Single-Axis Rotation of a Rigid Body
Description and Assumptions
This model is applicable to a single rigid body that is rotating about a fixed axis or else both rotating and translating in such a way that its angular momentum is a one-dimensional vector (usually taken to lie along the z-axis). As a special case, this model is also useful to constrain the forces acting on a static rigid body. It is a subclass of the Angular Momentum and External Torque about a Single Axis model defined by the constraint that the system consists of only one rigid body which has a fixed mass and a fixed moment of inertia for rotations about the fixed axis or its center of mass.
Problem Cues
This model is useful for a stationary object (the special case of statics). In that case, both the linear acceleration a and the angular acceleration α are zero, and there is the additional freedom that the axis can be placed at any point in the object. For accelerating objects, the model is commonly used in cases where a single object is placed in a situation where the forces are well understood, such as a cylinder rolling down an inclined plane or a sphere rolling along level ground. Often, the linear and angular accelerations will be related by the rolling without slipping condition.
Learning Objectives
Students will be assumed to understand this model who can:
- State appropriate choices for the axis of rotation for cases of a static object, an object rotating about a fixed axle, and an object that is both rotating and translating.
- Calculate the moment of inertia of a rigid body constructed of simple shapes like spheres and cylinders about any specified axis of rotation.
- Compute the torque resulting from a force applied at a known position about an arbitrary axis of rotation.
- Relate the net torque acting on a rigid body to its angular acceleration.
Model
Compatible Systems
One rigid body whose moment of inertia calculated about the chosen axis of rotation is constant.
Relevant Interactions
Only external torques need be considered, as internal torques do not produce angular accelerations. Since torque is needed, forces must be specified not only by their magnitude and direction, but also by either their point of application or moment arm with respect to the axis of rotation. Allowed choices of the axis of rotation will depend upon the specific
circumstances of the problem:
- If the body is static, the axis of rotation may be placed at any location.
- If the body is rotating about a fixed axis, the axis of rotation must be chosen to coincide with that axis.
- If the body is free to translate as well as rotate, the axis must pass through the body's center of mass.
Laws of Change
\begin
[ \sum \tau_
= I_
\alpha]\end
Diagrammatic Representations
Relevant Examples
ExamplesInvolvingaFixedAxis"> Examples Involving a Fixed Axis
ExamplesInvolvingaRotatingandTranslatingRigidBody"> Examples Involving a Rotating and Translating Rigid Body
ExamplesInvolvingaStaticRigidBody"> Examples Involving a Static Rigid Body
AllExamplesUsingthisModel"> All Examples Using this Model
Photos courtesy:
Wikimedia Commons
by Johann H. Addicks
U.S. Navy
by Mass Communications Specialist 3rd Class Walter M. Wayman