Rotation around a fixed axis is a special case of
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
al motion. The fixed-
axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as
wobbling or
precession. According to
Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible; if two rotations are forced at the same time, a new axis of rotation will appear.
This article assumes that the rotation is also stable, such that no
torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
is required to keep it going. The
kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fiel ...
and
dynamics of rotation around a fixed axis of a rigid body are mathematically much simpler than those for
free rotation of a rigid body; they are entirely analogous to those of
linear motion along a single fixed direction, which is not true for ''free rotation of a rigid body''. The expressions for the
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
of the object, and for the forces on the parts of the object, are also simpler for rotation around a fixed axis, than for general rotational motion. 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.
Translation and rotation
A ''rigid body'' is an object of 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. A change in the position of a rigid body is more complicated to describe. It can be regarded as a combination of two distinct types of motion: translational motion and circular motion.
Purely ''
translational motion'' occurs when every particle of the body has the same instantaneous velocity as every other particle; then the path traced out by any particle is exactly parallel to the path traced out by every other particle in the body. Under translational motion, the change in the position of a rigid body is specified completely by three coordinates such as ''x'', ''y'', and ''z'' giving the
displacement of any point, such as the center of mass, fixed to the rigid body.
Purely ''rotational motion'' occurs if every particle in the body moves in a circle about a single line. This line is called the axis of rotation. Then the radius
vectors from the axis to all particles undergo the same angular displacement at the same time. The axis of rotation need not go through the body. 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.
Any displacement of a rigid body may be arrived at by first subjecting the body to a displacement followed by a rotation, or conversely, to a rotation followed by a displacement. 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
:
where ''M'' is the total mass of the system and ''a''
cm is the acceleration of the center of mass. There remains the matter of describing the rotation of the body about the center of mass and relating it to the external forces acting on the body. The kinematics and dynamics of ''rotational motion around a single axis'' resemble the kinematics and dynamics of translational motion; rotational motion around a single axis even has a work-energy theorem analogous to that of particle dynamics.
Kinematics
Angular displacement
Given a particle that moves along the circumference of a circle of radius
, having moved an arc length
, its angular position is
relative to its initial position, where
.
In mathematics and physics it is conventional to treat the
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
, a unit of plane angle, as 1, often omitting it. Units are converted as follows:
:
An angular displacement is a change in angular position:
:
where
is the angular displacement,
is the initial angular position and
is the final angular position.
Angular velocity
Change in angular displacement per unit time is called angular velocity with direction along the axis of rotation. The symbol for angular velocity is
and the units are typically rad s
−1. Angular speed is the magnitude of angular velocity.
:
The instantaneous angular velocity is given by
:
Using the formula for angular position and letting
, we have also
:
where
is the translational speed of the particle.
Angular velocity and
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
are related by
:
.
Angular acceleration
A changing angular velocity indicates the presence of an angular acceleration in rigid body, typically measured in rad s
−2. The average angular acceleration
over a time interval Δ''t'' is given by
:
The instantaneous acceleration ''α''(''t'') is given by
:
Thus, the angular acceleration is the rate of change of the angular velocity, just as acceleration is the rate of change of velocity.
The translational acceleration of a point on the object rotating is given by
:
where ''r'' is the radius or distance from the axis of rotation. This is also the
tangential component of acceleration: it is tangential to the direction of motion of the point. If this component is 0, the motion is
uniform circular motion, and the velocity changes in direction only.
The radial acceleration (perpendicular to direction of motion) is given by
:
.
It is directed towards the center of the rotational motion, and is often called the ''centripetal acceleration''.
The angular acceleration is caused by the
torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
, which can have a positive or negative value in accordance with the convention of positive and negative angular frequency. 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 moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...
:
.
Equations of kinematics
When the angular acceleration is constant, the five quantities angular displacement
, initial angular velocity
, final angular velocity
, angular acceleration
, and time
can be related by four
equations of kinematics:
:
Dynamics
Moment of inertia
The moment of inertia of an object, symbolized by
, is a measure of the object's resistance to changes to its rotation. The moment of inertia is measured in kilogram metre² (kg m
2). It depends on the object's mass: increasing the mass of an object increases the moment of inertia. It also depends on the distribution of the mass: distributing the mass further from the center of rotation increases the moment of inertia by a greater degree. For a single particle of mass
a distance
from the axis of rotation, the moment of inertia is given by
:
Torque
Torque
is the twisting effect of a force F applied to a rotating object which is at position r from its axis of rotation. Mathematically,
:
where × denotes the
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
. A net torque acting upon an object will produce an angular acceleration of the object according to
:
just as F = ''m''a in linear dynamics.
The work done by a torque acting on an object equals the magnitude of the torque times the angle through which the torque is applied:
:
The power of a torque is equal to the work done by the torque per unit time, hence:
:
Angular momentum
The angular momentum
is a measure of the difficulty of bringing a rotating object to rest. It is given by
:
for all particles in the object.
Angular momentum is the product of moment of inertia and angular velocity:
:
just as p = ''m''v in linear dynamics.
The analog of linear momentum in rotational motion is angular momentum. The greater the angular momentum of the spinning object such as a top, the greater its tendency to continue to spin.
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.
Like linear momentum, angular momentum is vector quantity, and its conservation implies that the direction of the spin axis tends to remain unchanged. For this reason, the spinning top remains upright whereas a stationary one falls over immediately.
The angular momentum equation can be used to relate the moment of the resultant force on a body about an axis (sometimes called torque), and the rate of rotation about that axis.
Torque and angular momentum are related according to
:
just as F = ''d''p/''dt'' in linear dynamics. 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
Figure skating is a sport in which individuals, pairs, or groups perform on figure skates on ice. It was the first winter sport to be included in the Olympic Games, when contested at the 1908 Olympics in London. The Olympic disciplines are me ...
: when pulling the arms closer to the body during a spin, the moment of inertia is decreased, and so the angular velocity is increased.
Kinetic energy
The kinetic energy
due to the rotation of the body is given by
:
just as
in linear dynamics.
Kinetic energy is the energy of motion. The amount of translational kinetic energy found in two variables: the mass of the object (
) and the speed of the object (
) as shown in the equation above. Kinetic energy must always be either zero or a positive value. While velocity can have either a positive or negative value, velocity squared will always be positive.
Vector expression
The above development is a special case of general rotational motion. In the general case, angular displacement, angular velocity, angular acceleration, and torque are considered to be vectors.
An angular displacement is considered to be a vector, pointing along the axis, of magnitude equal to that of
. A
right-hand rule is used to find which way it points along the axis; if the fingers of the right hand are curled to point in the way that the object has rotated, then the thumb of the right hand points in the direction of the vector.
The
angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an object ...
vector also points along the
axis of rotation in the same way as the angular displacements it causes. If a disk spins counterclockwise as seen from above, its angular velocity vector points upwards. Similarly, the
angular acceleration vector points along the axis of rotation in the same direction that the angular velocity would point if the angular acceleration were maintained for a long time.
The torque vector points along the axis around which the torque tends to cause rotation. To maintain rotation around a fixed axis, the total torque vector has to be along the axis, so that it only changes the magnitude and not the direction of the angular velocity vector. In the case of a hinge, only the component of the torque vector along the axis has an effect on the rotation, other forces and torques are compensated by the structure.
Examples and applications
Constant angular speed
The simplest case of rotation around a fixed axis is that of constant angular speed. Then the total torque is zero. For the example of the Earth rotating around its axis, there is very little friction. For a
fan, the motor applies a torque to compensate for friction. 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. The angle of rotation is a linear function of time, which modulo 360° is a periodic function.
An example of this is the
two-body problem with
circular orbits.
Centripetal force
Internal
tensile stress provides the
centripetal force that keeps a spinning object together. A
rigid body model neglects the accompanying
strain. If the body is not rigid this strain will cause it to change shape. This is expressed as the object changing shape due to the "
centrifugal force
In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
".
Celestial bodies rotating about each other often have
elliptic orbits. The special case of
circular orbits is an example of a rotation around a fixed axis: this axis is the line through the
center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
perpendicular to the plane of motion. The centripetal force is provided by
gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
, see also
two-body problem. 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. (It will, however, tend to become
oblate.) For example, a spinning celestial body of water must take at least 3 hours and 18 minutes to rotate, regardless of size, or the water will separate. If the density of the fluid is higher the time can be less. See
orbital period
The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting pla ...
.
See also
*
Anatomical terms of motion
*
Artificial gravity by rotation
*
Axle
An axle or axletree is a central shaft for a rotating wheel or gear. On wheeled vehicles, the axle may be fixed to the wheels, rotating with them, or fixed to the vehicle, with the wheels rotating around the axle. In the former case, beari ...
*
Axial precession
*
Axial tilt
In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and orb ...
*
Axis–angle representation
*
Carousel
A carousel or carrousel (mainly North American English), merry-go-round (international), roundabout (British English), or hurdy-gurdy (an old term in Australian English, in SA) is a type of amusement ride consisting of a rotating circular pl ...
,
Ferris wheel
*
Center pin
A bogie or railroad truck holds the wheel sets of a rail vehicle.
Axlebox
An ''axle box'', also known as a ''journal box'' in North America, is the mechanical subassembly on each end of the axles under a railway wagon, coach or locomotive; ...
*
Centrifugal force
In Newtonian mechanics, the centrifugal force is an inertial force (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
*
Centrifuge
A centrifuge is a device that uses centrifugal force to separate various components of a fluid. This is achieved by spinning the fluid at high speed within a container, thereby separating fluids of different densities (e.g. cream from milk) or ...
*
Centripetal force
*
Circular motion
*
Coriolis effect
*
Fictitious force
A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame.
It is related to Newton's second law of motion, which trea ...
*
Flywheel
A flywheel is a mechanical device which uses the conservation of angular momentum to store rotational energy; a form of kinetic energy proportional to the product of its moment of inertia and the square of its rotational speed. In particular, as ...
*
Gyration
*
Instant centre of rotation
*
Linear-rotational analogs
*
Revolutions per minute
Revolutions per minute (abbreviated rpm, RPM, rev/min, r/min, or with the notation min−1) is a unit of rotational speed or rotational frequency for rotating machines.
Standards
ISO 80000-3:2019 defines a unit of rotation as the dimensio ...
*
Optical axis
An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens, microscope or telescopic sight.
The optical axis is an imaginary line that defines the path along which light pro ...
*
Revolving door
A revolving door typically consists of three or four doors that hang on a central shaft and rotate around a vertical axis within a cylindrical enclosure. Revolving doors are energy efficient as they, acting as an airlock, prevent drafts, thus de ...
*
Rigid body angular momentum
*
Rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
:R = \begin
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\ ...
*
Rotational speed
*
Rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which ...
*
Run-out
*
Spin
References
*''
Fundamentals of Physics
''Fundamentals of Physics'' is a calculus-based physics textbook by David Halliday, Robert Resnick, and Jearl Walker. The textbook is currently in its 12th edition (published October, 2021).
The current version is a revised version of the ori ...
'' Extended 7th Edition by
Halliday,
Resnick and
Walker.
*''Concepts of Physics'' Volume 1, by
H. C. Verma, 1st edition, {{ISBN, 81-7709-187-5
Celestial mechanics
Euclidean symmetries
Rotation