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Interactive module First-year engineering dynamics

Cylindrical Coordinates

A crane slews, a robot arm extends, a collar slides along a spinning rod, a satellite swings round the Earth. These motions are driven or measured from a point or an axis, so they are simplest in polar and cylindrical coordinates \((r, \theta, z)\). Eight short lessons, with animated figures and questions that check your work, take you from a particle's position to its velocity, its acceleration and the forces that cause it.

Start Lesson 1

8 lessons
260 min of lessons
Works offline

A cylindrical robot turns its arm at a constant \(2\ \text{rad/s}\), extends it at \(0.5\ \text{m/s}\) and lifts it at \(0.3\ \text{m/s}\). Every joint runs steadily, yet the gripper accelerates: the red arrow \(\avec\) points toward the column and tilts forward, in the direction of rotation. Lesson 5 names the two parts, the centripetal and the Coriolis acceleration.

Learning outcomes

By the end of the module you can:

Recognize motions that are driven or measured from a point or an axis, and choose rectangular, path or cylindrical components for a problem. (Lessons 1, 8)
Locate a particle in polar and cylindrical coordinates, convert to and from \((x, y, z)\) with \(\theta\) in the correct quadrant, and write its position vector. (Lesson 2)
Explain how the unit vectors \(\er\) and \(\et\) turn as the particle moves, and resolve a force into radial and transverse components. (Lesson 3)
Find the velocity and acceleration of a particle from \(r\), \(\theta\), \(z\) and their rates, and interpret the centripetal and Coriolis terms. (Lessons 4–5)
Analyze a particle that follows a path \(r = f(\theta)\), such as a cam groove or a spiral, and find the angle \(\psi\) of its tangent. (Lesson 6)
Draw free-body and kinetic diagrams and apply \(\sum F_r = m a_r\), \(\sum F_\theta = m a_\theta\), \(\sum F_z = m a_z\) to find forces and accelerations. (Lesson 7)
Find the forces on a particle pushed along a curved guide, and use \(r^2\dot\theta = \text{constant}\) for central-force motion such as orbits. (Lesson 8)

How to use this module

Before you start: you should be comfortable with derivatives (the product and chain rules), radians and the sine and cosine of common angles, 2D vectors (components, magnitude and the dot product), and free-body diagrams with \(\sum\Fvec = m\avec\). You do not need to know polar coordinates already: Lesson 2 teaches what you need. The module uses the notation of Hibbeler's Engineering Mechanics: Dynamics; if your book writes \(\mathbf{e}_r\) and \(\mathbf{e}_\theta\), those are the same vectors as \(\er\) and \(\et\).

Work through the lessons in order. The times allow for working the examples on paper and doing the checks. A suggested plan in four sittings:

Lessons 1–380 min

Why polar and cylindrical coordinates, the position of a particle, and the unit vectors that turn with it.
Lessons 4–6100 min

The kinematics: velocity, acceleration with its centripetal and Coriolis terms, and particles guided along a path \(r = f(\theta)\).
Lessons 7–880 min

The kinetics: equations of motion with free-body diagrams, then guides, cords through a hole and orbits.
Practice Lab and Self-Check Quiz65 min

About 30 min of practice on the topics you found hardest, then the 35-minute quiz.

About 5½ h in total:260 min of lessons, 30 min of practice, 35 min for the quiz

Get the most out of it

Keep pencil and paper handy
Try each worked example yourself before you open its solution, and sketch the free-body diagrams.
Do the checks
Every lesson has “Check your understanding” questions with hints and full solutions. You can type answers like 3pi/4 or 2sqrt(3).
Practice until it is routine
The Practice Lab generates unlimited problems, each with a worked solution.
Test yourself at the end
The Self-Check Quiz gives a score, solutions and a results page you can print or save as a PDF.
Track your progress
Tick “Mark this lesson complete” at the end of each lesson. A ✓ appears here and in Contents; it is saved in this browser only.
Keep the formulas close
Print the Formula Sheet, and look up terms in the Glossary.

Lessons

Each lesson has interactive figures, worked examples and questions with instant feedback.

Practice, tools and reference

Use these alongside the lessons, or on their own when you revise.

Requirements

Any modern browser. Works offline.

A current version of Chrome, Edge, Firefox or Safari, with JavaScript on. A laptop or desktop screen works best; the pages also fit tablets and phones.
No internet connection, installation or account. Everything is inside this folder.
The 3D figures need WebGL (hardware acceleration), which is on by default in these browsers.
Your progress is saved in this browser on this device. Private or incognito windows do not keep it.

Something not working? Read README.txt in the module folder for how to open the module and fix common problems.