Curvilinear Motion

Curvilinear motion describes a particle travelling along a curved path in space rather than a straight line, as seen in rectilinear motion. Unlike rotation about a fixed axis, there is no rigid body and no stationary axis.

Even with a constant speed, the direction of the velocity vector is constantly changing, which means acceleration is always present.

Coordinate Systems

Curvilinear motion can be described in a number of coordinate systems depending on how the path is described or constrained. A coordinate system can be chosen in order to simplify a problem.

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Vector Definitions

Curvilinear motion is described using vectors as we need to account for the magnitude and direction of curved motion.

Kinematic Quantity	Vector Notation	Derivative Definition
Position	$\vec{r}(t)=x\hat{i}+y\hat{j}+z\hat{k}$	—
Velocity	$\vec{v}(t)=v_x\hat{i}+v_y\hat{j}+v_z\hat{k}$	$\vec{v}=\frac{d\vec{r}}{dt}=\dot{\vec{r}}$
Acceleration	$\vec{a}(t)=a_x\hat{i}+a_y\hat{j}+a_z\hat{k}$	$\vec{a}=\frac{d\vec{v}}{dt}=\frac{d^2\vec{r}}{d{t}^2}=\ddot{\vec{r}}$

The magnitude of velocity (speed) is given by:
$|\vec{v}|=\sqrt{v_x^2+v_y^2+v_z^2}$

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Rectangular Coordinates ($x,y,z$)

Rectangular coordinates (also known as cartesian coordinates) are best used when paths are independent along perpendicular axes. Motion can be split into $x,y,z$ directions and be treated as 1D in each respective axis.

Quote bq

The reference axes do not change direction over time.
Unit vectors ($\hat{i},\hat{j},\hat{k}$) are constant in magnitude and direction.

Each kinematic quantity is split into components for each direction.
$a_x=\ddot{x},a_y=\ddot{y}$

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Normal & Tangential Coordinates ($n,t$)

Normal and tangential coordinates are best used when the physical path of motion is known. Motion is split into normal & tangential components.

The unit vectors (${\hat{u}}_n,{\hat{u}}_t$) move with the particle. ${\hat{u}}_n$ points towards the centre of curvature ($\rho$) and ${\hat{u}}_t$ is tangent to the path. The reference frame constantly moves and rotates with the particle.

Radius of Curvature

The centre of curvature ($O^{\prime }$) is an imaginary point that lies on the concave side of the curve. The $n$-axis has positive direction towards $O^{\prime }$

The radius of curvature ($\rho$) is perpendicular distance from curve to $O^{\prime }$.

An advantage of this coordinate system is that it isolates speed changes from the change in direction of the path. $a_t$ is known as tangential acceleration and $a_n$ is known as centripetal acceleration.

Parameter	Formula	Description
Position (scalar)	$s$	Distance along the curve from a fixed reference point.
Velocity Vector	$\vec{v}=v{\hat{u}}_t$	Tangent to path, no normal velocity.
Total Acceleration	$\vec{a}=a_t{\hat{u}}_t+a_n{\hat{u}}_n$	Combined vector tracking changes in both speed and path direction.
Tangential Acceleration	$a_t=\frac{dv}{dt}=\dot{v}$	Parallel to travel; purely responsible for changing the object’s speed.
Normal Acceleration	$a_n=\frac{v^2}{\rho }$	Perpendicular to travel; purely responsible for changing direction.

Path Angle

The path angle ($\beta$) is the angle the tangent vector makes with a fixed horizontal axis. Its time-derivative ($\dot{\beta }$) represents the angular velocity of the velocity vector (how fast the particle is turning).

Particle’s speed along a curve is directly proportional to how sharply the curve bends ($\rho$) and how fast its direction changes ($\dot{\beta }$).
$v=\rho \dot{\beta }$
Normal acceleration can also be written in terms of the path angle:
$a_n=\rho {\dot{\beta }}^2=v\dot{\beta }$
Path angle formulas are useful for when we need to link path geometry and/or angular rotation to acceleration.

Special Cases of Motion

1. Particle moves along a straight line
   $\rho \to \infty ⟹a_n=0⟹a=\dot{v}$
2. Particle moves with constant speed along curve
   $a_t=0⟹a=\frac{v^2}{\rho }$
3. Tangential acceleration ($a_t$) is constant
   $\begin{array}{rl}s& =s_o+v_{o}t+\frac{1}{2}(a_t{)}_c{t}^2\\ v& =v_o+(a_t{)}_{c}t\\ v^2& =v_o^2+2(a_t{)}_c(s-s_o)\end{array}$
4. Particle moves along path $y=f(x)$
   $\rho =\frac{[1+(dy/dx{)}^2{]}^{3/2}}{|d^{2}y/d{x}^2|}$

Three-Dimensional Normal & Tangential Motion

In 2D, the normal and tangential axes lie together in what is known as the osculating plane. When modelling curvilinear motion of a particle in 3D, we use a binormal axis ($b$) to represent the third dimension.

The binormal axis is perpendicular to the osculating plane. Its unit vector $({\hat{u}}_b)$ is given by the cross product of the other two unit vectors:
${\hat{u}}_b={\hat{u}}_t\times {\hat{u}}_n$

There is no motion, velocity or acceleration in the binormal axis as the reference frame will instead rotate to account for change in direction.

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Polar & Cylindrical Coordinates ($r,\theta ,z$)

Polar coordinates (2D) and cylindrical coordinates (3D) are best used when the motion of a particle is tracked from a central fixed point (origin/pole) rather than along its own path.

The unit vectors for polar and cylindrical coordinates are attached to the particle. ${\hat{u}}_r$ points radially outward from $O$ to the particle. $\widehat{u_{\theta }}$ is perpendicular to ${\hat{u}}_r$ in the direction of increasing $\theta$. As these terms rotate with the particle, differentiation produces additional terms for acceleration.

Unit Vector Notation

Traditionally, $\hat{e}$ is used for the notation of unit vectors. However, you will find most notation in this vault will use $\hat{u}$ instead, with $u$ for unit.

Parameter	Formula	Description
Radius	$\vec{r}=r{\hat{u}}_r$	Distance from $O$ to particle.
Radial Velocity	$v_r=\dot{r}$	Rate of motion towards/away from $O$.
Transverse Velocity	$v_{\theta }=r\dot{\theta }$	Rate of motion $\perp$ to $r$, due to rotation.
Radial Acceleration	$a_r=\ddot{r}-r{\dot{\theta }}^2$	Linear + centripetal acceleration.
Transverse Acceleration	$a_{\theta }=r\ddot{\theta }+2\dot{r}\dot{\theta }$	Tangential + Coriolis term.

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Combining components above, we get total velocity and acceleration:
$\vec{v}=\dot{r}\widehat{u_r}+r\dot{\theta }\widehat{u_{\theta }}$
$\vec{a}=(\ddot{r}-r{\dot{\theta }}^2)\widehat{u_r}+(r\ddot{\theta }+2\dot{r}\dot{\theta })\widehat{u_{\theta }}$

Special Cases -- Fixed Axis Rotation

If $r$ is constant -> $\dot{r}=\ddot{r}=0$.
Equations simplify $(\dot{\theta }=\omega ,\ddot{\theta }=\alpha )$:

$a_r=-r{\omega }^2{a}_{\theta }=r\alpha$

Polar coordinates are a more general representation that reduces to fixed axis rotation when $r$ stops changing.

$a_r\to a_n,a_{\theta }\to a_t$

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Cylindrical Coordinates (3D)

Cylindrical coordinates are an expansion of polar coordinates into 3D by adding a $z$-axis. The $z$-axis is treated the same way as rectangular $z$, meaning it is independent and non-rotation.

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