轉動力學實驗(一) Rotational Motion

Rigid Object ( 剛體 ) A rigid object is one that is nondeformable The relative locations of all particles making up the object remain constant. All real objects are deformable to some extent, but the rigid object model is very useful in many situations where the deformation is negligible Q

Angular Position The arc length s and r are related: s = q r

Conversions Comparing degrees and radians Converting from degrees to radians

Angular Displacement and Angular Speed Average Angular Speed Instantaneous Angular Speed ( rad/s or s-1 )

Angular Acceleration Average angular acceleration Instantaneous angular acceleration ( rad/s2 or s-2 )

Directions of w and a The directions are given by the right-hand rule out of the plane The directions are given by the right-hand rule into the plane

Relationship Between Angular and Linear Quantities Displacements Tangential Speeds

Relationship Between Angular and Linear Quantities Tangential Acceleration Centripetal Acceleration

Torque ( 力矩 ) Torque, t , is the tendency of a force to rotate an object about some axis. Torque is a vector t = r F sin f = F d

轉動力學實驗原理

Moments of Inertia of Various Rigid Objects

理想實驗 (忽略軸承之摩擦力矩) T m mg 轉盤 滑輪 轉動( Rotational Motion) 滑輪 (轉動慣量小，忽略) 轉盤 平移運動(Translational Motion) 砝碼 滑輪 T m 轉動( Rotational Motion) 滑輪 (轉動慣量小，忽略) 轉盤 mg 理想實驗 (忽略軸承之摩擦力矩)

修正軸承之摩擦力矩 平移運動達到平衡狀態 利用掛勾與黏土 T m0g 轉動達到平衡狀態 m0=掛勾與粘土的質量

軸承之摩擦力矩 T 轉盤 滑輪 轉動( Rotational Motion) 滑輪 (轉動慣量小，忽略) 轉盤 平移運動(Translational Motion) 砝碼+掛勾+黏土 滑輪 T 轉動( Rotational Motion) 滑輪 (轉動慣量小，忽略) 轉盤 軸承之摩擦力矩 (m砝碼+m0)g

角動量守恆實驗

Angular Momentum(角動量) The instantaneous angular momentum of a particle relative to the origin O is defined as the cross product of the particle’s instantaneous position vector and its instantaneous linear momentum

Conservation of Angular Momentum

Angular Momentum of a System of Particles The total angular momentum of a system of particles is defined as the vector sum of the angular momenta of the individual particles Differentiating with respect to time

Angular Momentum of a Rotating Rigid Object To find the angular momentum of the entire object, add the angular momenta of all the individual particles

End of Lecture

Perpendicular-Axis Theorem (垂直軸定理) Iz = Ix + Iy ri Prove it Hint :

圓盤 R M 利用垂直軸定理，系統的軸之轉動慣量 = ?

Parallel-Axis Theorem (平行軸定理) For an arbitrary axis, the parallel-axis theorem often simplifies calculations Ip = ICM + MD 2 Ip is about any axis parallel to the axis through the center of mass of the object ICM is about the axis through the center of mass D is the distance from the center of mass axis to the arbitrary axis D C.M. P

y  C.M. P x y x  y  y C.M. r  △m x  C.M. R rP P x rP r  R

rP r  R C.M. P x y x  y  r  △m R rP

r1 r1  R r2 r2  C.M. P x y x  y  △m2 r2  r1  △m1 r2 R r1

C.M. P x y x  y  IP = ICM + MR2 (Parallel-Axis Theorem)

Perpendicular-Axis Theorem (垂直軸定理) Iz = Ix + Iy ri Prove it Hint :

圓盤 R M 利用垂直軸定理，系統的軸之轉動慣量 = ?

Moments of Inertia of Various Rigid Objects