物理与光电工程学院：《大学物理》课程PPT教学课件（讲稿，英文第三版）Chap10&11 Rotational Motion About Fixed Axis

Chapter 10 &11 Rotational motion about a Fixed Axis Torque, Rotational Inertia and Law Work and Rotational Kinetic Energy Angular Momentum and Conservation of Angular Momentum

Chapter 10 & 11 Rotational Motion About a Fixed Axis • Torque, Rotational Inertia and Law • Work and Rotational Kinetic Energy • Angular Momentum and Conservation of Angular Momentum

1. Angular Quantities( P235-238 Angular position 0 2 Angular displacement(角位移):△6=62-61 Where ae is positive for counterclockwise rotation and negative for clockwise rotation Always:0=6t △ede 3.Angular velocity: @=lim 4→+0△tdt Right-hand-rule: When the fingers of the right hand are curled around the rotation axis and point in the direction of the rotation then the thumb points in the direction of

1. Angular Quantities (P235-238) 1.Angular position  : 2.Angular Displacement(角位移): Where  is positive for counterclockwise rotation and negative for clockwise rotation. Always:  = (t)  = 2 −1 t t t d d lim 0    =   =  → 3.Angular Velocity : Right-hand-rule: When the fingers of the right hand are curled around the rotation axis and point in the direction of the rotation, then the thumb points in the direction of . (p241) 

4.Angular Acceleration △ada a= m A→>0△tdt

t t t d d lim 0    =   =  → 4.Angular Acceleration :

2. Relation of Linear and Angular Variables The distance along a circular arc S=Or (radian me as ure ) The linear speed (with r held constant): ds de or 一[V=o〃 dt dt The tangential component at: a =ar The radial component an 2

r dt d dt ds  = v=r or 2. Relation of Linear and Angular Variables The distance along a circular arc: The linear speed (with r held constant): s = r (radian measure). The tangential component at : r t = The radial component an : 2 2 v r r v an = = =

3. Kinematics Equations for Uniformly Accelerated Rotational Motion(p238) Equations of Motion for Constant Linear Acceleration are analogous to the ones of Constant Angular acceleration Linear Missing Angular Equation variable Equation y=vo t at X 0=0+at x-xo=lot t-ai 6-6 Oot ta at 2 12=v2+2c(x-x0) 02=02+2a(-t C -6=(0o+0) X-x 0=v 2 6-0.=0t2q

 - 0  = 0 +t   - 0 = 0 t + t 2 t 2 = 0 2 +2( -0 )   - 0 = (0 + )t 0  - 0 = t - t 2 v = v0 + at x -x0 x - x0 = v0 t + at2 v v 2 = v0 2 + 2a(x - x0 ) t x - x0 = (v0 + v)t a x - x0 = vt - at2 v0 Equations of Motion for Constant Linear Acceleration are analogous to the ones of Constant Angular Acceleration: Linear Missing Angular Equation variable Equation 2 1 2 1 2 1 2 1 2 1 2 1 3. Kinematics Equations for Uniformly Accelerated Rotational Motion(p238)

4 Purely rotational motion Any point moves around one same axis. Axis of rotation 5. Torque on an axis The torque about a given axis is defined as: z=R1F=RF⊥ T=RFSing =R×F

4 Purely Rotational motion Any point moves around one same axis. Axis of rotation  = R F 5. Torque on an axis  = RF sin  = R⊥ F The torque about a given axis is defined as: = RF⊥

6.N- Law for rotation(定轴转动定律) I=la N-II Law for Rotation Net torque on a rigid body equals to the product of rotational inertia and angular acceleration it causes about same axis ∑△m 2 the moment of inertia or rotational inertia(转动惯量) on the axis The moment of inertia lis a measure of the rotational inertia of a body, which plays the same role for rotational motion that mass does for translational motion

6. N-II Law for Rotation (定轴转动定律)  net = I N-II Law for Rotation —— Net torque on a rigid body equals to the product of rotational inertia and angular acceleration it causes about same axis. the moment of inertia or rotational inertia (转动惯量) on the axis. =  2 i i I m r The moment of inertia I is a measure of the rotational inertia of a body, which plays the same role for rotational motion that mass does for translational motion

7. The Calculation of rotational Inertia of Body I=rdm where dm=ods 质量连续分布刚体转动惯量的计算方法: ①确定刚体的质量密度。 ②建立坐标系,坐标原点为轴。 ③确定质量元dlm ④由定义计算

    = dl dS dV dm    where, . I = r dm2 7. The Calculation of Rotational Inertia of Body 质量连续分布刚体转动惯量的计算方法： ①.确定刚体的质量密度。 ②.建立坐标系，坐标原点为轴。 ③.确定质量元dm。 ④.由定义计算

8 Solving Problems in Rotational dynamics net Ia N-II Law for Rotation 1. As always, draw a clear and complete diagram. 2. Draw a free-body diagram 3. Identify the axis of rotation and calculate the torques about it. 4. Apply newtons second law for rotation and for translation 5. Solve the resulting equations

8 Solving Problems in Rotational Dynamics  net = I N-II Law for Rotation 1. As always, draw a clear and complete diagram. 2. Draw a free-body diagram. 3. Identify the axis of rotation and calculate the torques about it. 4. Apply Newton’s second law for rotation and for translation. 5. Solve the resulting equations

9. Rotational Kinetic Energy(P254-255) K=-lo 力矩的功W=Md6 work-kinetic energy theorem for rigid bod rotated about a fixed axis. W 今~1=4 Where=J。zd the work done on a rigid body during a finite angular displacement equals to the change of its rotational kinetic energy

. 2 1 2 K = I 9. Rotational Kinetic Energy (P254-255)  = 2 1 d   力矩的功 W M  W = I f − I i = K 2 2 2 1 2 1    = f i W d   where   work-kinetic energy theorem for rigid body rotated about a fixed axis: —– the work done on a rigid body during a finite angular displacement equals to the change of its rotational kinetic energy