Kinematics of Rotational Motion — Revision Notes

⚡ 30-Second Revision

Angular Displacement — $heta$ (rad)
Angular Velocity — $omega = \frac{d\theta}{dt}$ (rad/s)
Angular Acceleration — $alpha = \frac{domega}{dt} = \frac{d^2\theta}{dt^2}$ (rad/s$^2$)
Kinematic Equations (constant $alpha$)

1. $omega = omega_0 + alpha t$ 2. $heta = omega_0 t + \frac{1}{2}alpha t^2$ 3. $omega^2 = omega_0^2 + 2alpha\theta$ 4. heta = left(\frac{omega_0 + omega}{2}\right)t

Linear-Angular Relations (at radius $r$)

- Arc length: $s = r\theta$ - Tangential velocity: $v_t = romega$ - Tangential acceleration: $a_t = ralpha$ - Centripetal acceleration: $a_c = romega^2 = \frac{v_t^2}{r}$

Conversions — $1,\text{rev} = 2pi,\text{rad}$, $1,\text{rpm} = \frac{2pi}{60},\text{rad/s}$

2-Minute Revision

Kinematics of Rotational Motion describes how rigid bodies rotate without considering the forces involved. The core concepts are angular displacement ($heta$), angular velocity ($omega$), and angular acceleration ($alpha$), which are direct analogs to linear displacement, velocity, and acceleration.

All angular quantities are measured in radians, rad/s, and rad/s$^2$ respectively. For constant angular acceleration, there are three main kinematic equations: $omega = omega_0 + alpha t$, $heta = omega_0 t + \frac{1}{2}alpha t^2$, and $omega^2 = omega_0^2 + 2alpha\theta$.

These equations are crucial for solving problems. Remember that linear quantities for a point on a rotating body are related to angular quantities by the radius $r$: $v_t = romega$ and $a_t = ralpha$.

Also, don't forget the centripetal acceleration $a_c = romega^2$, which is always present for circular motion and is directed towards the center. Always convert rpm to rad/s before calculations.

5-Minute Revision

To master rotational kinematics, focus on the fundamental definitions and their linear analogies. Angular displacement ($heta$) is the angle rotated, measured in radians. Angular velocity ($omega$) is the rate of change of $heta$, measured in rad/s, and angular acceleration ($alpha$) is the rate of change of $omega$, in rad/s$^2$. These are vector quantities, with direction along the axis of rotation determined by the right-hand rule.

For constant angular acceleration, the kinematic equations are your primary tools:

1
$omega = omega_0 + alpha t$ (Final angular velocity)
2
$heta = omega_0 t + \frac{1}{2}alpha t^2$ (Angular displacement)
3
$omega^2 = omega_0^2 + 2alpha\theta$ (Velocity-displacement relation)

Crucially, you must be able to relate linear motion of a point on the rotating body to its angular motion. For a point at radius $r$:

Linear distance $s = r\theta$
Tangential velocity $v_t = romega$
Tangential acceleration $a_t = ralpha$

Additionally, any object in circular motion experiences centripetal acceleration $a_c = romega^2 = v_t^2/r$, directed towards the center. The total linear acceleration of a point is the vector sum of $a_t$ and $a_c$. Always ensure units are consistent, especially converting rpm to rad/s ($1,\text{rpm} = \frac{2pi}{60},\text{rad/s}$). Practice problems involving these conversions and multi-stage motion (e.g., acceleration then constant velocity then deceleration) to solidify your understanding.

Prelims Revision Notes

1
Angular Variables — Define $heta$ (rad), $omega$ (rad/s), $alpha$ (rad/s$^2$). Remember their vector nature (right-hand rule, along axis of rotation).
2
Conversions — $1,\text{revolution} = 2pi,\text{radians}$. $1,\text{rpm} = \frac{2pi}{60},\text{rad/s}$. Always convert to radians for calculations.
3
Kinematic Equations (Constant $alpha$)

* $omega = omega_0 + alpha t$ * $heta = omega_0 t + \frac{1}{2}alpha t^2$ * $omega^2 = omega_0^2 + 2alpha\theta$ * heta = left(\frac{omega_0 + omega}{2}\right)t (useful for average angular velocity)

1
Linear-Angular Relationships (for a point at radius $r$)

* Arc length: $s = r\theta$ * Tangential velocity: $v_t = romega$. Direction is tangential to the circle. * Tangential acceleration: $a_t = ralpha$. Direction is tangential to the circle. * Centripetal acceleration: $a_c = romega^2 = \frac{v_t^2}{r}$. Direction is towards the center of the circle. * Total linear acceleration: $vec{a} = vec{a_t} + vec{a_c}$. Magnitude $a = sqrt{a_t^2 + a_c^2}$.

1
Rolling without slipping — For an object rolling without slipping, the point of contact with the ground is instantaneously at rest. The center of mass velocity $v_{CM} = Romega$ and acceleration $a_{CM} = Ralpha$, where $R$ is the radius of the rolling object.
2
Graphs — Understand analogies to linear motion graphs. Slope of $heta-t$ is $omega$. Slope of $omega-t$ is $alpha$. Area under $omega-t$ is $heta$.

Vyyuha Quick Recall

To remember the rotational kinematic equations, just recall the linear ones and swap variables:

Linear: Some Ugly Animals Trot Very Fast $s leftrightarrow \theta$ $u leftrightarrow omega_0$ $v leftrightarrow omega$ $a leftrightarrow alpha$ $t leftrightarrow t$

1
$v = u + at implies omega = omega_0 + alpha t$
2
$s = ut + \frac{1}{2}at^2 implies \theta = omega_0 t + \frac{1}{2}alpha t^2$
3
$v^2 = u^2 + 2as implies omega^2 = omega_0^2 + 2alpha\theta$