Rotational Kinematics

Summary

Rotational motion is a type of motion where an object experiences angular displacement, velocity and/or acceleration

We can break the motion into linear and tangential motion.

Key equations:

General equations:
$Δ θ = ω Δ t$
$Δ ω = α Δ t$

Read Time

⏱ 2 mins

Definition

Rotational kinematics describes the motion of angular displacement, velocity and acceleration. This is used to describe an unknown function (like angular displacement), knowing one more of its derivatives (angular velocity/acceleration). Rotational kinematics assumes the velocity or acceleration is always constant from point a to b or can be accurately described as the average of that function.

Deriving Rotational Kinematics

Assumptions

Assume the angular velocity and acceleration are always constant from time $t_{1}$ to $t_{2}$ . As well assume the following:

Angular velocity is defined as $ω = \frac{d θ}{d t}$
Angular acceleration is defined as $α = \frac{d ω}{d t}$
Angular displacement ( $θ$ ) is the area under angular velocity
Angular velocity is the area under angular acceleration

If the goal was to find the change in displacement know angular velocity over $t_{1}, t_{2}$

\begin{matrix} d t (\frac{d θ}{d t} = ω) \\ d θ = ω d t \\ \int_{θ (t_{1})}^{θ (t_{2})} d θ = \int_{t_{1}}^{t_{2}} ω d t \\ Δ θ = ω \int_{t_{1}}^{t_{2}} d t \\ Δ θ = ω Δ t \end{matrix}

If the goal was to find the change in angular velocity knowing angular acceleration over
$t_{1}, t_{2}$

\begin{matrix} d t (\frac{d ω}{d t} = α) \\ \int_{ω (t_{1})}^{ω (t_{2})} d ω = \int_{t_{1}}^{t_{2}} α d t \\ Δ ω = α Δ t \end{matrix}

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