Rotation around a fixed axis
This type of motion excludes the possibility of the instantaneous axis of rotation changing its orientation and cannot describe such phenomena as wobbling or precession.
For these reasons, rotation around a fixed axis is typically taught in introductory physics courses after students have mastered linear motion; the full generality of rotational motion is not usually taught in introductory physics classes.
A rigid body is an object of a finite extent in which all the distances between the component particles are constant.
No truly rigid body exists; external forces can deform any solid.
For our purposes, then, a rigid body is a solid which requires large forces to deform it appreciably.
A change in the position of a particle in three-dimensional space can be completely specified by three coordinates.
Purely rotational motion occurs if every particle in the body moves in a circle about a single line.
Then the radius vectors from the axis to all particles undergo the same angular displacement at the same time.
In general, any rotation can be specified completely by the three angular displacements with respect to the rectangular-coordinate axes x, y, and z.
Any change in the position of the rigid body is thus completely described by three translational and three rotational coordinates.
We already know that for any collection of particles—whether at rest with respect to one another, as in a rigid body, or in relative motion, like the exploding fragments of a shell, the acceleration of the center of mass is given by
In mathematics and physics it is conventional to treat the radian, a unit of plane angle, as 1, often omitting it.
The relationship between torque and angular acceleration (how difficult it is to start, stop, or otherwise change rotation) is given by the moment of inertia:
The angular momentum of a rotating body is proportional to its mass and to how rapidly it is turning.
In addition, the angular momentum depends on how the mass is distributed relative to the axis of rotation: the further away the mass is located from the axis of rotation, the greater the angular momentum.
A flat disk such as a record turntable has less angular momentum than a hollow cylinder of the same mass and velocity of rotation.
For this reason, the spinning top remains upright whereas a stationary one falls over immediately.
In the absence of an external torque, the angular momentum of a body remains constant.
The conservation of angular momentum is notably demonstrated in figure skating: when pulling the arms closer to the body during a spin, the moment of inertia is decreased, and so the angular velocity is increased.
The amount of translational kinetic energy found in two variables: the mass of the object (
An angular displacement is considered to be a vector, pointing along the axis, of magnitude equal to that of
If a disk spins counterclockwise as seen from above, its angular velocity vector points upwards.
Only two numbers, not three, are needed to define the direction of a unit vector e rooted at the origin because the magnitude of e is constrained.
For example, the elevation and azimuth angles of e suffice to locate it in any particular Cartesian coordinate frame.
The simplest case of rotation not around a fixed axis is that of constant angular speed.
Similar to the fan, equipment found in the mass production manufacturing industry demonstrate rotation around a fixed axis effectively.
For example, a multi-spindle lathe is used to rotate the material on its axis to effectively increase the productivity of cutting, deformation and turning operations.
Internal tensile stress provides the centripetal force that keeps a spinning object together.
This is expressed as the object changing shape due to the "centrifugal force".
This usually also applies for a spinning celestial body, so it need not be solid to keep together unless the angular speed is too high in relation to its density.
