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Angular velocity

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	Angular velocity
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	Topics Angular velocity Topic Home In physicsPhysics Physics , the most fundamental physical science, is concerned with the underlying principles of the natural world.... , the angular velocity is a vectorVector (spatial) In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a... quantity (more precisely, a pseudovectorPseudovector Summary In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but gains an... ) which specifies the angular speed, and axis about which an object is rotating. The SISi Si, si, or SI may stand for:... unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, degrees per hour, etc. When measured in cycles or rotations per unit time (e.g. revolutions per minuteRevolutions per minute Examples* The sweep of a second hand of an analogue clock or watch rotates at an average of 1 rpm.... ), it is often called the rotational velocity and its magnitude the rotational speedRotational speed Rotational speed indicates, for example, how fast a motor is running.... . Angular velocity is usually represented by the symbol omegaOmega Omega is the 24th and last letter of the Greek alphabet.... (O or ?). The direction of the angular velocity vector is perpendicular to the plane of rotation, in a direction which is usually specified by the right hand rule. The angular velocity of a particle Two dimensions The angular velocity of a particle in a 2-dimensional plane is the easiest to understand. As shown in the figure on the right (typically expressing the angular measures f and ? in radianRadian The radian is a unit of plane angle.... s), if we draw a line from the origin (O) to the particle (P), then the velocity vector () of the particle will have a component along the radius ( - the radial component) and a component perpendicular to the radius ( - the tangential component). A radial motion produces no rotation of the particle (relative to the origin), so for purposes of finding the angular velocity the parallel (radial) component can be ignored. Therefore, the rotation is completely produced by the tangential motion (like that of a particle moving along a circumference), and the angular velocity is completely determined by the perpendicular (tangential) component. It can be seen that the rate of change of the angular position of the particle is related to the tangential velocity by: Utilizing ?, the angle between vectors ' and v, or equivalently as the angle between vectors r and v, gives: Combining the above two equations and defining the angular velocity as ?=df/dt yields: In two dimensions the angular velocity is a single number which has no direction. A single number which has no direction is either a scalarScalar A scalar may be:* A scalar , a number as opposed to a vector, function, or other object... or a pseudoscalarPseudoscalar In mathematics and physics, a pseudoscalar is a quantity that behaves more or less like a scalar, except that it transforms ... , the difference being that a scalar does not change its sign when the x and y axes are exchanged (or inverted), while a pseudoscalar does. The angle as well as the angular velocity is a pseudoscalar. The positive direction of rotation is taken, by convention, to be in the direction towards the y axis from the x axis. If the axes are inverted, but the sense of a rotation does not, then the sign of the angle of rotation, and therefore the angular velocity as well, will change. It is important to note that the pseudoscalar angular velocity of a particle depends upon the choice of the origin and upon the orientation of the coordinate axes. The equation allows the angular velocity to be found. Three dimensions In three dimensions, the angular velocity becomes a bit more complicated. The angular velocity in this case is generally thought of as a vector, or more precisely, a pseudovectorPseudovector In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but gains an... . It now has not only a magnitude, but a direction as well. The magnitude is the angular speed, and the direction describes the axis of rotation. The right hand rule indicates the positive direction of the angular velocity pseudovector, namely: If you curl the fingers of your right hand to follow the direction of the rotation, then the direction of the angular velocity vector is indicated by your right thumb. Just as in the two dimensional case, a particle will have a component of its velocity along the radius from the origin to the particle, and another component perpendicular to that radius. The combination of the origin point and the perpendicular component of the velocity defines a plane of rotation in which the behavior of the particle (for that instant) appears just as it does in the two dimensional case. The axis of rotation is then a line perpendicular to this plane, and this axis defined the direction of the angular velocity pseudovector, while the magnitude is the same as the pseudoscalar value found in the 2-dimensional case. Define a unit vector which points in the direction of the angular velocity pseudovector. The angular velocity may be written in a manner similar to that for two dimensions: which, by the definition of the cross productCross product Overview In mathematics, the cross product is a binary operation on vectors in a three-dimensional Euclidean space.... , can be written: Higher dimensions In general, the angular velocity in an n-dimensional space is the time derivative of the angular displacement tensor which is a second rank skew-symmetric tensor. This tensor will have n(n-1)/2 independent components and this number is the dimension of the Lie algebraLie algebra In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups a... of the Lie groupLie group In mathematics, a Lie group is a continuous group, in the sense that the group elements have the topology of a manifold, an... of rotations of an n-dimensional inner product space. It turns out that in three dimensional space angular velocity can be represented by vector because number of independent components is equal to number of dimensions of space. Angular velocity of a rigid body In order to deal with the motion of a rigid bodyRigid body Overview In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected.... , it is best to consider a coordinate system that is fixed with respect to the rigid body, and to study the coordinate transformations between this coordinate and the fixed "laboratory" system. As shown in the figure on the right, the lab system's origin is at point O, the rigid body system origin is at O' and the vector from O to O' is R. A particle (i) in the rigid body is located at point P and the vector position of this particle is R_i in the lab frame, and at position r_i in the body frame. It is seen that the position of the particle can be written: The defining characteristic of a rigid body is that the distance between any two points in a rigid body is unchanging in time. This means that the length of the vector is unchanging. By Euler's rotation theoremEuler's rotation theorem Euler's rotation theorem states that, in 3D space, for any two coordinate systems with a common origin, there is a single ei... , we may replace the vector with where is a rotation matrixRotation matrix A rotation matrix is a matrix which when multiplied by a vector has the effect of changing the direction of the vector but n... and is the position of the particle at some fixed point in time, say t=0. This replacement is useful, because now it is only the rotation matrix which is changing in time and not the reference vector , as the rigid body rotates about point O'. The position of the particle is now written as: Taking the time derivative yields the velocity of the particle: where V_i is the velocity of the particle (in the lab frame) and V is the velocity of O' (the origin of the rigid body frame). Since is a rotation matrix its inverse is its transpose. So we substitute : Continue by taking the time derivitve of : Applying the formula (AB)^T = B^TA^T: is the negative of its transpose. Therefore it is a skew symmetric 3x3 matrixSkew-symmetric matrix In linear algebra, a skew-symmetric matrix is a square matrix A whose transpose is also its negative; that is, it ... . We can therefore take its dual to get a 3 dimensional vector. is called the angular velocity tensorAngular velocity tensor In physics, the angular velocity tensor is defined as a matrix T such that:... . If we take the dual of this tensor, matrix multiplication is replaced by the cross product. Its dual is called the angular velocity pseudovector, ?. Substituting ? into the above velocity expression: It can be seen that the velocity of a point in a rigid body can be divided into two terms - the velocity of a reference point fixed in the rigid body plus the cross product term involving the angular velocity of the particle with respect to the reference point. This angular velocity is the "spin" angular velocity of the rigid body as opposed to the angular velocity of the reference point O' about the origin O. It is an important point that the spin angular velocity of every particle in the rigid body is the same, and that the spin angular velocity is independent of the choice of the origin of the rigid body system or of the lab system. In other words, it is a physically real quantity which is a property of the rigid body, independent of one's choice of coordinate system. The angular velocity of the reference point about the origin of the lab frame will, however, depend on these choices of coordinate system. It is often convenient to choose the center of massCenter of mass In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the system's mass b... of the rigid body as the origin of the rigid body system, since a considerable mathematical simplification occurs in the expression for the angular momentumAngular momentum In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the... of the rigid body. See also Angular frequencyAngular frequency RadianPulsation ... Angular accelerationAngular acceleration Overview Angular acceleration is the rate of change of angular velocity over time.... Angular momentumFacts About Angular momentum In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the... Areal velocityAreal velocity Areal velocity is the rate at which area is swept by the position vector of a point which moves along a curve.... IsometryIsometry In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism be... Lie algebraLie algebra Overview In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups a... Orthogonal groupOrthogonal group In mathematics, the orthogonal group of degree n over a field F) is the group of n-by-n orthogonal matrices ... Rigid body dynamicsRigid body dynamics In physics, Rigid body dynamics differs from particle dynamics in that the body takes up space and can rotate, which introdu... Rotation groupRotation group In mechanics and geometry, the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space,...