Lesson 1 · 20 min

Why Cylindrical Coordinates?

A crane slews and runs its trolley out, a collar slides along a spinning rod, a radar dish follows an aircraft. Each of these motions is measured from a fixed point or axis, so the natural description is a distance \(\colR{r}\), an angle \(\colT{\theta}\) and, in space, a height \(\colZ{z}\).

Learning objectives

Recognize motions that are driven or measured from a fixed point or axis: rotating arms, cranes, cams, radar tracking and orbits.
Describe what \(\colR{r}\), \(\colT{\theta}\) and \(\colZ{z}\) measure, using a tower crane as the example.
Compare rectangular, path and cylindrical descriptions of motion, and choose the one that suits a problem.
Explain why a particle can accelerate even when every joint of the machine moving it runs at a constant rate.

Motion measured from a point or an axis

In particle dynamics you have already described curvilinear motion with rectangular components (\(x\), \(y\)) and with path components (tangential and normal). A third description is best whenever the motion is driven or measured from a fixed point \(O\) or a fixed axis: you give the particle's distance \(r\) from \(O\) (or from the axis) and the angle \(\theta\) of the line from \(O\) to the particle. In a plane these are polar coordinates; add the height \(z\) along the axis and they become cylindrical coordinates.

Collar on a rotating rod

The rod sets \(\theta\); the collar slides along it, changing \(r\). Governors, centrifuges and the slotted arms of mechanisms work this way.

Cylindrical robot

Its three joints are a turning base (\(\theta\)), a lifting carriage (\(z\)) and a telescoping arm (\(r\)). Tower cranes work the same way.

Radar tracking

A radar measures the range \(r\) to an aircraft and the angle \(\theta\) of its line of sight, and how fast each one changes. Those are polar coordinates and their rates.

Slotted arm and cam groove

The groove fixes the path, \(r = f(\theta)\); the turning arm fixes \(\theta(t)\). Cams, indexing mechanisms and spiral feeders are built on this idea.

Orbits

Gravity always pulls toward the planet's center, so it has a radial component only. In \(r\) and \(\theta\) that single fact explains how orbits speed up near the planet (Lesson 8).

Spiral slides and ramps

A helix keeps \(r\) fixed while \(\theta\) grows and \(z\) falls steadily. Parking-garage ramps and spiral chutes are helices too.

In every case one or two of the three numbers stay constant or change in a simple way, while \(x\) and \(y\) change in a complicated way. That is the reason to use cylindrical coordinates: they turn the description of the motion, and later the equations of motion, into something simple.

A tower crane works in \((r, \theta, z)\)

A tower crane has three motions, and each one changes exactly one cylindrical coordinate. Put the origin at the foot of the mast, with the \(z\)-axis up the mast and the \(x\)-axis along a fixed reference direction on the site, so \(\theta = 0\) means the jib points along \(+x\). The slew motor turns the jib through the angle \(\colT{\theta}\). The trolley runs along the jib to the radius \(\colR{r}\). The hoist winds the rope to put the hook at the height \(\colZ{z}\). The hook is the point \((\colR{r},\ \colT{\theta},\ \colZ{z})\).

Figure 1.1 A tower crane with a 40 m jib (schematic; the hook is drawn oversize). The orange trolley sets \(r\), the violet slewing ring turns the jib through \(\theta\), and the blue hook block carries the hook \(P\) at height \(z\). The construction lines under the hook show all three. Turn on Trace path, then move one slider at a time: each control alone moves the hook along a radial line, a horizontal circle or a vertical line.

One control, one coordinate
Control	Changes	Hook path when only this control moves
Trolley	\(r\) only	A horizontal line straight out from the mast
Slew	\(\theta\) only	A horizontal circle around the mast
Hoist	\(z\) only	A vertical line

The Cartesian readout in Figure 1.1 uses \(x = r\cos\theta\) and \(y = r\sin\theta\), which Lesson 2 develops. Watch it while you slew: \(x\) and \(y\) both change, even though only one motor is running.

Example 1.1 — Slewing a load

A tower crane holds a pallet of bricks at a trolley radius of \(30\ \text{m}\), \(12\ \text{m}\) above the ground, with the jib at \(\theta = \tfrac{\pi}{6}\ (30^\circ)\). The operator slews at a steady rate to \(\theta = \tfrac{2\pi}{3}\ (120^\circ)\) in \(25\ \text{s}\), without moving the trolley or the hoist.

(a) Write the start and end positions in cylindrical and in Cartesian coordinates. (b) How far does the load travel? (c) What are the slew rate \(\dot\theta\) and the speed of the load?

Show solution

(a) In cylindrical coordinates (lengths in m) only \(\theta\) changes:

\[ \text{start } \left(30,\ \tfrac{\pi}{6},\ 12\right) \ \to\ \text{end } \left(30,\ \tfrac{2\pi}{3},\ 12\right) \]

In Cartesian coordinates, use \(x = r\cos\theta\) and \(y = r\sin\theta\):

\[ \begin{aligned} \text{start } &(30\cos\tfrac{\pi}{6},\ 30\sin\tfrac{\pi}{6},\ 12) = (15\sqrt{3},\ 15,\ 12) \approx (25.98,\ 15,\ 12) \\ \text{end } &(30\cos\tfrac{2\pi}{3},\ 30\sin\tfrac{2\pi}{3},\ 12) = (-15,\ 15\sqrt{3},\ 12) \approx (-15,\ 25.98,\ 12) \end{aligned} \]

\(x\) falls by about \(40.98\ \text{m}\) while \(y\) rises by about \(10.98\ \text{m}\): two coordinates change for a move that one motor makes.

(b) The load moves along a horizontal circle of radius \(30\ \text{m}\) through \(\Delta\theta = \tfrac{2\pi}{3} - \tfrac{\pi}{6} = \tfrac{\pi}{2}\). With the angle in radians, the arc length is

\[ s = r\,\Delta\theta = 30 \cdot \tfrac{\pi}{2} = 15\pi \approx 47.12\ \text{m} \]

(c) The slew rate is the angle turned per second, and the speed is the distance per second:

\[ \begin{aligned} \dot\theta &= \frac{\Delta\theta}{\Delta t} = \frac{\pi/2}{25} = \frac{\pi}{50} \approx 0.06283\ \text{rad/s} \\ v &= \frac{s}{\Delta t} = \frac{15\pi}{25} \approx 1.885\ \text{m/s} \end{aligned} \]

Notice that \(v = r\dot\theta = 30 \times 0.06283 \approx 1.885\ \text{m/s}\). The speed of going round is the radius times the angular rate. Lesson 4 shows that \(r\dot\theta\) is one of the two velocity components in polar coordinates, and why a second one, \(\dot r\), appears when the trolley moves too.

Three ways to describe the same motion

Each coordinate system has its own unit vectors, and the choice of system decides how simple the analysis is. This module uses the notation of Hibbeler's Engineering Mechanics: Dynamics: \(\ihat\) and \(\jhat\) for rectangular, \(\mathbf{u}_t\) and \(\mathbf{u}_n\) for path, and \(\er\), \(\et\) and \(\ez\) for cylindrical coordinates.

Rectangular \((x, y)\)

Fixed directions \(\ihat\) and \(\jhat\) that never turn. Best when the forces act in fixed directions, as for a projectile under gravity.

Path \((t, n)\)

\(\mathbf{u}_t\) along the path and \(\mathbf{u}_n\) toward its center of curvature. Best when the path is known and you care about speed along it, as for a car on a curve.

Polar and cylindrical \((r, \theta, z)\)

\(\er\) away from the axis, \(\et\) around it, \(\ez\) along it. Best when the motion is driven or measured from a fixed point or axis, as for a rotating arm or a radar.

All three describe the same velocity and acceleration vectors; only the components differ. The skill is to pick the system in which the given information and the forces look simplest.

Steady joints, yet the particle accelerates

Here is the surprise that drives the rest of the module. In Figure 1.2 a rod spins at a constant rate and a collar slides out along it at a constant speed. Neither motion changes its rate, yet the collar accelerates, and its acceleration is not even along the rod.

Figure 1.2 Top view of a rod (gray) spinning about \(O\) at a constant \(\dot\theta = 1\ \text{rad/s}\), with a collar \(P\) sliding out along it at a constant \(\dot r = 0.15\ \text{m/s}\). Press Play or drag \(P\) along its path. The green arrow is the velocity \(\vvec\) and the red arrow the acceleration \(\avec\). Their orange and violet components point along and across the rod. The acceleration has a part toward \(O\) and a part across the rod, although both rates are constant.

The reason is that the directions "along the rod" and "across the rod" turn with the rod. In the language of the next lessons, the unit vectors \(\er\) and \(\et\) change direction as \(\theta\) changes, and every change of a vector, including a change of direction, is an acceleration. Lesson 3 shows exactly how fast they turn, and Lesson 5 names the two parts you see here: the centripetal acceleration toward \(O\) and the Coriolis acceleration across the rod.

Module roadmap

The module follows the order of a first-year dynamics course: describe the position, then the velocity and acceleration (kinematics), then relate them to the forces (kinetics). Each lesson takes 20 to 40 minutes, including the examples and checks.

Describing position

Lesson 1: Why Cylindrical Coordinates?20 min · you are here

Motion measured from a point or an axis, and why \((r, \theta, z)\) fits it.
Lesson 2: Position in Polar and Cylindrical Coordinates30 min

\((r, \theta)\) and \((r, \theta, z)\), converting to and from \(x\), \(y\), and motion given as \(r(t)\), \(\theta(t)\), \(z(t)\).

Kinematics: velocity and acceleration

Lesson 3: Unit Vectors That Turn30 min

\(\er\), \(\et\), \(\ez\), their time derivatives, and radial and transverse components of a vector.
Lesson 4: Velocity30 min

\(\vvec = \dot r\,\er + r\dot\theta\,\et + \dot z\,\ez\): robot arms, radar tracking and speed.
Lesson 5: Acceleration35 min

The centripetal and Coriolis terms, and motions given as functions of time.
Lesson 6: Paths Given as \(r = f(\theta)\)35 min

Spirals, cams and circles: the chain rule for \(\dot r\) and \(\ddot r\), and the tangent angle \(\psi\).

Kinetics: forces and motion

Lesson 7: Equations of Motion40 min

\(\sum F_r = m a_r\), \(\sum F_\theta = m a_\theta\), \(\sum F_z = m a_z\), with free-body diagrams.
Lesson 8: Kinetics Applications40 min

Particles pushed along curved guides, and central-force motion such as orbits.

After the lessons, test yourself in the Practice Lab, the Self-Check Quiz and the Printable Worksheet. Experiment freely in the Motion Explorer, and keep the Formula Sheet and Glossary open while you work.

Check your understanding

Key takeaways

Polar coordinates \((r, \theta)\) give a particle's distance from a fixed point \(O\) and the angle of the line from \(O\) to the particle. Cylindrical coordinates add the height \(z\) along a fixed axis.
They suit motion that is driven or measured from a point or axis: rotating arms, cranes, cams and slotted guides, radar tracking and orbits.
A tower crane's trolley, slew and hoist each change exactly one of \(r\), \(\theta\) and \(z\).
The unit vectors \(\er\) and \(\et\) turn with the particle. That is why a collar on a steadily spinning rod accelerates even when both rates are constant.
Next, Lesson 2 sets up the position: polar and cylindrical coordinates, conversions, and motion given as functions of time.