# Rotational Motion: Moment of Inertia

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Experiment 8 Rotational Motion: Moment of Inertia 8.1 Objectives Familiarize yourself with the concept of moment of inertia, I, which plays the same role in the description of the rotation of a rigid body as mass plays in the description of linear motion. Investigate how changing the moment of inertia of a body affects its rotational motion. 8.2 Introduction In physics, we encounter various types of motion, primarily linear or rotational. Today we will investigate rotational motion and measure one of the most important quantities pertaining to that: the moment of inertia. The way mass is distributed greatly affects how easily an object can rotate. For example, if you are sitting in a spinning office chair and extend your arms out away from your body, your rotation speed will slow down. If you then pull your arms back in as close as possible, you will start to rotate much faster than when your arms were extended. This illustrates that not only mass but also how it is distributed (i.e. moment of inertia) affects rotational motion. 83

2 8. Rotational Motion: Moment of Inertia 8.3 Key Concepts You can find a summary on-line at Hyperphysics. 1 moment of inertia, torque, angular acceleration Look for keywords: 8.4 Theory If we apply a single unbalanced force, F, to an object, the object will undergo a linear acceleration, a, which is determined by the unbalanced force acting on the object and the mass, m, of the object. Newton s Second Law expresses this relationship: F = ma where the mass is a measure of an object s inertia, or its resistance to being accelerated. In rotational motion, torque (represented by the Greek letter τ), is the rotational equivalent of force. Torque, roughly speaking, measures how effective a force is at causing an object to rotate about a pivot point. Torque, τ is defined as: τ = rfsin(θ) (8.1) where r is the lever arm, F is the force and θ is the angle between the force and the lever arm. In this lab the lever arm, r, will be the radius at which the force is applied (i.e. the radius of the axle). The force will be applied tangentially (i.e. perpendicular) to the radius so θ is 90 and sin(90 )=1 making Eq. 8.1 become τ = rf. Newton s Second law applied to rotational motion says that a single unbalanced torque, τ, on an object produces an angular acceleration, α, which depends not only on the mass of the object but on how that mass is distributed, called the moment of inertia, I. The equation which is analogous to F = ma for an object that is accelerating rotationally is τ = Iα (8.2) where the units of τ are in Newton-meters (N*m), α is in radians/sec 2 and I is in kg-m A radian is an angle measure based upon the circumference of a circle C =2πr where r is the radius of the circle. A complete circle (360 )issaidtohave2π radians. Therefore, a 1/4 circle (90 )isπ/2 radians. 84 Last updated August 20, 2015

3 8.4. Theory The moment of inertia, I, isameasureofthewaythemassis distributed on the object and determines its resistance to angular acceleration. Every rigid object has a definite moment of inertia about a particular axis of rotation. The moment of inertia of a collection of masses is given by: I =Σm i r 2 i (8.3) If there is only one mass (as shown in Fig. 8.1) theneq.8.3 becomes I = mr 2. (In order to simplify the calculation we have assumed that the mass m is a point at the end of the lever arm r and the rod it is attached to is massless.) To illustrate, we will calculate the moment of inertia for a mass of 2 kg at the end of a massless rod that is 2 m in length: I = mr 2 = (2 kg)(2 m) 2 =8kgm 2 If a force of 5 N were applied to the mass perpendicular to the rod (to make the lever arm equal to r) the torque is given by: τ = rf =(2m)(5N)=10Nm By Eq. 8.2 we can now calculate the angular acceleration: α = τ I = 10 N m =1.25 rad/sec2 8kgm2 Figure 8.1: One point mass m on a massless rod of radius r (I = mr 2 ). Last updated August 20,

4 8. Rotational Motion: Moment of Inertia The moment of inertia of a more complicated object is found by adding up the moments of each individual piece. For example, the moment of inertia of the system shown in Fig. 8.2 is found by adding up the moments of each mass so Eq. 8.3 becomes I = m 1 r1 2 + m 2r2 2. (Note that Fig. 8.2 is equivalent to the sum of two Fig. 8.1 components.) Figure 8.2: Two point masses on a massless rod (I = m 1 r m 2r 2 2 ). 8.5 In today s lab Today you will measure the moment of inertia for several different mass distributions. You will make a plot of the moment of inertia, I, vs. r 2, the radii that your masses were placed at, and determine if the moment of inertia does indeed depend on both mass and its distribution. Your setup will resemble Fig. 8.2 where the rigid body consists of two cylinders, which are placed on a metallic rod at varying radii from the axis of rotation. You will assume the rod is massless and will place the masses, m 1 and m 2,an equal distance from the center pivot point so that r 1 = r 2 = r. For this case Eq. 8.3 then becomes: I = m 1 r m 2r 2 2 =(m 1 + m 2 )r 2 (8.4) 86 Last updated August 20, 2015

5 8.5. In today s lab A schematic of the equipment you will be using today is shown in Fig The masses and rod are supported by a rotating platform attached to a central pulley and nearly frictionless air bearings. If the two masses are placed on the axis of rotation (so r = 0), then the measured moment of inertia I is the moment of inertia of the rotating apparatus alone plus the moment of inertia of each of the two cylinders about an axis through their own centers of mass, which we ll call I 0. So when the masses are placed at r =0,I = I 0. Now if the two masses are each placed a distance r from the axis of rotation Eq. 8.4 becomes: I =(m 1 + m 2 )r 2 + I 0 (8.5) Compare Eq. 8.5 to the equation for a straight line: y = mx + b Notice that a plot of I vs. r 2 should be a straight line. The slope of this line is the sum of the masses (m 1 + m 2 )andthey intercept is I 0. Figure 8.3: Schematic of the moment of inertia apparatus. To verify that the moment of inertia, I, does indeed depend on how the masses are distributed you will use the apparatus shown in Fig. 8.3 to calculate the angular acceleration, α, and torque, τ, of the rotating masses andthenuseeq.8.2 to calculate I. You will plot your measured I vs. r 2 Last updated August 20,

7 8.6. Equipment where the floor is defined to be at a height of y = 0. Since the final position of your mass will be the floor, y =0,andEq.8.7 can be rearranged to find the linear acceleration, a: a = 2h t 2 (8.8) where t is the time it takes the weight to fall to the floor. Using your measured values for the time and height you can calculate the linear acceleration a which can then be used to calculate the angular acceleration α that will be used in Eq Next you need to find the torque τ so you can calculate I using Eq Remember that torque was defined in Eq. 8.1 as the force applied times the lever arm when the force is applied tangentially: τ = rf (8.9) In this lab the force (F ) comes from the tension (T ) in the string that is acting perpendicular to the lever arm r, which here is the radius of the central axle (R), as shown in Fig So for this experiment Eq. 8.9 becomes: τ = R T (8.10) The tension in the string, T, comes from the weight which is hanging off the side pulley (see Fig. 8.3). By drawing the free-body diagram acting on the hanging mass and using Newton s Second Law, the tension in the string is: T = Mg Ma (8.11) where M is the amount of mass hanging below the side pulley, g is the acceleration due to gravity and a is the linear acceleration given in Eq Now you have all of the pieces to calculate the moment of inertia I of the rigid body using Eq. 8.2 (I = τ/α). 8.6 Equipment Two cylindrical masses Hanger with mass Air bearing with central pulley String Last updated August 20,

8 8. Rotational Motion: Moment of Inertia 8.7 Procedure In this experiment, you will change the moment of inertia of the rotating body by changing how the mass is distributed on the rotating body. You will place two cylindrical masses at four different radii, such that r = r 1 = r 2 in each case, on a metallic rod. You will then use your measurements to calculate the moment of inertia (I) for each of the four radial positions of the cylindrical masses (r). The sum of the two cylindrical masses (m 1 + m 2 ) can then be found from a graph of I versus r Measure and record the masses of the hanging mass, M, andthetwo cylinders, m 1 and m Place the cylinders on the horizontal rod such that the axes of the cylinders are along the horizontal rod as far away from the center as possible (as shown in Fig. 8.5). Make sure the thumbscrew on each cylinder is tightened. The center of mass of each cylinder MUST be the same distance (r) from the axis of rotation (i.e. r 1 = r 2 in Fig. 8.3). Figure 8.5: View of the apparatus with 2 masses at the same radius r. 3. Estimate the uncertainty in r (called δr). This should include both the uncertainty in reading your ruler and the uncertainty in locating the cylinder s center of mass. 90 Last updated August 20, 2015

10 8. Rotational Motion: Moment of Inertia 13. For the last trial you will place the two cylinders at r = 0. To do this, you will use the vertical bar on the support (see Fig. 8.6). When you place the cylinders on the vertical bar, make sure they are oriented the same way as in your previous trials, i.e. with the axes of the two cylinders perpendicular to the vertical bar. As before, make sure to tighten the thumbscrews on the cylinders. Follow the above procedure to calculate the moment of inertia of the body with the two cylinders at r = 0. Include this data in your data table. Figure 8.6: View of apparatus with 2 masses at radius r = Transfer your data into KaleidaGraph and make a plot of I vs. r 2. Your data points should have both horizontal and vertical error bars. Fit your data with a best fit line making sure to display the fit values for slope and y-intercept with their uncertainties. 8.8 Checklist 1. Excel spreadsheets (both data and formula views) 2. Plot of I vs. r 2 with both horizontal and vertical error bars, a best fit line and observations. 3. Answers to questions 92 Last updated August 20, 2015

11 8.9. Questions 8.9 Questions 1. The moment of inertia of a body depends not only on its mass, but also on how the mass is distributed. Does your data support this? Why or why not? 2. In your plot of I vs. r 2,whydidyouuser 2 and not r in the plot? What are the units of the slope of I vs. r 2? Last updated August 20,

12 8. Rotational Motion: Moment of Inertia 3. Discuss the consistency of the slope of the plot of I vs. r 2 with the value you measured for (m 1 + m 2 ). If they are not consistent, suggest possible reasons why. The uncertainty in your measured mass is given by: δ(m 1 + m 2 )=δ(m 1 )+δ(m 2 ) 4. In the procedure, you were given that R =1.27 ± 0.01 cm. Using only the experimental apparatus and a meter stick, how would you verify this radius with an uncertainty of less than or equal to 0.01 cm? (Hint: You cannot get this uncertainty by holding the meter stick next to the axle and measuring the diameter. Also note that the string is a part of the apparatus.) 94 Last updated August 20, 2015

13 Sheet A B C D E F G H I J K L M N Measure grey fields Calculate yellow fields Use standard deviation from exp. 2 Value Uncertainty Units Rotating mass m 1 : gm Rotating mass m 2 : gm Falling mass M : gm Radius of axle R: cm Gravitational acceleration g: 980 cm/sec 2 Starting elevation of M h: cm Uncertainty in time below sec Moment of Inertia Determination of the moment of inertia Mount the masses on the rod and measure the time it takes the mass M to fall to the floor for 4 different positions of the masses along the rod. Calculate the linear acceleration a, the angular acceleration α, the tension in the string, the torque and the moment of inertia (I) and their associated uncertainities for each configuration from the angular acceleration using equation using equations 1 through 4 in the tables below Trial r δr t a δa T δt α δα Torque δ(torque) cm cm sec cm/sec 2 cm/sec 2 dyne dyne rad/sec 2 rad/sec 2 dyne-cm dyne-cm Trial r 2 δr 2 I I cm 2 cm 2 gm*cm 2 gm*cm Page 1

### Solution Derivations for Capa #11

Solution Derivations for Capa #11 1) A horizontal circular platform (M = 128.1 kg, r = 3.11 m) rotates about a frictionless vertical axle. A student (m = 68.3 kg) walks slowly from the rim of the platform

### Midterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m

Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of

### Angular acceleration α

Angular Acceleration Angular acceleration α measures how rapidly the angular velocity is changing: Slide 7-0 Linear and Circular Motion Compared Slide 7- Linear and Circular Kinematics Compared Slide 7-

### PHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013

PHYSICS HOMEWORK SOLUTION #0 April 8, 203 0. Find the net torque on the wheel in the figure below about the axle through O, taking a = 6.0 cm and b = 30.0 cm. A torque that s produced by a force can be

### Lecture Presentation Chapter 7 Rotational Motion

Lecture Presentation Chapter 7 Rotational Motion Suggested Videos for Chapter 7 Prelecture Videos Describing Rotational Motion Moment of Inertia and Center of Gravity Newton s Second Law for Rotation Class

### 3600 s 1 h. 24 h 1 day. 1 day

Week 7 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution

### Physics 40 Lab 1: Tests of Newton s Second Law

Physics 40 Lab 1: Tests of Newton s Second Law January 28 th, 2008, Section 2 Lynda Williams Lab Partners: Madonna, Hilary Clinton & Angie Jolie Abstract Our primary objective was to test the validity

### Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam

Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam INSTRUCTIONS: Use a pencil #2 to fill your scantron. Write your code number and bubble it in under "EXAM NUMBER;" an entry

### PHY121 #8 Midterm I 3.06.2013

PHY11 #8 Midterm I 3.06.013 AP Physics- Newton s Laws AP Exam Multiple Choice Questions #1 #4 1. When the frictionless system shown above is accelerated by an applied force of magnitude F, the tension

### 11. Rotation Translational Motion: Rotational Motion:

11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational

### C B A T 3 T 2 T 1. 1. What is the magnitude of the force T 1? A) 37.5 N B) 75.0 N C) 113 N D) 157 N E) 192 N

Three boxes are connected by massless strings and are resting on a frictionless table. Each box has a mass of 15 kg, and the tension T 1 in the right string is accelerating the boxes to the right at a

### Chapter 3.8 & 6 Solutions

Chapter 3.8 & 6 Solutions P3.37. Prepare: We are asked to find period, speed and acceleration. Period and frequency are inverses according to Equation 3.26. To find speed we need to know the distance traveled

### PHYS 2425 Engineering Physics I EXPERIMENT 9 SIMPLE HARMONIC MOTION

PHYS 2425 Engineering Physics I EXPERIMENT 9 SIMPLE HARMONIC MOTION I. INTRODUCTION The objective of this experiment is the study of oscillatory motion. In particular the springmass system and the simple

### State Newton's second law of motion for a particle, defining carefully each term used.

5 Question 1. [Marks 20] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding

### Fundamental Mechanics: Supplementary Exercises

Phys 131 Fall 2015 Fundamental Mechanics: Supplementary Exercises 1 Motion diagrams: horizontal motion A car moves to the right. For an initial period it slows down and after that it speeds up. Which of

### 5. Forces and Motion-I. Force is an interaction that causes the acceleration of a body. A vector quantity.

5. Forces and Motion-I 1 Force is an interaction that causes the acceleration of a body. A vector quantity. Newton's First Law: Consider a body on which no net force acts. If the body is at rest, it will

### Physics: Principles and Applications, 6e Giancoli Chapter 4 Dynamics: Newton's Laws of Motion

Physics: Principles and Applications, 6e Giancoli Chapter 4 Dynamics: Newton's Laws of Motion Conceptual Questions 1) Which of Newton's laws best explains why motorists should buckle-up? A) the first law

### SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS

SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering

### EDUH 1017 - SPORTS MECHANICS

4277(a) Semester 2, 2011 Page 1 of 9 THE UNIVERSITY OF SYDNEY EDUH 1017 - SPORTS MECHANICS NOVEMBER 2011 Time allowed: TWO Hours Total marks: 90 MARKS INSTRUCTIONS All questions are to be answered. Use

### Newton s Law of Motion

chapter 5 Newton s Law of Motion Static system 1. Hanging two identical masses Context in the textbook: Section 5.3, combination of forces, Example 4. Vertical motion without friction 2. Elevator: Decelerating

### 8.012 Physics I: Classical Mechanics Fall 2008

MIT OpenCourseWare http://ocw.mit.edu 8.012 Physics I: Classical Mechanics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE

### Chapter 11 Equilibrium

11.1 The First Condition of Equilibrium The first condition of equilibrium deals with the forces that cause possible translations of a body. The simplest way to define the translational equilibrium of

### Lecture 6. Weight. Tension. Normal Force. Static Friction. Cutnell+Johnson: 4.8-4.12, second half of section 4.7

Lecture 6 Weight Tension Normal Force Static Friction Cutnell+Johnson: 4.8-4.12, second half of section 4.7 In this lecture, I m going to discuss four different kinds of forces: weight, tension, the normal

### VELOCITY, ACCELERATION, FORCE

VELOCITY, ACCELERATION, FORCE velocity Velocity v is a vector, with units of meters per second ( m s ). Velocity indicates the rate of change of the object s position ( r ); i.e., velocity tells you how

### HW Set VI page 1 of 9 PHYSICS 1401 (1) homework solutions

HW Set VI page 1 of 9 10-30 A 10 g bullet moving directly upward at 1000 m/s strikes and passes through the center of mass of a 5.0 kg block initially at rest (Fig. 10-33 ). The bullet emerges from the

### 2. To set the number of data points that will be collected, type n.

Force and Motion In this experiment, you will explore the relationship between force and motion. You are given a car with tabs, a string, a pully, a weight hanger, some weights, and the laser gate you

### AP Physics: Rotational Dynamics 2

Name: Assignment Due Date: March 30, 2012 AP Physics: Rotational Dynamics 2 Problem A solid cylinder with mass M, radius R, and rotational inertia 1 2 MR2 rolls without slipping down the inclined plane

### Columbia University Department of Physics QUALIFYING EXAMINATION

Columbia University Department of Physics QUALIFYING EXAMINATION Monday, January 13, 2014 1:00PM to 3:00PM Classical Physics Section 1. Classical Mechanics Two hours are permitted for the completion of

### 1 of 7 9/5/2009 6:12 PM

1 of 7 9/5/2009 6:12 PM Chapter 2 Homework Due: 9:00am on Tuesday, September 8, 2009 Note: To understand how points are awarded, read your instructor's Grading Policy. [Return to Standard Assignment View]

### G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M

G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M CONTENTS Foreword... 2 Forces... 3 Circular Orbits... 8 Energy... 10 Angular Momentum... 13 FOREWORD

### Problem Set #8 Solutions

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.01L: Physics I November 7, 2015 Prof. Alan Guth Problem Set #8 Solutions Due by 11:00 am on Friday, November 6 in the bins at the intersection

### The Effects of Wheelbase and Track on Vehicle Dynamics. Automotive vehicles move by delivering rotational forces from the engine to

The Effects of Wheelbase and Track on Vehicle Dynamics Automotive vehicles move by delivering rotational forces from the engine to wheels. The wheels push in the opposite direction of the motion of the

### Solving Equations Involving Parallel and Perpendicular Lines Examples

Solving Equations Involving Parallel and Perpendicular Lines Examples. The graphs of y = x, y = x, and y = x + are lines that have the same slope. They are parallel lines. Definition of Parallel Lines

### USING MS EXCEL FOR DATA ANALYSIS AND SIMULATION

USING MS EXCEL FOR DATA ANALYSIS AND SIMULATION Ian Cooper School of Physics The University of Sydney i.cooper@physics.usyd.edu.au Introduction The numerical calculations performed by scientists and engineers

### P211 Midterm 2 Spring 2004 Form D

1. An archer pulls his bow string back 0.4 m by exerting a force that increases uniformly from zero to 230 N. The equivalent spring constant of the bow is: A. 115 N/m B. 575 N/m C. 1150 N/m D. 287.5 N/m

### VISCOSITY OF A LIQUID. To determine the viscosity of a lubricating oil. Time permitting, the temperature variation of viscosity can also be studied.

VISCOSITY OF A LIQUID August 19, 004 OBJECTIVE: EQUIPMENT: To determine the viscosity of a lubricating oil. Time permitting, the temperature variation of viscosity can also be studied. Viscosity apparatus

### Chapter 7: Momentum and Impulse

Chapter 7: Momentum and Impulse 1. When a baseball bat hits the ball, the impulse delivered to the ball is increased by A. follow through on the swing. B. rapidly stopping the bat after impact. C. letting

### Mechanics lecture 7 Moment of a force, torque, equilibrium of a body

G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and

### 14 Engineering physics

Option B 14 Engineering physics ESSENTIAL IDEAS The basic laws of mechanics have an extension when equivalent principles are applied to rotation. Actual objects have dimensions and they require an expansion

### Study Guide for Mechanics Lab Final

Study Guide for Mechanics Lab Final This study guide is provided to help you prepare for the lab final. The lab final consists of multiple-choice questions, usually 2 for each unit, and 4 work-out problems

### The Bullet-Block Mystery

LivePhoto IVV Physics Activity 1 Name: Date: 1. Introduction The Bullet-Block Mystery Suppose a vertically mounted 22 Gauge rifle fires a bullet upwards into a block of wood (shown in Fig. 1a). If the

### AP1 Dynamics. Answer: (D) foot applies 200 newton force to nose; nose applies an equal force to the foot. Basic application of Newton s 3rd Law.

1. A mixed martial artist kicks his opponent in the nose with a force of 200 newtons. Identify the action-reaction force pairs in this interchange. (A) foot applies 200 newton force to nose; nose applies

### Unit - 6 Vibrations of Two Degree of Freedom Systems

Unit - 6 Vibrations of Two Degree of Freedom Systems Dr. T. Jagadish. Professor for Post Graduation, Department of Mechanical Engineering, Bangalore Institute of Technology, Bangalore Introduction A two

### EDUMECH Mechatronic Instructional Systems. Ball on Beam System

EDUMECH Mechatronic Instructional Systems Ball on Beam System Product of Shandor Motion Systems Written by Robert Hirsch Ph.D. 998-9 All Rights Reserved. 999 Shandor Motion Systems, Ball on Beam Instructional

### Experiment #1, Analyze Data using Excel, Calculator and Graphs.

Physics 182 - Fall 2014 - Experiment #1 1 Experiment #1, Analyze Data using Excel, Calculator and Graphs. 1 Purpose (5 Points, Including Title. Points apply to your lab report.) Before we start measuring

### HW Set II page 1 of 9 PHYSICS 1401 (1) homework solutions

HW Set II page 1 of 9 4-50 When a large star becomes a supernova, its core may be compressed so tightly that it becomes a neutron star, with a radius of about 20 km (about the size of the San Francisco

### Physics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives

Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring

### Application Information

Moog Components Group manufactures a comprehensive line of brush-type and brushless motors, as well as brushless controllers. The purpose of this document is to provide a guide for the selection and application

### WindWise Education. 2 nd. T ransforming the Energy of Wind into Powerful Minds. editi. A Curriculum for Grades 6 12

WindWise Education T ransforming the Energy of Wind into Powerful Minds A Curriculum for Grades 6 12 Notice Except for educational use by an individual teacher in a classroom setting this work may not

### Kinematic Physics for Simulation and Game Programming

Kinematic Phsics for Simulation and Game Programming Mike Baile mjb@cs.oregonstate.edu phsics-kinematic.ppt mjb October, 05 SI Phsics Units (International Sstem of Units) Quantit Units Linear position

### Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

### Problem 6.40 and 6.41 Kleppner and Kolenkow Notes by: Rishikesh Vaidya, Physics Group, BITS-Pilani

Problem 6.40 and 6.4 Kleppner and Kolenkow Notes by: Rishikesh Vaidya, Physics Group, BITS-Pilani 6.40 A wheel with fine teeth is attached to the end of a spring with constant k and unstretched length

### B) 286 m C) 325 m D) 367 m Answer: B

Practice Midterm 1 1) When a parachutist jumps from an airplane, he eventually reaches a constant speed, called the terminal velocity. This means that A) the acceleration is equal to g. B) the force of

### Midterm Exam 1 October 2, 2012

Midterm Exam 1 October 2, 2012 Name: Instructions 1. This examination is closed book and closed notes. All your belongings except a pen or pencil and a calculator should be put away and your bookbag should

### Simple Machines. Figure 2: Basic design for a mousetrap vehicle

Mousetrap Vehicles Figure 1: This sample mousetrap-powered vehicle has a large drive wheel and a small axle. The vehicle will move slowly and travel a long distance for each turn of the wheel. 1 People

### Review Assessment: Lec 02 Quiz

COURSES > PHYSICS GUEST SITE > CONTROL PANEL > 1ST SEM. QUIZZES > REVIEW ASSESSMENT: LEC 02 QUIZ Review Assessment: Lec 02 Quiz Name: Status : Score: Instructions: Lec 02 Quiz Completed 20 out of 100 points

### Using mechanical energy for daily

unit 3 Using mechanical energy for daily activities Physics Chapter 3 Using mechanical energy for daily activities Competency Uses mechanical energy for day-to-day activities Competency level 3.1 Investigates

### A) N > W B) N = W C) N < W. speed v. Answer: N = W

CTN-12. Consider a person standing in an elevator that is moving upward at constant speed. The magnitude of the upward normal force, N, exerted by the elevator floor on the person's feet is (larger than/same

Answer, Key { Homework 6 { Rubin H Landau 1 This print-out should have 24 questions. Check that it is complete before leaving the printer. Also, multiple-choice questions may continue on the next column

### AP Physics C: Mechanics: Syllabus 1

AP Physics C: Mechanics: Syllabus 1 Scoring Components Page(s) SC1 The course covers instruction in kinematics. 3, 5 SC2 The course covers instruction in Newton s laws of 3 4 motion. SC3 The course covers

### Statics problem solving strategies, hints and tricks

Statics problem solving strategies, hints and tricks Contents 1 Solving a problem in 7 steps 3 1.1 To read.............................................. 3 1.2 To draw..............................................

### State Newton's second law of motion for a particle, defining carefully each term used.

5 Question 1. [Marks 28] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding

### Physics 160 Biomechanics. Angular Kinematics

Physics 160 Biomechanics Angular Kinematics Questions to think about Why do batters slide their hands up the handle of the bat to lay down a bunt but not to drive the ball? Why might an athletic trainer

### 0 Introduction to Data Analysis Using an Excel Spreadsheet

Experiment 0 Introduction to Data Analysis Using an Excel Spreadsheet I. Purpose The purpose of this introductory lab is to teach you a few basic things about how to use an EXCEL 2010 spreadsheet to do

### Lecture 07: Work and Kinetic Energy. Physics 2210 Fall Semester 2014

Lecture 07: Work and Kinetic Energy Physics 2210 Fall Semester 2014 Announcements Schedule next few weeks: 9/08 Unit 3 9/10 Unit 4 9/15 Unit 5 (guest lecturer) 9/17 Unit 6 (guest lecturer) 9/22 Unit 7,

### ( ) where W is work, f(x) is force as a function of distance, and x is distance.

Work by Integration 1. Finding the work required to stretch a spring 2. Finding the work required to wind a wire around a drum 3. Finding the work required to pump liquid from a tank 4. Finding the work

### Modeling Mechanical Systems

chp3 1 Modeling Mechanical Systems Dr. Nhut Ho ME584 chp3 2 Agenda Idealized Modeling Elements Modeling Method and Examples Lagrange s Equation Case study: Feasibility Study of a Mobile Robot Design Matlab

### (Equation 1) to determine the cord s characteristics. Hooke s Law represents the

Using Hooke s Law to Solve for Length of Bungee Cord Needed for Egg Drop Introduction This experiment is the second part of a three- part experiment. The first two lead up to the final in which we aim

### Kyu-Jung Kim Mechanical Engineering Department, California State Polytechnic University, Pomona, U.S.A.

MECHANICS: STATICS AND DYNAMICS Kyu-Jung Kim Mechanical Engineering Department, California State Polytechnic University, Pomona, U.S.A. Keywords: mechanics, statics, dynamics, equilibrium, kinematics,

### How to Write a Formal Lab Report

Union College Physics and Astronomy How to Write a Formal Lab Report A formal lab report is essentially a scaled-down version of a scientific paper, reporting on the results of an experiment that you and

### AP Physics 1 and 2 Lab Investigations

AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks

### 30 minutes in class, 2 hours to make the first time

Asking questions and defining problems Developing and using models Planning and carrying out investigations 30 minutes in class, 2 hours to make the first time 3 12 x 24 x ¾ inch plywood boards 1 x 12

### PAScar Accessory Track Set (1.2m version)

Includes Teacher's Notes and Typical Experiment Results Instruction Manual and Experiment Guide for the PASCO scientific Model ME-6955 012-07557A 1/01 PAScar Accessory Track Set (1.2m version) Model ME-9435

### Recitation Week 4 Chapter 5

Recitation Week 4 Chapter 5 Problem 5.5. A bag of cement whose weight is hangs in equilibrium from three wires shown in igure P5.4. wo of the wires make angles θ = 60.0 and θ = 40.0 with the horizontal.

### Solutions to Exercises, Section 5.1

Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

### Dear Accelerated Pre-Calculus Student:

Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also

### Exam 2 is at 7 pm tomorrow Conflict is at 5:15 pm in 151 Loomis

* By request, but I m not vouching for these since I didn t write them Exam 2 is at 7 pm tomorrow Conflict is at 5:15 pm in 151 Loomis There are extra office hours today & tomorrow Lots of practice exams

### The Grange School Maths Department. Mechanics 1 OCR Past Papers

The Grange School Maths Department Mechanics 1 OCR Past Papers June 2005 2 1 A light inextensible string has its ends attached to two fixed points A and B. The point A is vertically above B. A smooth ring

### Physics C AP Review Packet Name: Mechanics

Physics C AP Review Packet Name: Mechanics Position(x, y or z) Displacement ( x, y or z) Change in position. Depends only on initial and final positions, not on path. Includes direction. x = vdt Velocity

### Tutorial 2: Using Excel in Data Analysis

Tutorial 2: Using Excel in Data Analysis This tutorial guide addresses several issues particularly relevant in the context of the level 1 Physics lab sessions at Durham: organising your work sheet neatly,

### Module 8 Lesson 4: Applications of Vectors

Module 8 Lesson 4: Applications of Vectors So now that you have learned the basic skills necessary to understand and operate with vectors, in this lesson, we will look at how to solve real world problems

### PHYSICS WORKBOOK 2013 EDITION WRITTEN BY: TOM PATERSON NJSPECIALEVENTS@SIXFLAGS.COM

PHYSICS WORKBOOK 2013 EDITION WRITTEN BY: TOM PATERSON NJSPECIALEVENTS@SIXFLAGS.COM SIX FLAGS GREAT ADVENTURE PHYSICS DAY WORKBOOK TABLE OF CONTENTS RESOURCE MATERIALS PAGE 2 RIDES INTRODUCTION AND LEARNING

### Torsion Tests. Subjects of interest

Chapter 10 Torsion Tests Subjects of interest Introduction/Objectives Mechanical properties in torsion Torsional stresses for large plastic strains Type of torsion failures Torsion test vs.tension test

### Orbital Mechanics. Angular Momentum

Orbital Mechanics The objects that orbit earth have only a few forces acting on them, the largest being the gravitational pull from the earth. The trajectories that satellites or rockets follow are largely

### Solid Mechanics. Stress. What you ll learn: Motivation

Solid Mechanics Stress What you ll learn: What is stress? Why stress is important? What are normal and shear stresses? What is strain? Hooke s law (relationship between stress and strain) Stress strain

### Free Fall: Observing and Analyzing the Free Fall Motion of a Bouncing Ping-Pong Ball and Calculating the Free Fall Acceleration (Teacher s Guide)

Free Fall: Observing and Analyzing the Free Fall Motion of a Bouncing Ping-Pong Ball and Calculating the Free Fall Acceleration (Teacher s Guide) 2012 WARD S Science v.11/12 OVERVIEW Students will measure

### DISPLACEMENT & VELOCITY

PHYSICS HOMEWORK #1 DISPLACEMENT & VELOCITY KINEMATICS d v average t v ins d t verysmall / error d t d t v a ave t 1. You walk exactly 50 steps North, turn around, and then walk exactly 400 steps South.

### Physics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal

Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3-D We have defined the velocit and acceleration of a particle as the first and second

### 8. As a cart travels around a horizontal circular track, the cart must undergo a change in (1) velocity (3) speed (2) inertia (4) weight

1. What is the average speed of an object that travels 6.00 meters north in 2.00 seconds and then travels 3.00 meters east in 1.00 second? 9.00 m/s 3.00 m/s 0.333 m/s 4.24 m/s 2. What is the distance traveled

### Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

I. A mechanical device shakes a ball-spring system vertically at its natural frequency. The ball is attached to a string, sending a harmonic wave in the positive x-direction. +x a) The ball, of mass M,

### (I) s(t) = s 0 v 0 (t t 0 ) + 1 2 a (t t 0) 2 (II). t 2 = t 0 + 2 v 0. At the time. E kin = 1 2 m v2 = 1 2 m (a (t t 0) v 0 ) 2

Mechanics Translational motions of a mass point One-dimensional motions on the linear air track LD Physics Leaflets P1.3.3.8 Uniformly accelerated motion with reversal of direction Recording and evaluating

### Dynamics of Rotational Motion

Chapter 10 Dynamics of Rotational Motion PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman Lectures by James Pazun Modified by P. Lam 5_31_2012 Goals for Chapter

### The Viscosity of Fluids

Experiment #11 The Viscosity of Fluids References: 1. Your first year physics textbook. 2. D. Tabor, Gases, Liquids and Solids: and Other States of Matter (Cambridge Press, 1991). 3. J.R. Van Wazer et

### Satellites and Space Stations

Satellites and Space Stations A satellite is an object or a body that revolves around another object, which is usually much larger in mass. Natural satellites include the planets, which revolve around

### Faraday s Law of Induction

Chapter 10 Faraday s Law of Induction 10.1 Faraday s Law of Induction...10-10.1.1 Magnetic Flux...10-3 10.1. Lenz s Law...10-5 10. Motional EMF...10-7 10.3 Induced Electric Field...10-10 10.4 Generators...10-1