STUDY GUIDE 4: Equilibrium, Angular Kinematics, and Dynamics

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1 PH 1110 Term C11 STUDY GUIDE 4: Equilibrium, Angular Kinematics, and Dynamics Objectives 25. Define torque. Solve problems involving objects in static equilibrium. 26. Define angular displacement, angular velocity and angular acceleration. Given the graph or the functional form of one of the quantities versus time, determine the graphs of the other two. Describe in words and equations the motion from an analysis of one or more of the graphs. 27. Define moment of inertia, and solve problems involving rotational motion of rigid bodies subject to a net torque. 28. Calculate the angular momentum, relative to a specified axis, of a point mass traveling in a straight line. 29. Calculate the angular momentum of a rigid body whose angular velocity is specified. 30. Solve problems using the law of conservation of angular momentum. Suggested Study Procedure for Chapter 11. Study Sec first; then Study Secs through Study Example 10.1, and then Examples 11.1 through Answer Discussion Questions 2, 9, 11, 12 in Chapt. 11. Do Exercises 10.1, 10.3, and then Exercises 11.5, 11.13, 11.15, Do Problems 49, 59, 65, 67, 73 in Chapt. 11. A. So far we've ignored ONE LITTLE DETAIL about motion: that objects can ROTATE as well as TRANSLATE. We'll TURN to that little detail in this last section of the course. Perhaps you will be happy to hear that we don't really need any new theory for this. We just need to recast all of the old familiar ideas into angular terms. That's one of the things we'll be illustrating and emphasizing a lot in lecture and conference meetings. B. We think that it is preferable to consider STATIC EQUILIBRIUM first before letting objects experience angular acceleration, so we're going to deviate just a bit from the order of topics in the text, first by studying Sec (for the definition of torque), and then by skipping to Secs through 11.3 (for the application of torque to static equilibrium situations). Note that the Problem-Solving Strategy on p. 359 applies to ALL of the work covered in Sec About all that changes from one problem to the next is the geometry of the situation! This is what takes some practice. Suggested Study Procedure for Chapter 9. Study Secs. 9.1 through 9.4. Study the first nine Examples in Chapt. 9, Examples 1 through 9. Answer Discussion Questions 3, 4, 6, 7, 10. Do Exercises 1, 7, 9, 13, 19, 23, 25, 39, 47. Do Problems 83, 85, 87, 89. A. As you study Secs. 9.1 through 9.4, keep in mind that ALL of the equations for rotational motion with constant angular acceleration are identical in form to those of one dimensional motion with constant

2 linear acceleration. Table 9.1 is intended to help you learn the new set of angular symbols and to see how they are interrelated in constant acceleration situations. B. MOMENT OF INERTIA plays a role in rotational motion similar to that of MASS in translational motion. We will emphasize the use of moment of inertia in rotational motion rather than the calculation of moment of inertia for different bodies. On the other hand, we will expect you to know the expression for the moment of inertia for our standard objects: the hoop, the uniform disk, the stick pivoted about one end, and the point particle at distance R from the rotation axis. Suggested Study Procedure for Chapter 10. Skim Sec. 10.1; Study Secs. 10.2, 10.4, 10.5, and Study Examples 2 and 3, and Examples 11 through 14. Answer Discussion Questions 2, 3, 8, 10, 20, 21, 22. Do Exercises 8, 13, 17, 27, 29, 40, 42, 43, 45. Do Problems 54, 64, 85, 87. A. Study Sec with particular care. This section shows how to apply Newton's laws to problems involving rotation. Notice that the Free-Body Diagram is still an indispensable part of our problemsolving strategy! B. Prior to this time we have ignored or neglected the effects of pulleys in problems other than their role in changing the direction of a force. Examples 10.2 and 10.3 show how a real pulley (with mass!) can affect a system. C. ANGULAR MOMENTUM deserves more time than we can assign to it! Not only is it important in understanding all sorts of everyday situations but also in the study of phenomena from the sub-atomic way up through the astronomical. For now, however, we have to content ourselves with Secs and 10.6, and Examples through 10.16, paying particular attention to the definition of angular momentum of a point particle (Eqn ), the definition of angular momentum of a rigid body (Eqn ), and the principle of ANGULAR MOMENTUM CONSERVATION (as shown in Examples through 10.14). If you can stay focused, this should be enough coverage to help you understand the inner workings of many everyday angular motion situations that may have previously puzzled you. LABORATORY WORK FOR STUDY GUIDE 4 Experiment #8 will give you practice in working with and doing calculations involving torque. Experiment #9 will give you practice in relating rotational to linear motion, and vice versa. Again, reading the instructions before going to the lab is helpful. See HOMEWORK ASSIGNMENTS FOR STUDY GUIDE 4. Homework Assignment #13 due by Noon on Tuesday, Feb. 22. Just as with Assignment #12, this assignment is to be submitted via the web. In your web browser, go to Then login using the username and password you have chosen. After logging in, click on Assignment List and select Assignment #13. If you need any review, you can always do that by repeating Assignment 0, a brief, noncredit tutorial on how to enter answers in Mastering Physics. In Assignment #13, you will get 4 chances to submit a correct answer. If your first answer is incorrect, you should consider making use of the hints.

3 Homework Assignment #14 due by Noon on Thursday, Feb. 24. Just as with Assignment #13, this assignment is to be submitted via the web. Follow the instructions for Assignment #13, except click on Assignment #14. Homework Assignment #15 due by Noon on Saturday, Feb. 26. Just as with Assignment #13, this assignment is to be submitted via the web. Follow the instructions for Assignment #13, except click on Assignment #15. Homework Assignment #16 due by Noon on Tuesday, Mar. 1. Just as with Assignment #13, this assignment is to be submitted via the web. Follow the instructions for Assignment #13, except click on Assignment #16. SUMMARY HOMEWORK FOR STUDY GUIDE 4 DUE IN Conference Tuesday, Mar 1. This assignment will be discussed in conference on Thursday, Mar 4 the day before Examination 4.

4 PH 1110 Summary Homework 4, C Term 2011 Total grade: /100 Name Section Number This assignment is worth 1.25% of your final grade, and it assesses your readiness for Exam 4. It is due in conference on Tuesday, March 1st and will be returned to you in conference Thursday, March 3rd, the day before Exam 4. Show your work; use the back sides of these pages if necessary. Answers should be reported to no more than three significant digits, as appropriate. This summary homework covers the majority of the material in Study Guide 4. The last page of this study guide has non-credit supplementary problems concerning the last topics of the course, which you should solve and bring to conference on Thursday, March 4th, when they will be discussed. 1a. [5 pts] In Experiment 8, you verified the conditions for static equilibrium. State these conditions here, in either sentences or equations. 1b. [20 pts] If three forces are acting on the stick that you used in Lab 8, and two forces and their position vectors are known and given below, find the third force and the torque that the third force exerts upon the stick, which is in equilibrium. The coordinate system is shown on the sketch for part c). The first two forces and their position vectors are: F 1 = ( i j) N F 2 = ( i 1.40 j ) N r 1 = 7.00 i cm r 2 = 10.0 i cm F 3 = 1c. [15 pts] Sketch the three force vectors on this diagram in their correct positions. The end-to-end length of the stick is 11.4 cm. Each of the scale marks below represents one centimeter. τ 3 = j i Points earned on this page:

5 Summary Homework 4, continued. Initials Section Number 2a. [5 pts] As you will see in Experiment 9, a string connecting a cart and a hanger goes over a pulley, as in the sketch below on the left. The pulley has tensions T 1 and T 2 acting upon it, as shown below in the center. Write an expression for the total torque vector on the pulley in terms of the shown variables. Assume no slip between the string and the pulley throughout this problem. r T 2 θ T 1 Experimental set-up Close-up of pulley τ tot = 2b. [5 pts] Given that the pulley has a moment of inertia of I = 1.00 x 10-4 kg m 2, T 1 = 4.40 N, T 2 = 4.39 N, and r = 2.50 cm, what is the magnitude of the angular acceleration of the pulley? 2c. [5 pts] What is the direction of the angular acceleration of the pulley? α = 2c. [5 pts] The pulley starts rotating at t = 0 s. At t = 10.0 s, through how many radians has it revolved? α = 2d. [5 pts] What is the angular speed at t = 10.0 s? θ = 2e. [5 pts] Of the variables τ tot, α, α, θ, and ω, which are vectors and which are scalars? ω = Vectors: Scalars: Points earned on this page:

6 Summary Homework 4, continued. Initials Section Number 3a. [5 pts] A pulley of moment of inertia I and radius r is fixed to a wall. A rope is wrapped around it and tied to a block of mass m. The block is initially held at rest, and then it is released. The block drops a distance h. The rope does not slip on the pulley, and the magnitude of the acceleration due to gravity is g. What is the work done by gravity on the system? You know I, r, m, h, and g. I, r m Pulley and block W g = 3b. [5 pts] What is the change in potential energy of the system? U = 3c. [5 pts] What is the change in kinetic energy of the system? K = 3d. [5 pts] Write the equation that expresses the no-slip condition the relationship between the linear and angular speeds. 3e. [5 pts] What is the speed v of the block in terms of I, r, m, h, and g? No-slip: 3f. [5 pts] What is the angular speed ω of the pulley in terms of I, r, m, h, and g? v = ω = Points earned on this page:

7 Supplementary Problems Solve these non-credit problems and bring them to conference on Tuesday, October 14 th. They are intended to give you practice on the material in the last part of Study Guide 4. Sa. A large turntable has moment of inertia I. Person A, mass m A stands at a distance halfway to the outer edge of the turntable of radius R, whereas Person B, mass m B, stands right at the outer edge of the turntable, as shown below. The turntable and people spin with angular speed ω 0. The people can be regarded as point masses for this problem. What is the angular speed ω 1 of the turntable after person A walks to the center of the table? A B Sb. With Person A at the center of the turntable, Person B jumps off in a direction parallel to his instantaneous velocity, with v B relative to the ground. What is angular speed ω 2 of the turntable now? To keep your expression simple, you may consider ω 1 to be a known value. Sc. Person B now returns to the side of the turntable and stands on the ground. He places his hands on the side of the turntable, causing a friction force with magnitude f. How many revolutions does the turntable make before coming to rest? To keep your expression simple, you may consider ω 2 to be a known value.

STUDY GUIDE 1: Introduction, Vectors, Kinematics

PH 1110 Term C12 STUDY GUIDE 1: Introduction, Vectors, Kinematics PH 1110 Study Guides are intended to help you achieve the objectives of this course as completely and as efficiently as possible. Each