普通物理 General Physics 11 - Rotational Motion II 郭艷光Yen-Kuang Kuo 國立彰化師大物理系暨光電科技研究所 電子郵件: ykuo@cc.ncue.edu.tw 網頁: http://ykuo.ncue.edu.tw

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Outline 11-1 What Is Physics? 11-2 Rolling as Translation and Rotation Combined 11-3 The Kinetic Energy of Rolling 11-4 The Forces of Rolling 11-5 The Yo-Yo 11-6 Torque Revisited 11-7 Angular Momentum 11-8 Newton’s Second Law in Angular Form 11-9 The Angular Momentum of a System of Particles 11-10 The Angular Momentum of a Rigid Body Rotating About a Fixed Axis 11-11 Conservation of Angular Momentum 11-12 Precession of a Gyroscope 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 11-1 What Is Physics? The most important application of physics is in the rolling motion of wheels and wheel-like objects. Today, like it or not, the world is filled with cars, trucks, motorcycles, bicycles, and other rolling vehicles. 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

11-2 Rolling as Translation and Rotation Combined A wheel of radius R rolling without slipping: O and P move forward by a distance s. The linear speed vcom of the center of the wheel (the center of mass of this uniform wheel) is (smooth rolling motion) 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

11-2 Rolling as Translation and Rotation Combined Rolling motion of a wheel is a combination of translational and rotational motions. At the top of the wheel, the velocity is 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-1 (a) A flywheel of radius 20 cm starts from rest, and has a constant angular acceleration of 60 rad/s2. Find the magnitude of the net linear acceleration of a point on the rim after 0.15 s. Solution: The tangential acceleration is 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-1 (b) The number of revolutions completed in 0.25 s. Solution: From The corresponds to 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

11-3 The Kinetic Energy of Rolling A rolling object has two types of kinetic energy: (1) Rotational kinetic energy : Due to its rotation about its center of mass. (2) Translational kinetic energy : Due to translation of its center of mass. 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

11-3 The Kinetic Energy of Rolling Proof of Equation: If we view the rolling as pure rotation about an axis through P, From the parallel-axis theorem, we have and 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-2 Approximate each wheel on the car Thrust SSC as a disk of uniform thickness and mass M = 170 kg, and assume smooth rolling. When the car’s speed was 1233 km/h, what was the kinetic energy of each wheel? 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-2 Solution: 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 11-4 The Forces of Rolling Friction and Rolling: If the wheel does not slide, the force is a static frictional force and the motion is smooth rolling. We can obtain linear acceleration and angular acceleration . (smooth rolling motion) 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 11-4 The Forces of Rolling Rolling down a Ramp: A round uniform body of mass M and radius R rolling smoothly down a ramp at angle , along an x axis. 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 11-4 The Forces of Rolling 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 11-4 The Forces of Rolling Cylinder: acom 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 11-4 The Forces of Rolling Hoop: acom 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-3 (a) A uniform ball, of mass M = 6.00 kg and radius R, rolls smoothly from rest down a ramp at angle = 30.0°. The ball descends a vertical height h = 1.20 m to reach the bottom of the ramp. What is its speed at the bottom? 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-3 (a) Solution: 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-3 (b) What are the magnitude and direction of the frictional force on the ball as it rolls down the ramp? Solution: 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-3 (b) 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 11-5 The Yo-Yo We use Newton’s second law to find an expression for the linear acceleration acom of a yo-yo rolling down a string. y acom 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 11-5 The Yo-Yo y acom 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 11-6 Torque Revisited 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-4 Three forces, each of magnitude 2.0 N, act on a particle. The particle is in the xz plane at point A given by position vector , where r = 3.0 m and . Force is parallel to the x axis . force and is parallel to the z and y axis. What is the torque, about the origin O, due to each force? 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-4 Solution: 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 11-7 Angular Momentum The angular momentum of this particle with respect to the origin O is a vector quantity defined as 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 11-7 Angular Momentum Angular momentum depends on the choice of the origin O. If the origin is shifted in general, we get a different value of . SI unit for angular momentum: Sometimes the equivalent is used. 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-5 Figure shows an overhead view of two particles moving at constant momentum along horizontal paths. Particle 1, with momentum magnitude p1 = 5.0 kg·m/s, has position vector and will pass 2.0 m from point O. Particle 2, with momentum magnitude p2 = 2.0 kg·m/s, has position vector and 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-5 will pass 4.0 m from point O. What are the magnitude and direction of the net angular momentum about point O of the two- particle system? Solution: 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

11-8 Newton’s Second Law in Angular Form The (vector) sum of all the torques acting on a particle is equal to the time rate of change of the angular momentum of that particle. Rotation form of Newton’s second law for a particle. Compare with 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

11-8 Newton’s Second Law in Angular Form Proof of Equation: 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-6 (a) A penguin of mass m falls from rest at point A, a horizontal distance D from the origin O of an xyz coordinate system. (The positive direction of the z axis is directly outward from the plane of the figure.) What is 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-6 (a) the angular momentum of the falling penguin about O? Solution: 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-6 (b) About the origin O, what is the torque on the penguin due to the gravitational force ? Solution: 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

11-9 The Angular Momentum of a System of Particles The angular momentum of particles: The time derivative of the angular momentum is: O m1 m3 m2 ℓ1 ℓ2 ℓ3 x y z 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-7 Two blocks with masses m1 = 3 kg and m2 = 1 kg are connected by a rope that passes over a pulley of radius R = 0.2 m and mass M = 4 kg. The moment of inertia of the pulley about its center is I = MR2/2. Find the linear acceleration of the block of mass m2 is at a distance R above the center of the pulley. 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-7 Solution: 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

11-10 The Angular Momentum of a Rigid Body Rotating About a Fixed Axis Angular momentum L: Lz is not a vector equation; it represents only the component of the angular momentum along the angular velocity. (Rigid body, fixed axis) 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

11-10 The Angular Momentum of a Rigid Body Rotating About a Fixed Axis Proof of Equation: 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

11-10 The Angular Momentum of a Rigid Body Rotating About a Fixed Axis 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

11-10 The Angular Momentum of a Rigid Body Rotating About a Fixed Axis 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-8 (a) George Washington Gale Ferris, Jr., a civil engineering graduate from Rensselaer Polytechnic Institute, built the original Ferris wheel for the 1893 World’s Columbian Exposition in Chicago. The wheel, an astounding engineering construction at the time, carried 36 wooden 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-8 (a) cars, each holding as many as 60 passengers, around a circle of radius R = 38 m. The mass of each car was about 1.1 × 104 kg. The mass of the wheel’s structure was about 6.0 × 105 kg, which was mostly in the circular grid from which the cars were suspended. The wheel made a complete rotation at an angular speed in about 2 min. Estimate the magnitude L of the angular momentum of 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-8 (a) the wheel and its passengers while the wheel rotated at . Solution: 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-8 (a) 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-8 (b) Assume that the fully loaded wheel is rotated from rest to in a time period What is the magnitude of the average net external torque acting on it during ? Solution: 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

11-11 Conservation of Angular Momentum Low of conservation of angular momentum: If the external torque on a system is zero, the total angular momentum is constant in magnitude and direction. or 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

11-11 Conservation of Angular Momentum If the component of the net external torque on a system along a certain axis is zero, then the component of the angular momentum of the system along that axis cannot change, no matter what changes take place within the system. 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

11-11 Conservation of Angular Momentum The spinning volunteer: 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

11-11 Conservation of Angular Momentum The rotation rate of Fig, (b) is faster 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-9 Fig (a) shows a student, again sitting on a stool that can rotate freely about a vertical axis. The student, initially at rest, is holding a bicycle wheel whose rim is loaded with lead and whose rotational inertia Iwh about its central axis is 1.2 kg·m2. The wheel is 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-9 rotating at an angular speed of 3.9 rev/s; as seen from overhead, the rotation is counterclockwise. The axis of the wheel is vertical, and the angular momentum of the wheel points vertically upward. The student now inverts the wheel ( Fig. 11-20b ) so that, as seen from overhead, it is rotating clockwise. Its angular momentum is now . The inversion results in the student, the 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-9 stool, and the wheel’s center rotating together as a composite rigid body about the stool’s rotation axis, with rotational inertia Ib = 6.8 kg ·m2. With what angular speed and in what direction does the composite body rotate after the inversion of the wheel? 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-9 Solution: 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-10 A cockroach with mass m rides on a disk of mass 6.00 m and radius R. The disk rotates like a merry-go-round around its central axis at angular speed = 1.50 rad/s. The cockroach is initially at radius r = 0.800 R, but then it crawls out to the rim of the disk. 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 Example 11-10 Treat the cockroach as a particle. What then is the angular speed? Solution: 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

11-12 Precession of a Gyroscope Translational Motion Rotational Motion 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

11-12 Precession of a Gyroscope Translational Motion Rotational Motion 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授 End of chapter 11! 2018/11/20 普通物理講義-11 / 國立彰化師範大學物理系/ 郭艷光教授