Radians per second squared |
Unit system |
SI derived unit |
Unit of |
Angular acceleration |
Symbol |
rad/s2 or rad⋅s−2 |
Classical mechanics |
Second law of motion
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Branches
- Applied
- Celestial
- Continuum
- Dynamics
- Kinematics
- Kinetics
- Statics
- Statistical
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Fundamentals
- Acceleration
- Angular momentum
- Couple
- D'Alembert's principle
- Energy
- Force
- Frame of reference
- Impulse
- Inertia / Moment of inertia
- Mass
Mechanical power
- Mechanical work
Moment
- Momentum
- Space
- Speed
- Time
- Torque
- Velocity
- Virtual work
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Formulations
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Newton's laws of motion
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Analytical mechanics
- Lagrangian mechanics
- Hamiltonian mechanics
- Routhian mechanics
- Hamilton–Jacobi equation
- Appell's equation of motion
- Udwadia–Kalaba equation
- Koopman–von Neumann mechanics
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Core topics
- Damping (ratio)
- Displacement
- Equations of motion
- Euler's laws of motion
- Fictitious force
- Friction
- Harmonic oscillator
- Inertial / Non-inertial reference frame
- Mechanics of planar particle motion
- Motion (linear)
- Newton's law of universal gravitation
- Newton's laws of motion
- Relative velocity
- Rigid body
- dynamics
- Euler's equations
- Simple harmonic motion
- Vibration
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Rotation
- Circular motion
- Rotating reference frame
- Centripetal force
- Centrifugal force
- Coriolis force
- Pendulum
- Tangential speed
- Rotational speed
- Angular acceleration / displacement / frequency / velocity
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Scientists
- Galileo
- Newton
- Kepler
- Horrocks
- Halley
- Euler
- d'Alembert
- Clairaut
- Lagrange
- Laplace
- Hamilton
- Poisson
- Daniel Bernoulli
- Johann Bernoulli
- Cauchy
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Angular acceleration is the rate of change of angular velocity. In SI units, it is measured in radians per second squared (rad/s2), and is usually denoted by the Greek letter alpha (α).[1]
Contents
- 1 Mathematical definition
- 2 Equations of motion
- 2.1 Constant acceleration
- 2.2 Non-constant acceleration
- 2.3 Relationship with Angular Momentum
- 3 See also
- 4 References
Mathematical definition
The angular acceleration can be defined as either:
- , or
- ,
where is the angular velocity, is the linear tangential acceleration, and , (usually defined as the radius of the circular path of which a point moving along), is the distance from the origin of the coordinate system that defines and to the point of interest.
Equations of motion
For two-dimensional rotational motion (constant ), Newton's second law can be adapted to describe the relation between torque and angular acceleration:
- ,
where is the total torque exerted on the body, and is the mass moment of inertia 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 equation 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.
Relationship with Angular Momentum
Angular acceleration can be obtained from angular momentum using the relation,[2]
Above relationship indicates that even when there is no change in angular momentum (i.e. no torques are being applied), the angular acceleration can still be non-zero. In fact, this will happen whenever the angular momentum and angular velocity point in different directions (i.e. rotational velocity axis is not the axis of symmetry).
See also
- Angular momentum
- Angular speed
- Angular velocity
- Rotation
- Spin
References
- ^ "Angular Velocity and Acceleration". Theory.uwinnipeg.ca. Retrieved 2015-04-13.
- ^ "An Introduction to Physically-Based Modeling" (PDF).
Classical mechanics SI units
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Linear/translational quantities |
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Angular/rotational quantities |
Dimensions |
1 |
L |
L2 |
Dimensions |
1 |
1 |
1 |
T |
time: t
s |
absement: A
m s |
|
T |
time: t
s |
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1 |
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distance: d, position: r, s, x, displacement
m |
area: A
m2 |
1 |
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angle: θ, angular displacement: θ
rad |
solid angle: Ω
rad2, sr |
T−1 |
frequency: f
s−1, Hz |
speed: v, velocity: v
m s−1 |
kinematic viscosity: ν,
specific angular momentum: h
m2 s−1 |
T−1 |
frequency: f
s−1, Hz |
angular speed: ω, angular velocity: ω
rad s−1 |
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T−2 |
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acceleration: a
m s−2 |
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T−2 |
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angular acceleration: α
rad s−2 |
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T−3 |
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jerk: j
m s−3 |
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T−3 |
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angular jerk: ζ
rad s−3 |
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M |
mass: m
kg |
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ML2 |
moment of inertia: I
kg m2 |
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MT−1 |
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momentum: p, impulse: J
kg m s−1, N s |
action: 𝒮, actergy: ℵ
kg m2 s−1, J s |
ML2T−1 |
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angular momentum: L, angular impulse: ΔL
kg m2 s−1 |
action: 𝒮, actergy: ℵ
kg m2 s−1, J s |
MT−2 |
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force: F, weight: Fg
kg m s−2, N |
energy: E, work: W
kg m2 s−2, J |
ML2T−2 |
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torque: τ, moment: M
kg m2 s−2, N m |
energy: E, work: W
kg m2 s−2, J |
MT−3 |
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yank: Y
kg m s−3, N s−1 |
power: P
kg m2 s−3, W |
ML2T−3 |
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rotatum: P
kg m2 s−3, N m s−1 |
power: P
kg m2 s−3, W |
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