Domain A — Basic Mechanics & Rigid-Body Kinematics

Vector kinematics in different coordinate systems, moving frames, and the transport theorem.

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Domain A: Vector Kinematics

Based on Chapter 1: Particle position description, angular velocity, moving frames, and the transport theorem.

1. Reference Frames and Vector Representation

The motion of a particle in space must be described relative to a reference frame.

A reference frame is fully defined by:

If a particle is located at a distance d from the origin of a reference frame R, its position vector can be written as a 3×1 column matrix whose entries are the scalar components along the frame's three unit vectors.

Key idea: different coordinate systems give different component values for the same vector. Before doing vector operations, always express all vectors in a common reference frame.

2. Coordinate Systems

2.1 Cartesian Coordinates

The Cartesian basis does not rotate, so its unit vectors î, ĵ, k̂ are constant in time.

2.2 Cylindrical Coordinates

Cylindrical coordinates are useful when a particle undergoes rotational motion, or when the dominant forces act radially.

The coordinate set is:

Two basis vectors lie in the plane perpendicular to the axis and rotate with angle θ; the third is fixed along the axis. For planar rotation, one of the three coordinates often remains zero.

Position Vector

\[ \mathbf{r} = r \,\hat{e}_r + z \,\hat{e}_z . \]

In full vector (Cartesian) form:

\[ \mathbf{r} = r \cos\theta \,\hat{\mathbf{i}} + r \sin\theta \,\hat{\mathbf{j}} + z \,\hat{\mathbf{k}} . \]

Unit Vectors

\[ \hat{e}_r = \cos\theta \,\hat{\mathbf{i}} + \sin\theta \,\hat{\mathbf{j}}, \quad \hat{e}_\theta = -\sin\theta \,\hat{\mathbf{i}} + \cos\theta \,\hat{\mathbf{j}}, \quad \hat{e}_z = \hat{\mathbf{k}} . \]

Velocity

\[ \mathbf{v} = \dot{r}\,\hat{e}_r + r \dot{\theta}\,\hat{e}_\theta + \dot{z}\,\hat{e}_z . \]

Acceleration

\[ \mathbf{a} = ( \ddot{r} - r \dot{\theta}^2 ) \hat{e}_r + ( r \ddot{\theta} + 2 \dot{r}\dot{\theta} ) \hat{e}_\theta + \ddot{z} \,\hat{e}_z . \]

2.3 Spherical Coordinates

Spherical coordinates are used when a particle moves on the surface of a sphere or orbits a central body.

The coordinate set is:

All three unit vectors depend on θ and ϕ and rotate as the particle moves.

Position Vector

\[ \mathbf{r} = \rho\,\hat{e}_\rho . \]

Expanded in Cartesian components:

\[ \mathbf{r} = \rho \sin\theta \cos\phi \,\hat{\mathbf{i}} + \rho \sin\theta \sin\phi \,\hat{\mathbf{j}} + \rho \cos\theta \,\hat{\mathbf{k}} . \]

Unit Vectors

\[ \begin{aligned} \hat{e}_\rho &= \sin\theta \cos\phi \,\hat{\mathbf{i}} + \sin\theta \sin\phi \,\hat{\mathbf{j}} + \cos\theta \,\hat{\mathbf{k}},\\[0.4em] \hat{e}_\theta &= \cos\theta \cos\phi \,\hat{\mathbf{i}} + \cos\theta \sin\phi \,\hat{\mathbf{j}} - \sin\theta \,\hat{\mathbf{k}},\\[0.4em] \hat{e}_\phi &= -\sin\phi \,\hat{\mathbf{i}} + \cos\phi \,\hat{\mathbf{j}} . \end{aligned} \]

Velocity

\[ \mathbf{v} = \dot{\rho}\,\hat{e}_\rho + \rho \dot{\theta}\,\hat{e}_\theta + \rho \sin\theta \,\dot{\phi}\,\hat{e}_\phi . \]

Acceleration

\[ \begin{aligned} \mathbf{a} =\,& ( \ddot{\rho} - \rho \dot{\theta}^2 - \rho \sin^2\theta \,\dot{\phi}^2 ) \hat{e}_\rho \\ &+ ( \rho \ddot{\theta} + 2 \dot{\rho}\dot{\theta} - \rho \sin\theta \cos\theta \,\dot{\phi}^2 ) \hat{e}_\theta \\ &+ ( \rho \sin\theta \,\ddot{\phi} + 2 \dot{\rho}\sin\theta \,\dot{\phi} + 2 \rho \cos\theta \,\dot{\theta}\dot{\phi} ) \hat{e}_\phi . \end{aligned} \]

These cylindrical and spherical expressions are standard in rigid-body, orbital, and spacecraft kinematics.

3. Angular Velocity Vector

The angular velocity vector describes how quickly the radius vector of a particle sweeps around an axis.

For a body rotating about a fixed axis with angle \( \theta(t) \), the angular velocity is

\[ \boldsymbol{\omega} = \dot{\theta}\,\hat{e}_{\text{axis}} . \]

The direction of \( \boldsymbol{\omega} \) follows the right-hand rule. Even if the magnitude of the radius is constant, a non-zero \( \boldsymbol{\omega} \) indicates rotational motion.

4. Kinematics in Moving Reference Frames

Consider two reference frames:

Let L be the vector from OU to OV, and let d and k be the position vectors of the particle from each origin. Then

\[ \mathbf{d} = \mathbf{L} + \mathbf{k}. \]

The same physical position can therefore be expressed seamlessly in either frame.

5. The Transport Theorem

Let \( N \) be an inertial frame and \( B \) a frame that rotates relative to \( N \) with angular velocity \( \boldsymbol{\omega}_{B/N} \). For any vector \( \mathbf{r} \) expressed in frame \( B \), the time derivatives are related by

\[ \left( \frac{d\mathbf{r}}{dt} \right)_N = \left( \frac{d\mathbf{r}}{dt} \right)_B + \boldsymbol{\omega}_{B/N} \times \mathbf{r}. \]

The derivative seen in the inertial frame equals the derivative seen in the rotating frame plus a correction due to the frame’s rotation. When applied to velocity, this term produces Coriolis, centripetal, and transverse contributions to the acceleration.

6. Velocity Relations Between Frames

Suppose the particle is observed in a moving frame V, whose origin moves relative to frame U with velocity \( \mathbf{v}_{O_V/U} \), and rotates with angular velocity \( \boldsymbol{\omega}_{V/U} \). Then the particle’s velocity in frame U is

\[ \mathbf{v}_{P/U} = \mathbf{v}_{O_V/U} + \left( \frac{d\mathbf{r}}{dt} \right)_V + \boldsymbol{\omega}_{V/U} \times \mathbf{r}. \]

where

7. Acceleration in a Moving Frame

Differentiating again and using the transport theorem gives the full acceleration expression:

\[ \mathbf{a}_{P/U} = \mathbf{a}_{O_V/U} + \left( \frac{d\mathbf{v}}{dt} \right)_V + 2\,\boldsymbol{\omega} \times \mathbf{v}_{\text{rel}} + \boldsymbol{\omega} \times ( \boldsymbol{\omega} \times \mathbf{r} ) + \dot{\boldsymbol{\omega}} \times \mathbf{r}. \]

which contains:

This is the complete kinematic description of motion in a rotating frame, and it underpins the dynamics of aircraft, spacecraft, and many robotic systems.

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Domain A.2 — Newtonian Mechanics →

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