There is a straight line of motionless movement in the rigid body, called rotation for short.This fixed line is called the rotation axis of the rigid body.Obviously, other points in the rigid body move in circles in planes perpendicular to the rotation axis, with the center on the rotation axis.
Any point in the rigid bodyQAnd its circumferential trajectory centerO''s connectionO'Q(Figure 1) is called the turning radius of the point.From fixed planeOzxAngle to rotation plane OzQφ, which can be used to determine the instantaneous position of the rigid body.cornerφOver timetThe change law of is called the rotation equation of rigid body. Write:
φ=f（t）
cornerφChange Δ ofφAnd corresponding time interval ΔtRatio Δ ofφ/Δt=ω*It is called average angular velocity.When Δt→ 0,ω*Limit of trendωIs called (instantaneous) angular velocity, i.e
Current angular velocityωOver timetWhen changing, its change ΔωAnd corresponding time interval ΔtRatio Δ ofω/Δt=ε*It is called average angular acceleration.When Δt→ 0,ε*Limit of trendεIs called (instantaneous) angular acceleration, namely
The angular velocity and angular acceleration of a rigid body can be expressed along the axis of rotationOz(Unit vector isk）The slip vector of.(Figure 2).Angular velocity vectorωAnd angular acceleration vectorεCan write separatelyω=ωk，ε=εk。
Rotate any point in the rigid bodyQLinear speed ofvbe equal tov=ω×r, andv=ω·O´Q。spotQLinear acceleration ofαFor:
α=αt+αn=ε×r+ω×v，
Andαt=ε·O´Q，αn=ω·O´Q。
In the above formularAny point on the shaftOTo pointQVector diameter of, andαtandαnPoints respectivelyQTangential and normal acceleration of (see acceleration).
The magnitude of the rigid body's moment of inertia is related to the following factors:
(1) Rigid bodies with the same shape and size have large mass and large moment of inertia;
(2) For rigid bodies with the same total mass, the farther the mass distribution is away from the axis, the greater the moment of inertia;
(3) For the same rigid body, if the rotation axis is different, the distribution of the mass to the axis is different, and the size of the moment of inertia is different.
