普通物理 General Physics 10 - Rotational Motion I 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-10 / 國立彰化師範大學物理系/ 郭艷光教授 Outline 10-1 What Is Physics? 10-2 The Rotational Variables 10-3 Are Angular Quantities Vectors? 10-4 Rotation with Constant Angular Acceleration 10-5 Relating the Linear and Angular Variables 10-6 Kinetic Energy of Rotation 10-7 Calculating the Rotational Inertia 10-8 Torque 10-9 Newton’s Second Law for Rotation 10-10 Work and Rotational Kinetic Energy 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 10-1 What Is Physics? In this chapter, we will study the rotational motion of rigid bodies about a fixed axis. We will also calculate the kinetic energy associated with rotation, write Newton’s second law for rotational motion, and introduce the work-kinetic energy for rotational motion. 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-2 The Rotational Variables Reference line: Fixed in the body, perpendicular to the rotation axis and rotating with the body. Angular Position : (Radian measure) 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-2 The Rotational Variables An angle defined in this way is measured in radians (rad) rather than in revolutions (rev) or degrees. 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-2 The Rotational Variables Angular Displacement : An angular displacement in the counterclockwise direction is positive, and one in the clockwise direction is negative. 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-2 The Rotational Variables Average Angular Velocity: Instantaneous Angular Velocity: : angular position t : angular position The magnitude of an angular velocity is called the angular speed. SI unit: radians/second 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-2 The Rotational Variables Average Angular Acceleration: Instantaneous Angular Acceleration: is the change in the angular velocity that occurs during the time interval . SI unit: radians/second2 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-1 (a) The disk is rotating about its central axis like a merry-go- round. The angular position of a reference line on the disk is given by , with t in seconds, in radians, and the zero angular position as indicated in the figure. Graph the angular position of the disk versus time from t = ̶ 3.0 s to t = 5.4 s. 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-1 (a) Sketch the disk and its angular position reference line at t = 2.0 s, 0 s, and 4.0 s, and when the curve crosses the t axis. Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-1 (a) 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-1 (b) At what time tmin does reach the minimum value shown in figure of page 10? What is that minimum value? Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-1 (c) Graph the angular velocity of the disk versus time from t = 3.0 s to t = 6.0 s. Sketch the disk and indicate the direction of turning and the sign of at t = 2.0 s, 4.0 s, and tmin. Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-1 (c) 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-1 (d) Use the results in parts (a) through (c) to describe the motion of the disk from t = 3.0 s to t = 6.0 s. Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-2 (a) A child’s top is spun with angular acceleration with t in seconds and α in radians per second-squared. At t = 0, the top has angular velocity 5 rad/s, and a reference line on it is at angular position θ= 2 rad. Obtain an expression for angular velocity of the top. 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-2 (a) Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-2 (b) Obtain an expression for the angular position of the top. Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-3 Are Angular Quantities Vectors? The direction of the ( vector ) angular velocity is given by the right-hand rule. 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-4 Rotation with Constant Angular Acceleration Translational Motion Rotational Motion 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-4 Rotation with Constant Angular Acceleration When the angular acceleration is constant, from and integrate We use , and again integrate And we find 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-3 (a) A grindstone rotates at constant angular acceleration α= 0.35 rad/s2. At time t = 0, it has an angular velocity of = ̶ 4.6 rad/s and a reference line on it is horizontal, at the angular position . At what time after t = 0 is the reference line at the angular position = 5.0 rev? 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-3 (a) Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-3 (b) Describe the grindstone’s rotation between t = 0 and t = 32 s. Solution: The wheel slows in its rotation in the negative direction. 2. Then reverses to rotate in the positive direction. 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-3 (c) At what time t does the grindstone momentarily stop? Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-4 (a) While you are operating a Rotor ( Sample Problem 6-8 ), you spot a passenger in acute distress and decrease the angular speed of the cylinder from 3.40 rad/s to 2.00 rad/s in 20.0 rev, at constant angular acceleration. What is the constant angular acceleration during this decrease in angular speed? 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-4 (a) Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-4 (b) How much time did the speed decrease take? Solutions: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-5 Relating the Linear and Angular Variables The Position: A rigid body rotates through an angle , a point within the body at a position r from the rotation axis moves a distance s along a circular arc, (radian measure) 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-5 Relating the Linear and Angular Variables The Speed: Differentiating with respect to time — with r held constant —leads to (radian measure) 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-5 Relating the Linear and Angular Variables The Acceleration: The tangential ( linear ) acceleration at The centripetal acceleration ar α: instantaneous angular acceleration 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-5 (a) We are to design the track for an induction roller coaster (which can be accelerated by magnetic forces even on a horizontal track). To create an initial thrill, we want each passenger to leave the loading point with acceleration g along the horizontal track. To increase the thrill, we also want that first section of track to form a circular arc, so that the passenger also experiences a 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-5 (a) centripetal acceleration. As the passenger accelerates along the arc, the magnitude of this centripetal acceleration increases alarmingly. When the magnitude a of the net acceleration reaches 4g at some point P and angle along the arc, we want the passenger then to move in a straight line, along a tangent to the arc. What angle 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-5 (a) should the arc subtend so that a is 4g at point P? Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-5 (b) What is the magnitude a of the passenger’s net acceleration at point P and after point P? Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-6 Kinetic Energy of Rotation ( rotational inertia ) 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-7 Calculating the Rotational Inertia 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-7 Calculating the Rotational Inertia Parallel-Axis Theorem: Let h be the perpendicular distance between the given axis and the axis through the center of mass (remember these two axes must be parallel). Then the rotational inertia I about the given axis is 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-7 Calculating the Rotational Inertia Proof of Parallel-Axis Theorem: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-6 (a) A rigid body consisting of two particles of mass m connected by a rod of length L and negligible mass. What is the rotational inertia Icom about an axis through the center of mass, perpendicular to the rod as shown? 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-6 (a) Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-6 (b) What is the rotational inertia I of the body about an axis through the left end of the rod and parallel to the first axis? Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-7 Four point masses lie at the corners of a rectangle with sides of length 3 m and 4 m. Find the moment of inertia about each of diagonals. Take M = 1 kg. 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-7 Solution: For each axis, two masses do not contribute to the moment of inertia. The other two are at the same distance 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-8 (a) A thin, uniform rod of mass M and length L, on an x axis with the origin at the rod’s center. What is the rotational inertia of the rod about the perpendicular rotation axis through the center? 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-8 (a) Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-8 (b) What is the rod’s rotational inertia I about a new rotation axis that is perpendicular to the rod and through the left end? Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-9 A solid sphere and a disk are released from the same point on an incline. Given that they roll without slipping, which has the greater speed at the bottom? Ignore dissipative effects. 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-9 Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 10-8 Torque The torque of a force F that acts at a distance r from the origin is defined to be ( : Lever arm ) 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-9 Newton’s Second Law for Rotation By analogy with Newton’s second law ( Fnet = ma ) for the acceleration a of a body of mass m due to a net force Fnet along a coordinate axis. We replace Fnet with , m with I, and a with , writing ( Newton’s second law for rotation ) 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-9 Newton’s Second Law for Rotation Proof of Equation: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-10 A uniform disk, with mass M = 2.5 kg and radius R = 20 cm, mounted on a fixed horizontal axle. A block with mass m = 1.2 kg hangs from a massless cord that is wrapped around the rim of the disk. Find the acceleration of the falling block, the angular acceleration of the disk, and the tension in the cord. The cord 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-10 does not slip, and there is no friction at the axle. Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-11 (a) A sphere of mass M and radius R that rolls without slipping down an incline. Its moment of inertia about a central axis is 0.4 MR2 . Find the linear acceleration of the CM. 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-11 (a) Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-11 (b) What is the minimum coefficient of friction required for the sphere to roll without slipping? Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-10 Work and Rotational Kinetic Energy Work And Kinetic Energy Theorem: A torque acts a rigid body in rotation, the body’s rotational kinetic energy can change. If is the torque, then we can have ( : body’s angular position ) 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-10 Work and Rotational Kinetic Energy Rate at which the work is done is the power The similarities in the equations for general and rotational motion: ( power, rotation about fixed axis ) General Rotation 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-10 Work and Rotational Kinetic Energy Proof of Equations: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

10-10 Work and Rotational Kinetic Energy 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-12 Let the disk in Example 10-10 start from rest at time t = 0. What is its rotational kinetic energy K at t = 2.5 s? Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-13 A tall, cylindrical chimney will fall over when its base is ruptured. Treat the chimney as a thin rod of length L = 55.0 m. At the instant it makes an angle of θ= 35.0° with the vertical, what is its angular speed ? 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-13 Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-13 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 Example 10-14 A motor rotates a pulley of radius 25 cm at 20 rpm. A rope around the pulley lifts a 50- kg block. What is the power output of the motor? Solution: 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授 End of chapter 10! 2018/12/5 普通物理講義-10 / 國立彰化師範大學物理系/ 郭艷光教授