![Rotating Sphere](https://upload.wikimedia.org/wikipedia/commons/0/02/Rotating_Sphere.gif)
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
An axis (plural ''axes'') is an imaginary line around which an object rotates or is symmetrical. Axis may also refer to:
Mathematics
* Axis of rotation: see rotation around a fixed axis
* Axis (mathematics), a designator for a Cartesian-coordinat ...
hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as
wobbling or
precession
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In othe ...
. According to
Euler's rotation theorem
In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed p ...
, 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 (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
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
Linear motion, also called rectilinear motion, is one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion, with co ...
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 accele ...
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
Linear motion, also called rectilinear motion, is one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion, with co ...
; the full generality of rotational motion is not usually taught in introductory physics classes.
Translation and rotation
![Worm Gear](https://upload.wikimedia.org/wikipedia/commons/c/c3/Worm_Gear.gif)
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
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every ...
'' 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
Displacement may refer to:
Physical sciences
Mathematics and Physics
* Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
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
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
s 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 c ...
, 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
In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the no ...
of acceleration: it is tangential to the direction of motion of the point. If this component is 0, the motion is
uniform circular motion
In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rot ...
, 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:
.
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 is ...
. 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
In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors.
Most of th ...
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 objec ...
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
In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with one another; the only force affecting each ...
with
circular orbit
A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.
Listed below is a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Here the centripetal force is ...
s.
Centripetal force
Internal
tensile stress
In continuum mechanics, stress is a physical quantity. It is a quantity that describes the magnitude of forces that cause deformation. Stress is defined as ''force per unit area''. When an object is pulled apart by a force it will cause elonga ...
provides the
centripetal force
A centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous c ...
that keeps a spinning object together. A
rigid body
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
model neglects the accompanying
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
. 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 orbit
In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, i ...
s. The special case of
circular orbit
A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.
Listed below is a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Here the centripetal force is ...
s 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
In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with one another; the only force affecting each ...
. 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 planets ...
.
See also
*
Anatomical terms of motion
Motion, the process of movement, is described using specific anatomical terms. Motion includes movement of organs, joints, limbs, and specific sections of the body. The terminology used describes this motion according to its direction relative ...
*
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, bearing ...
*
Axial precession
In astronomy, axial precession is a gravity-induced, slow, and continuous change in the orientation of an astronomical body's rotational axis. In the absence of precession, the astronomical body's orbit would show axial parallelism. In particu ...
*
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 orbi ...
*
Axis–angle representation
In mathematics, the axis–angle representation of a rotation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector indicating the direction of an axis of rotation, and an angle describing the magnitu ...
*
Carousel
A carousel or carrousel (mainly North American English), merry-go-round (List of sovereign states, international), roundabout (British English), or hurdy-gurdy (an old term in Australian English, in South Australia, SA) is a type of amusement ...
,
Ferris wheel
A Ferris wheel (also called a Giant Wheel or an observation wheel) is an amusement ride consisting of a rotating upright wheel with multiple passenger-carrying components (commonly referred to as passenger cars, cabins, tubs, gondolas, capsules ...
*
Center pin
A bogie or railroad truck holds the Wheelset (rail transport), 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 wa ...
*
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
A centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous c ...
*
Circular motion
In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of ro ...
*
Coriolis effect
In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...
*
Fictitious force
*
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, ass ...
*
Gyration
In geometry, a gyration is a rotation in a discrete subgroup of symmetries of the Euclidean plane such that the subgroup does not also contain a reflection symmetry whose axis passes through the center of rotational symmetry. In the orbifold co ...
*
Instant centre of rotation
The instant center of rotation (also, instantaneous velocity center, instantaneous center, or instant center) is the point fixed to a body undergoing planar movement that has zero velocity at a particular instant of time. At this instant, the vel ...
*
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 dimensionl ...
*
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 propaga ...
*
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
\en ...
*
Rotational speed
Rotational frequency (also known as rotational speed or rate of rotation) of an object Rotation around a fixed axis, rotating around an axis is the frequency of rotation of the object. Its unit is revolution per minute (rpm), cycle per second ...
*
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 i ...
*
Run-out
Run-out or runout is an inaccuracy of rotating mechanical systems, specifically that the tool or shaft does not rotate exactly in line with the main axis. For example; when drilling, run-out will result in a larger hole than the drill's nomina ...
*
Spin
References
*''
Fundamentals of Physics'' Extended 7th Edition by
Halliday
Halliday or Haliday is a surname. Notable people with the surname include:
*Alexander Henry Haliday (1807–1870), Irish entomologist
*Andrew Halliday (journalist) (1830–1877), British journalist and dramatist
*Andrew Halliday (physician) (1782& ...
,
Resnick and
Walker
Walker or The Walker may refer to:
People
* Walker (given name)
*Walker (surname)
* Walker (Brazilian footballer) (born 1982), Brazilian footballer
Places
In the United States
*Walker, Arizona, in Yavapai County
*Walker, Mono County, California ...
.
*''Concepts of Physics'' Volume 1, by
H. C. Verma
Harish Chandra Verma (born 03 April 1952), popularly known as hcv, is an Indian experimental physicist, author and emeritus professor of the Indian Institute of Technology Kanpur (IIT Kanpur). In 2021, he was awarded the Padma Shri, the fourth ...
, 1st edition, {{ISBN, 81-7709-187-5
Celestial mechanics
Euclidean symmetries
Rotation