Forces and Momentums

motion

Position

The position ($x$) is a vector quantity representing a point in space from a reference origin position.

Time Derivatives of motions

Displacement

The displacement ($s$ or $\Delta x$) is a vector quantity representing the change in position as a translation:

$$\begin{aligned} s = \Delta x = x_f - x_i \end{aligned}$$

Velocity

The velocity ($v$) is a vector quantity representing the change in displacement with time:

$$\begin{aligned} v = \frac{ds}{dt} \end{aligned}$$

Acceleration

The acceleration ($a$) is a vector quantity representing the change in velocity with time:

$$\begin{aligned} a = \frac{dv}{dt} \end{aligned}$$

Rigid Body

A rigid body is a solid with no deformation regardless of any external forces or moments applied.

Angular Motion

Unit vector for Radial Coordinates

The radial unit vector ($\bm{\hat{r}}$) is the directional vector with magnitude $1$ from the center radially outside.

$$\begin{aligned} \bm{\hat{r}} = \frac{\bm{r}}{|\bm{r}|} \end{aligned}$$

Centripetal acceleration

The centripetal acceleration ($a$) is the acceleration directed radially to the center ($-\bm{\hat{r}}$), keeping the body in circular motion:

$$\begin{aligned} a = -\frac{v^2}{r}\bm{\hat{r}} = -\omega^2 \bm{r}\bm{\hat{r}} = -\frac{4\pi^2r}{T^2}\bm{\hat{r}} \end{aligned}$$

warning

The IB only consider the magnitude of centripetal acceleration, therefore is shown in the equation booklet as:

$$\begin{aligned} a = \frac{v^2}{r} = \omega^2 r = \frac{4\pi^2r}{T^2} \end{aligned}$$

Centripetal Force

The centripetal force is any force applied on a mass towards the center ($-\bm{\hat{r}}$), leading to a centripetal acceleration $\bm{a}$:

$$\begin{aligned} F = m\bm{a} = -\frac{mv^2}{r}\bm{\hat{r}} = -m\omega^2 \bm{r}\bm{\hat{r}} = -\frac{4\pi^2mr}{T^2}\bm{\hat{r}} \end{aligned}$$

warning

As mentioned above, IB only consider the magnitude of centripetal force:

$$\begin{aligned} F = ma = \frac{mv^2}{r} = m\omega^2 r = \frac{4\pi^2mr}{T^2} \end{aligned}$$

Angular Position

The angular position ($\theta$) of a rigid body is a representation of the object's orientation by the angle between a reference position and the current position.

Time Derivatives of Angular Motions

Angular Displacement

The angular displacement ($\Delta \theta$) of a rigid body is the change in angular position $\theta$ measured from the center:

$$\begin{aligned} \Delta \theta = \theta_f - \theta_i \end{aligned}$$

Angular velocity

The angular velocity ($\omega$) is the change of angular displacement $\Delta \theta$ of a rigid body with time:

$$\begin{aligned} \omega = \frac{d\theta}{dt} \end{aligned}$$

Angular acceleration

The angular acceleration ($\alpha$) is the change of angular velocity of a rigid body with time:

$$\begin{aligned} \alpha = \frac{d\omega}{dt} \end{aligned}$$

Moment of Inertia

The measure of a solid body's resistance to angular acceleration.

warning

The IB syllabus only consider the moment of inertia of rigid body with approximated geometry as a system of discrete particles, as well as moment of inertia that is determined by closed-form expressions.

Angular Momentum

The angular momentum ($L$) is the product of moment of inertia $I$ and angular velocity $\omega$:

$$\begin{aligned} L = I\omega \end{aligned}$$

Work

The work ($W$) done by a force $F$ on a point mass with a movement of displacement $s$ is:

$$\begin{aligned} W = F \cdot s = |F||s|cos\theta, \quad \theta = \frac{F \cdot s}{|F||s|} \end{aligned}$$

info

For a force that varies at different position, the line integral of $F$ across a surface $C$ is:

$$\begin{aligned} W = \int_C F \cdot ds \end{aligned}$$