Angular acceleration is the rate of change of
angular velocityIn physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
over
timeTime is a component of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects...
. In
SISi, si, or SI may refer to :- Places :* Mount Si, a mountain in state of Washington* Si County, county in Anhui, China* Si River, a river in China* Slovenia, a European nation Si, si, or SI may refer to (all SI unless otherwise stated):- Places :* Mount Si, a mountain in state of Washington* Si...
units, it is measured in
radianThe radian is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees, or about 57°17′45″. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level....
s per
secondThe second , sometimes abbreviated sec., is the name of a unit of time, and is the International System of Units base unit of time...
squared (rad/s
2), and is usually denoted by the Greek letter
alphaAlpha is the first letter of the Greek alphabet. In the system of Greek numerals it has a value of 1. It was derived from the Phoenician letter Aleph...
(
α).
The angular acceleration can be defined as either:
, or
,
where is the angular velocity, is the linear tangential acceleration, and r is the
radius of curvatureThe distance from the center of a sphere or ellipsoid to its surface is its radius. The equivalent "surface radius" that is described by radial distances at points along the body's surface is its radius of curvature .With a sphere, the radius of curvature equals the...
.
For
rotational motionA rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A three-dimensional object rotates around a line called an axis. If the axis of rotation is within the body, the body is said to rotate upon itself, or spin—which implies...
, Newton's second law can be adapted to describe the relation between
torqueTorque, also called moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
and angular acceleration:
,
where is the total torque exerted on the body, and is the mass
moment of inertiaMoment of inertia, also called mass moment of inertia or the angular mass, is a measure of an object's resistance to changes in its rotation rate. It is the rotational analog of mass. That is, it is the inertia of a rigid rotating body with respect to its rotation...
of the body.
For all constant values of the torque, , of an object, the angular acceleration will also be constant.
Discussion
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Angular acceleration is the rate of change of
angular velocityIn physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
over
timeTime is a component of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects...
. In
SISi, si, or SI may refer to :- Places :* Mount Si, a mountain in state of Washington* Si County, county in Anhui, China* Si River, a river in China* Slovenia, a European nation Si, si, or SI may refer to (all SI unless otherwise stated):- Places :* Mount Si, a mountain in state of Washington* Si...
units, it is measured in
radianThe radian is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees, or about 57°17′45″. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level....
s per
secondThe second , sometimes abbreviated sec., is the name of a unit of time, and is the International System of Units base unit of time...
squared (rad/s
2), and is usually denoted by the Greek letter
alphaAlpha is the first letter of the Greek alphabet. In the system of Greek numerals it has a value of 1. It was derived from the Phoenician letter Aleph...
(
α).
Mathematical definition
The angular acceleration can be defined as either:
, or
,
where is the angular velocity, is the linear tangential acceleration, and r is the
radius of curvatureThe distance from the center of a sphere or ellipsoid to its surface is its radius. The equivalent "surface radius" that is described by radial distances at points along the body's surface is its radius of curvature .With a sphere, the radius of curvature equals the...
.
Equations of motion
For
rotational motionA rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A three-dimensional object rotates around a line called an axis. If the axis of rotation is within the body, the body is said to rotate upon itself, or spin—which implies...
, Newton's second law can be adapted to describe the relation between
torqueTorque, also called moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
and angular acceleration:
,
where is the total torque exerted on the body, and is the mass
moment of inertiaMoment of inertia, also called mass moment of inertia or the angular mass, is a measure of an object's resistance to changes in its rotation rate. It is the rotational analog of mass. That is, it is the inertia of a rigid rotating body with respect to its rotation...
of the body.
Constant acceleration
For all constant values of the torque, , of an object, the angular acceleration will also be constant. For this special case of constant angular acceleration, the above equation will produce a definitive, constant value for the angular acceleration:
Non-constant acceleration
For any non-constant torque, the angular acceleration of an object will change with time. The equation becomes a differential equation instead of a constant value. This
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
is known as the equation of motion of the system and can completely describe the motion of the object. It is also the best way to calculate the angular velocity.
See also
- Angular momentum
Angular momentum is a quantity that is useful in describing the rotational state of a physical system. For a rigid body rotating around an axis of symmetry , the angular momentum can be expressed as the product of the body's moment of inertia and its angular velocity...
- Angular speed
- Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
- Rotation
A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A three-dimensional object rotates around a line called an axis. If the axis of rotation is within the body, the body is said to rotate upon itself, or spin—which implies...
- Spin
In particle physics and quantum mechanics, spin is a fundamental characteristic property of elementary particles including the force carriers , composite particles , and atomic nuclei....