Chapter 8- Rotational Motion

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1 Chapter 8- Rotational Motion

2 Assignment 9 Textbook (Giancoli, 6 th edition), Chapter 5 and 8: Due on Thursday, November 20, 2008 The assignment has been posted at sign/assignment9.pdf

3 Old assignments and midterm exams (solutions have been posted on the web) can be picked up in my office (LB-212)

4 All marks, including assignments, have been posted on the web. Please, verify that all your marks have been entered in the list.

5 Chapter 8 Angular Quantities Constant Angular Acceleration Rolling Motion (Without Slipping) Centripetal Forces Torque Rotational Dynamics; Torque and Rotational Inertia Rotational Kinetic Energy Angular Momentum and Its Conservation

6 Recalling Last Lecture

7 Angular Quantities In purely rotational motion, all points on the object move in circles around the axis of rotation ( O ) which is perpendicular to this slide. The radius of the circle is r. All points on a straight line drawn through the axis move through the same angle in the same time. The angle θ in radians is defined: (8-1) Where r = radius of the circle l = arc length covered by the angle θ The angular displacement is what characterizes the rotational motion.

8 Angular Quantities One radian is defined such that it corresponds to an arc of circle equal to the radius of the circle. Or, if we use eq. 8.1: Radians are dimensionless. (8-2)

9 Angular Quantities (8-3) Angular displacement: (8-4)

10 Angular Quantities Average angular velocity and instantaneous angular velocity: (8-5) (8-6) Average angular acceleration and instantaneous angular acceleration: (8-7) (8-8) Both the velocity and acceleration are the same for any point in the object.

11 Angular Quantities At each angular position, point P will have a linear velocity whose directions are tangent to its circular path. (8-9) Tangential linear acceleration: (8-10) Centripetal acceleration (radial direction towards the center of rotation): (8-11) (8-12) Total acceleration: (8-13)

12 Angular Quantities

13 Angular Quantities The frequency is the number of complete revolutions per second: (8-14) Frequencies are measured in hertz. The time required for a complete revolution is called period, or in other words: period is the time one revolution takes: (8-15)

14 Constant Angular Acceleration The equations of motion for constant angular acceleration are the same as those for linear motion, with the substitution of the angular quantities for the linear ones.

15 Today

16 Centripetal Forces (Section 5-2, textbook) We can now go back to chapter 5 and discuss the concept of centripetal forces. For an object to be in uniform circular motion, there must be a net force acting on it. In fact, we have discussed that there is a radial (centripetal) force associated to a particle (point) moving in circular motion. According to Newton s second law, there should exist a net force F rnet such that: (8-16) F Rnet is known as centripetal force. The centripetal force points in the direction of the radial acceleration (inwards). We can use eq ( ) and rewrite 8-16 as: (8-17)

17 Centripetal Forces (Section 5-2, textbook) A misconception is to believe that there exist a force (centrifugal force) pointing outwards acting on an object when it moves in a circular path. An example is the case of a person swinging a ball on the end of a string as shown in the figure. The force on the person s hand is the reaction of the string to the inward pull the person exerts on it to produce the circular motion. When the string is released, the ball flies away tangentially as depicted in the figure b below. So, there was no forces in the outward direction, otherwise we would have motion in this direction after the string is released (figure a).

18 Centripetal Forces (Section 5-2, textbook) Note: If a particle is developing a circular motion at constant angular velocity ω, its angular acceleration α is zero(eq. 8-8), and the tangential acceleration a tan is consequently zero (eq. 8-10). The net force will be exclusively due to the radial acceleration and will point inward On the other hand, if there is a non-zero angular acceleration α, then there is a tangential acceleration a tan The net force will NOT be pointing inward. (We will come back to it when we discuss on torque)

19 Linear Momentum Problem 5-7 (textbook) A ball on the end of a string is revolved at a uniform rate in a vertical circle of radius 72.0 cm, as shown in Fig If its speed is 4.00 m/s and its mass is kg, calculate the tension in the string when the ball is (a) at the top of its path, and (b) at the bottom of its path.

20 Linear Momentum Problem 5-7 A free-body diagram is shown in the figure. Since the object is moving in a circle with a constant speed, the net force on the object at any point must point to the center of the circle. (a) Take positive to be downward. Write Newton s 2 nd law in the downward direction. T1 2 F mg F ma mv r R T1 R = + = = ( ) ( ) F m v r g ( 4.00 m s) = = = This is a downward force, as expected kg 9.80 m s 3.73 N m

21 Linear Momentum Problem 5-7 A free-body diagram is shown in the figure. Since the object is moving in a circle with a constant speed, the net force on the object at any point must point to the center of the circle. (b) Take positive to be upward. Write Newton s 2 nd law in the upward direction. T 1 2 F F mg ma mv r R T 2 = = = ( ) ( ) F m v r g This is a upward force, as expected. ( 4.00 m s) kg 9.80 m s 9.61 N m 2 2 = + = + =

22 Rolling Motion (Without Slipping) When a wheel or radius R rolls without slipping along a flat straight path, the points of the wheel in contact with the surface are instantaneously at rest and the wheel rotates about a rotation axis through the contact point. In fact, the wheel is experiencing both a translational and rotational motion. P The center C of the wheel does not rotate relative to the contact point. It undergoes only translational motion with velocity. r R ω A point P at a distance r from the center of rotation (point of contact) will undergo a rotation with angular velocity given by eq. 8-9: (nonslip condition for the speed of any point at a distance r). The center of the wheel moves with a linear speed given by: (nonslip condition for the speed of the center of the wheel)

23 Rolling Motion (Without Slipping) The linear acceleration of the center of the wheel will be given by: (nonslip condition for the acceleration of the center of the wheel) P r R ω θ

24 Rolling Motion (Without Slipping) Let s now look at this problem from a different perspective. So far we have placed ourselves on a reference system which is at rest relative to the surface. Let s now assume a reference system at rest with respect to the center of the wheel. In this system, the surface (and therefore the point of contact) moves with a velocity. As the wheel rotates about its center through and angle θ with angular velocity ω, the point of contact between the wheel and the surface will have moved a distance l (arc of circle) given by eq. 8-1: ω R The center of the wheel remains directly over the point of contact, therefore it also moves by the same distance. R θ l l

25 Rotational Kinetic Energy and Moment of Inertia An object can be seen as made of many individual tiny particles located at different positions. In the real word, this is actually the way things are made. The object can then be interpreted as a system of particles. We have already seen that the total kinetic energy of a system is the sum of the kinetic energy of each of its constituents. So, the energy kinetic of this object can be written as: The object is undergoing a rotational motion, we can use eq. 8-9 (noting that the angular velocity is the same for all particles) to write:

26 Rotational Kinetic Energy and Moment of Inertia Then (8-18) We can now define the following quantity: (8-19) Such that: (8-20) The quantity I is known as moment of inertia.

27 Rotational Kinetic Energy and Moment of Inertia Note that in the case of the wheel discussed in the rolling motion without slipping few slides ago, the wheel also undergoes translational motion. The total kinetic energy will be then the sum of the kinetic energy due to its linear and angular motion: (8-21) P r R Where ω is the total mass of the object. θ

28 Rotational Kinetic Energy and Moment of Inertia Moment of inertia depends not only on the mass of each particle, but also on their distribution (position r) in the object. These two objects have the same mass, but the one on the left has a greater rotational inertia, as so much of its mass is far from the axis of rotation. We will come back to moment of inertia when we discuss about torque.

29 Rotational Kinetic Energy and Moment of Inertia The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation Compare (f) and (g), for example. We will come back to moment of inertia when we discuss about torque.

30 Linear Momentum Problem 8-45 (textbook) A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3.3 m/s. Calculate its total kinetic energy.

31 Linear Momentum Problem 8-45 The total kinetic energy is the sum of the translational and rotational kinetic energies. Since the ball is rolling without slipping, the angular velocity is given by ω=v R The rotational inertia of a sphere about an axis through its center is I = 2 5 mr 2 (see table presented two slides ago) v KE KE KE mv I mv mr mv R = + = + ω = + = total trans rot ( )( ) = kg 3.3 m s = 56 J 2

Chapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.

Chapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc. Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems