The connecting line O'Q (Figure 1) between any point Q in a rigid body and its circular trajectory center O' is called the rotation radius of this point. The rotation angle φ from the fixed plane Ozx to the rotation plane OzQ can be used to determine the instantaneous position of the rigid body. The change law of rotation angle φ with time t is called rigid body rotation equation.
φ=f(t)
The ratio Δ φ/Δ t = ω * between the change of rotation angle Δ φ and the corresponding time interval Δ t is called the average angular velocity. When Δ t→ 0, the limit ω * is called (instantaneous) angular velocity, that is,
When the angular velocity ω changes with time t, the ratio of the change δ ω to the corresponding time interval δ t δ ω/δ t = ε * is called the average angular acceleration. When Δ t→ 0, the limit ε * is called (instantaneous) angular acceleration, i.e.
Both angular velocity and angular acceleration of a rigid body can be expressed as a sliding vector (unit vector is k) along the rotation axis Oz. (Figure 2). Angular velocity vector ω and angular acceleration vector ε can be written as ω=ωk and ε=εk respectively.
The linear velocity v of any point q in a rotating rigid body is equal to v=ω×r, and V = ω o? Q. The linear acceleration α of point Q is:
α=αt+αn=ε×r+ω×v，
And α t = ε o? q，αn=ω O？ Ask.
In the above formula, R is the vector diameter from any point O to point Q on the rotation axis, and αt and αn are the tangential and normal accelerations of point Q, respectively (see acceleration).
The moment of inertia of a rigid body 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 from the axis, the greater the moment of inertia;
(3) For the same rigid body, with different rotating shafts, the distribution of mass on the shaft is different, and the magnitude of moment of inertia is also different.