Rotational Motion
Description and Assumptions
This model applies to a rigid body which is executing pure rotation confined to the xy plane about the origin.
Problem Cues
Problems in rotational motion often feature an object which is constrained to rotate about some axle or pivot point. Additionally, the motion of any rigid body which can be treated using the Angular Momentum and External Torque about a Single Axis model can be described as translation of the center of mass plus pure rotation about the center of mass.
Learning Objectives
Students will be assumed to understand this model who can:
- Describe what it means for a system to execute pure rotation.
- Convert from tangential (linear) quantities to the corresponding angular quantities using the radius of the motion.
- Explain the dependence of angular quantities and of tangential quantities describing the motion of a point on the radius of the point from the axis of rotation.
- Define tangential and centripetal acceleration for an object in rotational motion.
- Relate centripetal acceleration to angular velocity.
- Give an expression for the total acceleration of any point in a rigid body executing rotational motion in terms of the angular acceleration of the body, the angular velocity of the body and the radius of the point from the axis of rotation.
- Summarize the analogies between angular motion with constant angular acceleration and linear motion with constant (linear) acceleration.
Model
Compatible Systems
This model applied to a single rigid body or to a single point particle constrained to move in a circular path.
Relevant Interactions
The system will be subject to a position-dependent centripetal acceleration, and may also be subject to an angular (or equivalently, tangential) acceleration.
Relevant Definitions
\begin
[ \vec
_
= \vec
\times \vec
= \omega r \;\hat
]
[ \vec
_
= \vec
\times \vec
= \alpha r \;\hat
]\end
\begin
[ \vec
_
= -\frac{v_
^{2}}
\hat
= -\omega^
r\;\hat
]\end
\begin
[ a = \sqrt{a_
^
+a_
{2}} = r\sqrt{\alpha
+\omega^{4}} ]\end
By definition, every point in an object undergoing pure rotation will have the same value for all angular quantities (θ, ω, α). The linear quantities (r, v, a), however, will vary with position in the object.
Laws of Change
\begin
[ \frac
= \alpha ]
[\frac
= \omega]
\end
\begin
[ \omega_
= \omega_
+\int_{t_{i}}^{t_{f}} \alpha \;dt]
[ \theta_
= \theta_
+\int_{t_{i}}^{t_{f}} \omega\;dt]\end
Note the analogy between these Laws of Change and those of the One-Dimensional Motion (General) model. Thus, for the case of constant angular acceleration, the integral form of these Laws are equivalent to:
\begin
[ \omega_
= \omega_
+ \alpha(t_
-t_
)]
[ \theta_
= \theta_
+ \frac
(\omega_
+\omega_
)(t_
-t_
)]
[ \theta_
= \theta_
+ \omega_
(t_
-t_
) +\frac
\alpha(t_
-t_
)^
]
[ \omega_
^
=\omega_
^
+ 2\alpha(\theta_
-\theta_
)]\end
Diagrammatic Representations
Consider, as a concrete example, the case where the angular velocity is constant, with θi = 0.5, ω = 5, and ti = 0. Then
\begin
[ \theta = \theta_
+ \omega t ]\end
and Angular Position as a function of time looks like this:
|
Angular Velocity as a function of time looks like this:
|
- Angular position versus time graph.
- Angular velocity versus time graph.
Relevant Examples
AllExamplesUsingtheModel"> All Examples Using the Model