Angular velocity

Japanese: 角速度 - かくそくど

Velocity is defined as the rate of instantaneous angular rotation of a mass or rigid body. In other words, when a line connecting a moving point on a plane with the origin changes angle dθ around the origin during an infinitesimal time dt , dθ / dt is the magnitude of the angular velocity of the moving point around the origin ( see figure ). Let n be the unit vector that passes through the origin, moves in the direction of the axis perpendicular to the plane of rotation, and is positive in the direction in which a right-handed screw advances due to the rotation. In this case, ω = nω is called the angular velocity vector with magnitude ω, or simply angular velocity.

When the angular velocity is ω, the velocity v from the origin O to the position vector r is given by the vector product ω× r . That is, at a certain moment, the motion of point P is a circular motion with the radius of the length of the perpendicular line from P to the axis of rotation ρ = r sin (the angle between OP and the axis of rotation) and the center of the perpendicular line's foot Q, so the magnitude of the velocity v is v = ωr sin. The direction is perpendicular to ω and r , the direction is the right-handed screw direction from ω to r , and v is equal to ω× r . At a point on the curve described by the moving point, the acceleration has components dv / dt ^and v2 /ρ in the tangential and normal directions. The former is called tangential acceleration and the latter is called normal acceleration. If the length of the velocity vector v along the space curve is s, the magnitude is ds / dt , the direction is the tangential direction, and ρ is the radius of curvature of the orbit ds / dθ . Tangential acceleration is the change in the magnitude of the velocity over time and is not related to the change in direction. Also, normal acceleration is not related to the change in the magnitude of the velocity, but depends on the velocity itself and the shape of the orbit. When the tangential acceleration is zero, the acceleration is perpendicular to the velocity, i.e., the orbit. The relationship with the angular velocity ω is that the velocity is ρω, so the normal acceleration is ρω ^2. This is also called the centripetal acceleration. The centrifugal force acts in the opposite direction to this acceleration.

[Mitsuo Muraoka]

[Reference] | Rotational motion

Angular velocity (figure)
©Shogakukan ">

Angular velocity (figure)

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

質点または剛体の瞬間的な角回転の割合として定義される速度。すなわち、平面上を運動する点と原点を結ぶ直線が、微小時間dt中に原点の周りに角度dθだけ変化したとき、dθ/dtが運動点の原点の周りの角速度の大きさである（図）。原点を通り回転面に垂直な軸方向で、かつ回転によって右ねじの進む向きを正とする単位ベクトルをnとする。そのときω＝nωを大きさωの角速度ベクトルあるいは単に角速度という。

　角速度ωのとき原点Oから位置ベクトルrにおける速度vはベクトル積ω×rで与えられる。すなわち、ある瞬間においてP点の運動は、Pから回転軸へ下ろした垂線の長さρ＝rsin（はOPと回転軸のなす角）を半径とし、垂線の足Qを中心とする円運動であるから、速度vの大きさはv＝ωrsinである。方向はωとrに垂直で、向きはωからrへ右ねじの進む向きで、vはω×rに等しい。運動点の描く曲線上の点において、加速度は接線および法線方向にdv/dt、v²/ρの成分をもつ。前者を接線加速度、後者を法線加速度とよぶ。速度ベクトルvは、空間曲線に沿っての長さをsとすれば、大きさはds/dtで、向きは接線方向、ρは軌道の曲率半径ds/dθである。接線加速度は速度の大きさの時間的変化であって方向の変化には関係ない。また法線加速度は速度の大きさの変化には関係なく速度自身と軌道の形に依存する。接線加速度がゼロの場合は、加速度は速度すなわち軌道に垂直になる。角速度ωとの関係は、速度がρωだから法線加速度はρω²である。これを向心加速度ともいう。遠心力はこの加速度と反対の向きに働く。