adjustable inclined surface to estimate a coefficient of static friction from the angle of impending slip, repeat the measurement, and discuss uncertainty and modeling limits.
Projectile video analysis: Film a safely tossed ball from the side, estimate its trajectory from video frames, fit a simple constant-gravity model, and discuss deviations caused by measurement error or air resistance.
Truss analysis project: Build or model a small planar truss, calculate selected member forces using joints or sections, and identify which members are in tension and compression.

Advanced

Rigid-body motion study: Analyze a rolling wheel, linkage, or robot arm using relative-velocity and relative-acceleration equations, then validate one predicted point velocity with video or simulation data.
Energy versus Newton-Euler comparison: Solve the same dynamics problem using both a force-and-acceleration method and a work-energy method, compare the intermediate unknowns each method requires, and explain which method is more efficient for the chosen question.
Vibration identification experiment: Measure the free oscillation of a safe spring-mass or pendulum system, estimate its natural frequency and damping from data, and compare the result with a theoretical model.
Dynamic modeling critique: Select a real engineering event such as elevator startup, vehicle braking, robotic pick-and-place motion, machine vibration, or structural impact; propose both a quasi-static and a dynamic model and argue quantitatively which model is justified.

Learning Assessment

Model selection assessment: Given a physical system with incomplete information, justify whether a particle, rigid-body, truss, beam, or vibration model is appropriate and identify the assumptions that most strongly affect the result.
Equilibrium transfer assessment: Analyze a multi-body structure by drawing separate free-body diagrams, solving for reactions or member forces, and explaining how internal forces move across the system boundary when components are isolated.
Dynamics method assessment: For a motion problem, decide whether Newton-Euler, work-energy, or impulse-momentum is the most efficient primary method and defend the choice before carrying out the solution.
Statics to dynamics assessment: Start from the dynamic equations of a planar rigid body and show how a specific equilibrium problem is recovered as a zero-acceleration case, including a physical interpretation of the limiting process.
Experimental validation assessment: Compare a theoretical mechanics prediction with measured or simulated data, quantify the discrepancy, and identify whether modeling assumptions, parameter uncertainty, or measurement error most plausibly explains it.
Engineering communication assessment: Present one complete solution in a professional format that includes the system boundary, diagrams, equations, units, sign conventions, assumptions, checks, and an interpretation understandable to another engineering student.

Evidence of Learning

Strong evidence of learning includes both correct results and defensible reasoning. You should be able to demonstrate:

Knowledge: equilibrium conditions, Newton's laws, kinematic relations, work-energy, impulse-momentum, rigid-body rotation, friction, truss assumptions, and introductory vibration concepts;
Modeling skill: selection of appropriate particles, rigid bodies, supports, coordinates, constraints, and system boundaries;
Representational skill: accurate free-body diagrams, kinematic diagrams, load diagrams, graphs, and clearly defined variables;
Analytical skill: correct use of equilibrium, Newton–Euler equations, energy methods, momentum methods, and mass moment of inertia;
Computational skill: organized symbolic or numerical calculations with consistent units and interpretable signs;
Experimental skill: acquisition and comparison of motion or force data with a mechanics model;
Communication products: a worked engineering analysis, a technical diagram, a short report or presentation, and a documented validation check;
Transfer achievement: the ability to recognize when a real situation is approximately static, clearly dynamic, or vibration-dominated and to justify the chosen analysis method.