三定轴转动的角速度和角加速度 1速度:定义:O=mn△(数量) △t→0△tdlt 若已知转动方程9=f() 则O=f(1)单位rads (t+△ 工程中常用单位: KKAp(t) n=转分r/min) 则n与o的关系为: 2Tn In n O =n≈(rad/s) 603010 16
16 三.定轴转动的角速度和角加速度 1.角速度： 工程中常用单位： n = 转/分(r / min) 则n与的关系为: ) n n n (rad/s 60 30 10 2 = = 则:= f (t) 单位 rad/s 若已知转动方程 = f(t) ( ) Δ Δ : lim Δ 0 定义 代数量 = = = → dt d t t
Kinematics 2. Angular acceleration: Denote the angular velocity at t as a and at t+At as a+Ao, the angular acceleration can be defined as △odod2 E=lim 9=f(t)unit: rad/s2(scalar) →>0△tdh A rigid body has accelerated rotation when e and o have the same signs, decelerated rotation when a and a have opposite signs. 3. Uniform rotation and rotation with uniform angular acceleration a body has uniform rotation when o=const. and rotation with constant angular acceleration when =const 0=00+Et frequently Similar to the motion used p=BottoT- of a particle. formulae Q2=00+2E
17 A rigid body has accelerated rotation when and have the same signs, decelerated rotation when and have opposite signs. 3. Uniform rotation and rotation with uniform angular acceleration A body has uniform rotation when =const. and rotation with constant angular acceleration when =const. = + = + = + 2 2 1 2 0 2 2 0 0 t t t Frequently used formulae Similar to the motion of a particle. 2. Angular acceleration: Denote the angular velocity at t as and at t +△t as +△, the angular acceleration can be defined as lim ( ) 2 2 0 f t dt d dt d t t = = = = = → unit:rad/s2 (scalar)
2角加速度: 设当t时刻为m,t+△t时刻为a+△O 角加速度=lm1=如-2=0=r0)单位:rads(代数量) t→>0 6与o方向一致为加速转动,与O方向相反为减速转动 3匀速转动和匀变速转动 当a=常数,为匀速转动;当=常数,为匀变速转动。 0=0+Et 常用公式19=o0t+1a2与点的运动相类似 l02=00+28P 18
18 2.角加速度: 设当t 时刻为 , t +△t 时刻为+△ 与方向一致为加速转动, 与 方向相反为减速转动 3.匀速转动和匀变速转动 当 =常数,为匀速转动；当 =常数,为匀变速转动。 = + = + = + 2 2 1 2 0 2 2 0 0 t t t 常用公式 与点的运动相类似。 : lim ( ) 2 2 0 f t dt d dt d t t = = = = = → 角加速度 单位:rad/s2 (代数量)
Kinematics 87-3 Velocity and acceleration of a point in a rigid body rotating about a fixed axis 1. Relationship between linear velocity and angular velocity a @, 8 are defined for the whole rigid body (including all points in it) △t内 v, a are defined for a given point in the body they are different for different point △S as ∴1 =m dt4→>0△t A1Q·R V=m OR →0∠ ∴=OR 19
19 , are defined for the whole rigid body (including all points in it)； v, a are defined for a given point in the body (they are different for different point). t S dt dS v t = = →0 lim R t R v t = = →0 limv=R §7–3 Velocity and acceleration of a point in a rigid body rotating about a fixed axis 1. Relationship between linear velocity and angular velocity
§7-3转动刚体内各点的速度和加速度 (即角量与线量的关系) 一线速度V和角速度之间的关系 O,E对整个刚体而言(各点都一样); ,a对刚体中某个点而言(各点不一样)。 △t内 ds: AS lim △S f tr→x0∠t v= lim ap R =OR A1→0t V=OR 20
20 , 对整个刚体而言(各点都一样)； v, a 对刚体中某个点而言(各点不一样)。 t S dt dS v t 0 lim → = = R t R v t = = →0 limv=R (即角量与线量的关系) §7-3 转动刚体内各点的速度和加速度 一.线速度V和角速度之间的关系