Short Version 10 Rotational Motion 10 Polar coord

Short Version : 短版: 10. Rotational Motion 10. 轉動

Polar coord ( r, ) 10. 1. Angular Velocity & Acceleration 角速度和加速度 Average angular

Angular Acceleration We shall restrict ourselves to rotations about a fixed axis. 角加速度以下限於繞固定轉軸的旋轉

Angular vs Linear 角性比線性 Table 10. 1 Angular & Linear Position, Velocity, & Acceleration

10. 3. Rotational Inertia & the Analog of Newton’s Law 轉動慣量和牛頓定律的比照把球在軸附近轉很輕鬆 Linear

Calculating the Rotational Inertia 計算轉動慣量 Rotational inertia of discrete masses 離散質點的轉動慣量 ri = perpendicular

Example 10. 4. Dumbbell 啞鈴 A dumbbell consists of 2 equal masses m =

Example 10. 5. Rod 桿 Find the rotational inertia of a uniform, narrow rod

Example 10. 6. Ring 環 Find the rotational inertia of a thin ring of

Example 10. 7. Disk 盤 Find the rotational inertia of a uniform disk of

Parallel - Axis Theorem 平行軸定理 Parallel - Axis Theorem: 平行軸定理軸通過球心軸移了 d =

Example 10. 9. Into the Well 到井裏 A solid cylinder of mass M &

GOT IT 懂嗎? 10. 4. Two masses m is connected by a string that

10. 4. Rotational Energy 轉動能 Rotational kinetic energy = sum of kinetic energies of

Energy & Work in Rotational Motion 旋轉運動的能和功 Work-energy theorem for rotations: 轉動的功-能定理：

10. 5. Rolling Motion Composite object: 複合物體：滾動 V = velocity of CM ui

Example 10. 12. Rolling Downhill 滾到山下 A solid ball of mass M and radius

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Short Version : 短版: 10. Rotational Motion 10. 轉動

Polar coord ( r, ) 10. 1. Angular Velocity & Acceleration 角速度和加速度 Average angular velocity 平均角速度 Right hand rule 右手法則. = angular displacement 角位移 ( positive if CCW 逆時針為正) // rotational axis 轉軸 in radians 1 rad = 360 / 2 = 57. 3 (Instantaneous) angular velocity Angular speed: 角速率 Circular motion: 圓周運動 Linear speed: 直線速率逆時針 in radians (瞬時)角速度

Angular Acceleration We shall restrict ourselves to rotations about a fixed axis. 角加速度以下限於繞固定轉軸的旋轉 (Instantaneous) angular acceleration (瞬時)角加速度 Trajectory of point on rotating rigid body is a circle ( r = const) 旋轉剛體上一點的軌跡是個圓圈 ( r 為常數)。 at a v ar Its velocity v is always tangential: 其速度 v必在切線方向： Its acceleration is in the plane of rotation ( ) : 它的加速度在轉動平面上 ( ) : Tangential component: 切線分量： Radial component: 徑分量： Mathematica

Angular vs Linear 角性比線性 Table 10. 1 Angular & Linear Position, Velocity, & Acceleration 表 10. 1 角和線位置，速度和加速度 Linear Quantity 線性量 Position 位置 x Velocity 速度 Acceleration 加速度 Eqs for Constant Linear Acceleration 等直線加速的方程式 Angular Quantity 角性量 Angular position 角位置 Angular velocity 角速度 Angular Acceleration 角加速度 Eqs for Constant Angular Acceleration 等角加速的方程式

10. 3. Rotational Inertia & the Analog of Newton’s Law 轉動慣量和牛頓定律的比照把球在軸附近轉很輕鬆 Linear acceleration: 直線加速遠一點就比較難 Rotating baton 旋轉中的指揮棒 (massless rod of length R + ball of mass m at 1 end): ( 無質量的棒長 R + 在一端質量為 m 的球 ) 轉軸 Tangential force on ball : 施於球在切線方向的力： = moment of inertia 轉動慣量 = rotational inertia 轉動慣量 of the baton (指揮棒的)

Calculating the Rotational Inertia 計算轉動慣量 Rotational inertia of discrete masses 離散質點的轉動慣量 ri = perpendicular distance of mass i to the rotational axis. 質點 i 離轉軸的垂直距離 Rotational inertia of continuous matter 連續物質的轉動慣量 r = perpendicular distance of point r to the rotational axis. 點 r 離轉軸的垂直距離 ( r) = density at point r. 點 r 處的密度質量單元 dm 提供轉動慣量 r 2 dm。

Example 10. 4. Dumbbell 啞鈴 A dumbbell consists of 2 equal masses m = 0. 64 kg on the ends of a massless rod of length L = 85 cm. 一個啞鈴包含一條無質量，長 L = 85 cm的桿子，和在其兩端兩個質量 m = 0. 64 kg 的墜子。 Calculate its rotational inertia about an axis ¼ of the way from one end & perpendicular to it. 設轉軸在離一端 ¼ 桿長處，且與桿垂直。求啞鈴對此軸的轉動慣量。 GOT IT懂嗎 ? 10. 2 Would I I會 (a) increase 比較大 (b) decrease 比較小 (c) stay the same 保持一樣 if the rotational axis were 若轉軸是 (b) (1) at the center of the rod 在桿的中點 (a) (2) at one end? ？在桿的端點

Example 10. 5. Rod 桿 Find the rotational inertia of a uniform, narrow rod of mass M and length L about an axis through its center & perpendicular to it. 一均勻長桿的質量為 M，長度為 L。求對於一垂直並通過它中點的軸的轉動慣量。質量單元的質量為 dm，長度為 dx。

Example 10. 6. Ring 環 Find the rotational inertia of a thin ring of radius R and mass M about the ring’s axis. 一幼環的半徑為 R ，質量為 M 。求對環軸的轉動慣量。 Pipe of radius R & length L : 半徑為 R ，長度為 L 的管子： I = MR 2 for any thin ring / pipe 任何幼環或管子

Example 10. 7. Disk 盤 Find the rotational inertia of a uniform disk of radius R & mass M about an axis through its center & perpendicular to it. 一均勻的盤子半徑是 R ，質量是 M 。求對一垂直且通過它中心的軸的轉動慣量。

Parallel - Axis Theorem 平行軸定理 Parallel - Axis Theorem: 平行軸定理軸通過球心軸移了 d = R 遠 Ex. Prove theorem for a set of particles. 習題：証明此定理適用於一組粒子

Example 10. 9. Into the Well 到井裏 A solid cylinder of mass M & radius R is mounted on a frictionless horizontal axle over a well. 一質量為 M，半徑為 R 的實心圓柱架在一口井上面無摩擦的水平軸上。 A rope of negligible mass is wrapped around the cylinder & supports a bucket of mass m. 一條質量可忽略的繩子捲在圓柱上，其末端吊着一個質量為 m 的桶子。 Find the bucket’s acceleration as it falls into the well. 求桶子掉到井裏時的加速度。 Let downward direction be positive. 設朝下的方向為正 Bucket 桶 : Cylinder 圓柱：

GOT IT 懂嗎? 10. 4. Two masses m is connected by a string that passes over a frictionless pulley of mass M. 一條繩子兩端分別縛了質塊 m ；中間則擺在一個無摩擦，質量為 M 的滑輪上。 One mass rests on a frictionless table; the other vertically. 一質塊停在一個無摩擦的桌面上；另一個則是垂直吊下。 Is the magnitude of the tension force in the vertical section of the string 繩子在垂直部份的張力會 (a) greater than, (b) equal to, or (c) less than (a)大於， (b)等於，或(c) 小於 that in the horizontal 在水平部份的 ? Explain 請解釋. (a): There must be a net torque to increase the pulley’s clockwise angular velocity. 必需有淨力距才能提高滑輪的順時針角速度。

10. 4. Rotational Energy 轉動能 Rotational kinetic energy = sum of kinetic energies of all mass elements, taken w. r. t the rotational axis. 轉動能 =所有質量單元對轉軸的動能的和。 Set of particles: 一組粒子：

Energy & Work in Rotational Motion 旋轉運動的能和功 Work-energy theorem for rotations: 轉動的功-能定理：

10. 5. Rolling Motion Composite object: 複合物體：滾動 V = velocity of CM ui = velocity relative to CM. 質心速度對質心的相對速度 Moving wheel: 運動中的輪子： w is w. r. t. axis thru CM w 是對通過質心的軸來算

Example 10. 12. Rolling Downhill 滾到山下 A solid ball of mass M and radius R starts from rest & rolls down a hill. 一個質量為 M ，半徑為 R 的實心球從靜止滾往山下。 Its center of mass drops a total distance h. 它的質心總共落下 h 遠。 Find the ball’s speed at the bottom of the hill. 求球在山底的速率。 Initially: 開始時： Finally: 最後時： Note: v is independent of M & R 注： v 與 M 和 R 無關 sliding ball 滑動的球