Chapter 10. Rotation

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Chapter 10. Rotation"

Transcription

1 Chapter 10 Rotation

2 Rotation Rotational Kinematics: Angular velocity and Angular Acceleration Rotational Kinetic Energy Moment of Inertia Newton s nd Law for Rotation Applications MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01

3 Angular Displacement y θ f θ θ i x θ is the angular position. Angular displacement: θ = θ f θ i Note: angles measured CW are negative and angles measured CCW are positive. θ is measured in radians. π radians = 360 = 1 revolution MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 3

4 y Arc Length arc length = s = r θ r θ θ f θ i x θ = s r θ is a ratio of two lengths; it is a dimensionless ratio! MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 4

5 Angular Speed The average and instantaneous angular velocities are: ω av = θ θ and ω = lim t t 0 t ω is measured in rads/sec. MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 5

6 Angular Speed θ f r y θ θ i An object moves along a circular path of radius r; what is its average speed? x Also, v = total distance r θ θ v av = = = r = total time t t rω (instantaneous values). rω av MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 6

7 Period and Frequency The time it takes to go one time around a closed path is called the period (T). v total distance total time av = = πr T Comparing to v = rω: π ω = = πf T f is called the frequency, the number of revolutions (or cycles) per second. MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 7

8 Comparison of Kinematic Equations Angular Linear Displacement Velocity Acceleration Velocity MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 8

9 Rotational Kinetic Energy KE = Iω = m v 1 1 R i i i ( ) m v = m r ω = m r ω i i i i i i i i i 1 1 = m r i i ω = Iω i MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 9

10 Estimating the Moment of Inertia Point particle assumption. The particle is at the center of mass of each rod segment. ML I = ML = ML 35 35ML = = = 0.34ML The actual value is I = (1/3) ML MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 10

11 Moment of Inertia: Thin Uniform Rod A favorite application from calculus class. λ M M I = x dm = x dx = x dx = x dx L L 1 M 1 I = L = ML 3 L 3 3 L 0 MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 11

12 MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 1

13 Parallel Axis Theorem The Parallel Axis theorem is used with the center of mass moments of inertia found in the table to extend those formulas to noncenter of mass applications I = I cm + Mh MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 13

14 Application of the Parallel Axis Theorem 1 cm 1 I = ML I = I cm L I = I cm + M Mh I = ML + ML I = ML MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 14

15 Torque A torque is caused by the application of a force, on an object, at a point other than its center of mass or its pivot point. hinge P u s h Q: Where on a door do you normally push to open it? A: Away from the hinge. A rotating (spinning) body will continue to rotate unless it is acted upon by a torque. MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 15

16 Torque Torque method 1: Top view of door F F Hinge end τ = rf r = the distance from the rotation axis (hinge) to the point where the force F is applied. F is the component of the force F that is perpendicular to the door (here it is Fsinθ). MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 16 θ F

17 The units of torque are Newton-meters (Nm) (not joules!). By convention: Torque When the applied force causes the object to rotate counterclockwise (CCW) then τ is positive. When the applied force causes the object to rotate clockwise (CW) then τ is negative. Be careful with placement of force vector components. MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 17

18 Torque Torque method : τ = r F r is called the lever arm and F is the magnitude of the applied force. Lever arm is the perpendicular distance to the line of action of the force from the pivot point or the axis of rotation. MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 18

19 Top view of door Torque F Hinge end r θ θ Lever arm r sinθ = r r = r sinθ The torque is: τ r F Line of action of the force = = rf sinθ Same as before MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 19

20 The Vector Cross Product τ = r F τ = r F sinθ = rfsinθ The magnitude of C C = ABsin(Φ) The direction of C is perpendicular to the plane of A and B. Physically it means the product of A and the portion of B that is perpendicular to A. MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 0

21 Vector Nature of the Cross Product In rotation the direction of the vectors can be determined by using the right-hand rule MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 1

22 The Right-Hand Rule Curl the fingers of your right hand so that they curl in the direction a point on the object moves, and your thumb will point in the direction of the angular momentum. Torque is an example of a vector cross product MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01

23 Torque (Disk) - F t - Component The radial component F r cannot cause a rotation. Only the tangential component, F t can cause a rotation. MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 3

24 Torque (Disk) - Lever Arm Method Sometimes the Lever Arm method is easier to implement, especially if there are several force vectors involved in the problem. MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 4

25 Torque Examples Hanging Sign Meter stick balanced at 5cm. A Winch and A Bucket Tension in String - Massive Pulley Atwood with Massive Pulley Two Blocks and a Pulley Bowling Ball Loop, Disk, Sphere and Block MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 5

26 Equilibrium The conditions for equilibrium are: F = 0 Linear motion τ = 0 Rotational motion For motion in a plane we now have three equations to satisfy. F = F = 0; τ = 0 x y z MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 6

27 A sign is supported by a uniform horizontal boom of length 3.00 m and weight 80.0 N. A cable, inclined at a 35 angle with the boom, is attached at a distance of.38 m from the hinge at the wall. The weight of the sign is 10.0 N. Using Torque What is the tension in the cable and what are the horizontal and vertical forces exerted on the boom by the hinge? MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 7

28 Using Torque This is important! You need two components for F, not just the expected perpendicular normal force. Apply the conditions for equilibrium to the bar: (1) () (3) y F y F X x F y = F x = F x F τ = w y T bar w L FBD for the bar: T cosθ = bar θ Should be CM w bar F F sb sb 0 + T F sb sinθ = 0 x ( L) + ( T sinθ ) x = 0 MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 8

29 Using Torque Equation (3) can be solved for T: T = = L wbar + F xsinθ 35 N sb ( L) Equation (1) can be solved for F x : F x = T cos θ = 88 N Equation () can be solved for F y : F y = w + F T sinθ bar sb =.00 N MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 9

30 A Winch and a Bucket Finally a massive pulley and massive string problem. The pulley has its mass concentrated in the rim. Consider it to be a loop for moment of inertia considerations. This example in the book takes a couple of short cuts that are not obvious. MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 30

31 A Winch and a Bucket The origin of the y-axis is placed at the level of the axis of the winch. The initial level of the handle of the bucket is also placed at the origin. This simplifies distance measurements for figuring the gravitational potential energy. MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 31

32 Tricks - A Winch and a Bucket For a hoop or any other object similar in construction all the mass is concentrated along the outside edge. For these objects the rotational KE and the linear KE are identical in form. Therefore the rotational KE of the winch and the cable on the winch are represented by their linear forms and no details are given. 1 1 v 1 KE R = Iω = (mr ) = mv = KE r The cable on the winch and the portion falling are combined together in one KE term without a word of explanation. MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 3

33 Torque Due to Gravity This is a thin dimensional object in the x-y plane. Assume that gravity acts at the center of mass. This can be used to determine the CM of a thin irregularly shaped object MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 33

34 Tension in a String - Massive Pulley Massive pulley- moment of inertia needed. Wheel bearing frictionless Non-slip condition Massless rope MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 34

35 Tension in a String - Massive Pulley No Torques Only torque causing force MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 35

36 Tension in a String - Massive Pulley Solve by applying Newton s Laws τ = TR = Iα y F = -T + mg = ma a = Rα mg - T TR R = R = T m I I mg T = mr 1+ I (1) () (3) a = a = mg - T m g I 1+ mr Solve () for a and solve (1) for α and then use (3) MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 36

37 Two Blocks and a Pulley - II Frictionless surface Massive pulley - Need the moment of inertia Tension not continuous across pulley - because pulley has mass. Friction on pulley - string does not slip. - Non-slip condition a t = Rα Massless string MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 37

38 Two Blocks and a Pulley - II F s is the pulley reaction to forces acting on it for the purpose of maintaining the equilibrium of the pulley center of mass. We include F s and m p g in our initial discussions of this system. In subsequent problem solving there is no need to include them. MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 38

39 Two Blocks and a Pulley - II T = m a; m g - T = m a 1 1 T R - T R = Iα 1 a = Rα If I = 0, the case of the massless pulley a = m 1 + m T = 1 1 m 1 + m T = 1 m 1 + m T = T 1 m g m m m g m g mg a = I m 1 + m + R m T = I m 1 + m + R 1 1 T = I m 1 + R I m 1 + m + R m g m g MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 39

40 A Bowling Ball Only KE and GPE in this problem MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 40

41 A Bowling Ball PE at the top = KE at the bottom Mgh = Mv + I ω v = cm 1+β 1 1 cm cm i v 1 1 cm cm Mgh = Mv + β MR R Mgh = Mv + β Mv gh = v 1 1 cm cm (1+β) cm gh β-values Sphere - Hollow = /3 Sphere - Solid = /5 MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 41

42 Loop vs Disk vs Sphere This is the classic race between similarly shaped objects of different mass distributions. We might want to add a frictionless block for comparison. MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 4

43 Loop vs Disk vs Sphere Loop: β = 1 Disk: β = 1/ Sphere: β = /5 Perpendicular length = Radius All moment of inertia formulas have a similar form. They are linear in the mass, quadratic in a characteristic length perpendicular to the axis of rotation and have a dimensionless factor β. I =βml MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 43

44 Loop vs Disk vs Sphere To determine the winner of the race we want to calculate the linear velocity of the cm of each of the objects. For the frictionless block we need only consider the vertical drop of its center of mass and set the change in GPE equal to the translational KE of its CM. v CM = gh MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 44

45 Loop vs Disk vs Sphere vs Block The block wins! Loop: β = 1 Disk: β = 1/ Sphere: β = /5 Block: β = 0 For the other three objects we also need to consider the vertical drop of its center of mass but we set the change in GPE equal to the translational KE of its CM PLUS the rotational KE. gh v CM = 1+ β MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 45

46 Instantaneous Axis of Rotation Instantaneous Axis of Rotation MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 46

47 Instantaneous Axis of Rotation MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 47

48 Rotational and Linear Analogs MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 48

49 Extra Slides MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised 10/13/01 49

= 2 5 MR2. I sphere = MR 2. I hoop = 1 2 MR2. I disk

= 2 5 MR2. I sphere = MR 2. I hoop = 1 2 MR2. I disk A sphere (green), a disk (blue), and a hoop (red0, each with mass M and radius R, all start from rest at the top of an inclined plane and roll to the bottom. Which object reaches the bottom first? (Use

More information

Translational vs Rotational. m x. Connection Δ = = = = = = Δ = = = = = = Δ =Δ = = = = = 2 / 1/2. Work

Translational vs Rotational. m x. Connection Δ = = = = = = Δ = = = = = = Δ =Δ = = = = = 2 / 1/2. Work Translational vs Rotational / / 1/ Δ m x v dx dt a dv dt F ma p mv KE mv Work Fd / / 1/ θ ω θ α ω τ α ω ω τθ Δ I d dt d dt I L I KE I Work / θ ω α τ Δ Δ c t s r v r a v r a r Fr L pr Connection Translational

More information

Angular velocity and angular acceleration CHAPTER 9 ROTATION. Angular velocity and angular acceleration. ! equations of rotational motion

Angular velocity and angular acceleration CHAPTER 9 ROTATION. Angular velocity and angular acceleration. ! equations of rotational motion Angular velocity and angular acceleration CHAPTER 9 ROTATION! r i ds i dθ θ i Angular velocity and angular acceleration! equations of rotational motion Torque and Moment of Inertia! Newton s nd Law for

More information

Rotational Kinetic Energy

Rotational Kinetic Energy Lecture 17, Chapter 10: Rotational Energy and Angular Momentum 1 Rotational Kinetic Energy Consider a rigid body rotating with an angular velocity ω about an axis. Clearly every point in the rigid body

More information

Physics 131: Lecture 21. Today s Agenda

Physics 131: Lecture 21. Today s Agenda Physics 131: Lecture 1 Today s Agenda Rotational dynamics Torque = I Angular Momentum Physics 01: Lecture 10, Pg 1 Newton s second law in rotation land Sum of the torques will equal the moment of inertia

More information

Exam 3 Practice Solutions

Exam 3 Practice Solutions Exam 3 Practice Solutions Multiple Choice 1. A thin hoop, a solid disk, and a solid sphere, each with the same mass and radius, are at rest at the top of an inclined plane. If all three are released at

More information

Chapter 10 Rotational Kinematics and Energy. Copyright 2010 Pearson Education, Inc.

Chapter 10 Rotational Kinematics and Energy. Copyright 2010 Pearson Education, Inc. Chapter 10 Rotational Kinematics and Energy Copyright 010 Pearson Education, Inc. 10-1 Angular Position, Velocity, and Acceleration Copyright 010 Pearson Education, Inc. 10-1 Angular Position, Velocity,

More information

= o + t = ot + ½ t 2 = o + 2

= o + t = ot + ½ t 2 = o + 2 Chapters 8-9 Rotational Kinematics and Dynamics Rotational motion Rotational motion refers to the motion of an object or system that spins about an axis. The axis of rotation is the line about which the

More information

AP Physics C: Rotation II. (Torque and Rotational Dynamics, Rolling Motion) Problems

AP Physics C: Rotation II. (Torque and Rotational Dynamics, Rolling Motion) Problems AP Physics C: Rotation II (Torque and Rotational Dynamics, Rolling Motion) Problems 1980M3. A billiard ball has mass M, radius R, and moment of inertia about the center of mass I c = 2 MR²/5 The ball is

More information

Chapter 6, Problem 18. Agenda. Rotational Inertia. Rotational Inertia. Calculating Moment of Inertia. Example: Hoop vs.

Chapter 6, Problem 18. Agenda. Rotational Inertia. Rotational Inertia. Calculating Moment of Inertia. Example: Hoop vs. Agenda Today: Homework quiz, moment of inertia and torque Thursday: Statics problems revisited, rolling motion Reading: Start Chapter 8 in the reading Have to cancel office hours today: will have extra

More information

Chap. 10: Rotational Motion

Chap. 10: Rotational Motion Chap. 10: Rotational Motion I. Rotational Kinematics II. Rotational Dynamics - Newton s Law for Rotation III. Angular Momentum Conservation (Chap. 10) 1 Newton s Laws for Rotation n e t I 3 rd part [N

More information

Physics 131: Lecture 21. Today s Agenda

Physics 131: Lecture 21. Today s Agenda Physics 131: Lecture 21 Today s Agenda Rotational dynamics Torque = I Angular Momentum Physics 201: Lecture 10, Pg 1 Newton s second law in rotation land Sum of the torques will equal the moment of inertia

More information

AP Physics. Harmonic Motion. Multiple Choice. Test E

AP Physics. Harmonic Motion. Multiple Choice. Test E AP Physics Harmonic Motion Multiple Choice Test E A 0.10-Kg block is attached to a spring, initially unstretched, of force constant k = 40 N m as shown below. The block is released from rest at t = 0 sec.

More information

Solution Only gravity is doing work. Since gravity is a conservative force mechanical energy is conserved:

Solution Only gravity is doing work. Since gravity is a conservative force mechanical energy is conserved: 8) roller coaster starts with a speed of 8.0 m/s at a point 45 m above the bottom of a dip (see figure). Neglecting friction, what will be the speed of the roller coaster at the top of the next slope,

More information

Slide 1 / 37. Rotational Motion

Slide 1 / 37. Rotational Motion Slide 1 / 37 Rotational Motion Slide 2 / 37 Angular Quantities An angle θ can be given by: where r is the radius and l is the arc length. This gives θ in radians. There are 360 in a circle or 2π radians.

More information

第 1 頁, 共 7 頁 Chap10 1. Test Bank, Question 3 One revolution per minute is about: 0.0524 rad/s 0.105 rad/s 0.95 rad/s 1.57 rad/s 6.28 rad/s 2. *Chapter 10, Problem 8 The angular acceleration of a wheel

More information

1.1. Rotational Kinematics Description Of Motion Of A Rotating Body

1.1. Rotational Kinematics Description Of Motion Of A Rotating Body PHY 19- PHYSICS III 1. Moment Of Inertia 1.1. Rotational Kinematics Description Of Motion Of A Rotating Body 1.1.1. Linear Kinematics Consider the case of linear kinematics; it concerns the description

More information

Physics A - PHY 2048C

Physics A - PHY 2048C Physics A - PHY 2048C and 11/15/2017 My Office Hours: Thursday 2:00-3:00 PM 212 Keen Building Warm-up Questions 1 Did you read Chapter 12 in the textbook on? 2 Must an object be rotating to have a moment

More information

Physics 221. Exam III Spring f S While the cylinder is rolling up, the frictional force is and the cylinder is rotating

Physics 221. Exam III Spring f S While the cylinder is rolling up, the frictional force is and the cylinder is rotating Physics 1. Exam III Spring 003 The situation below refers to the next three questions: A solid cylinder of radius R and mass M with initial velocity v 0 rolls without slipping up the inclined plane. N

More information

1301W.600 Lecture 16. November 6, 2017

1301W.600 Lecture 16. November 6, 2017 1301W.600 Lecture 16 November 6, 2017 You are Cordially Invited to the Physics Open House Friday, November 17 th, 2017 4:30-8:00 PM Tate Hall, Room B20 Time to apply for a major? Consider Physics!! Program

More information

Chapter 8. Rotational Motion

Chapter 8. Rotational Motion Chapter 8 Rotational Motion Rotational Work and Energy W = Fs = s = rθ Frθ Consider the work done in rotating a wheel with a tangential force, F, by an angle θ. τ = Fr W =τθ Rotational Work and Energy

More information

Concept Question: Normal Force

Concept Question: Normal Force Concept Question: Normal Force Consider a person standing in an elevator that is accelerating upward. The upward normal force N exerted by the elevator floor on the person is 1. larger than 2. identical

More information

TutorBreeze.com 7. ROTATIONAL MOTION. 3. If the angular velocity of a spinning body points out of the page, then describe how is the body spinning?

TutorBreeze.com 7. ROTATIONAL MOTION. 3. If the angular velocity of a spinning body points out of the page, then describe how is the body spinning? 1. rpm is about rad/s. 7. ROTATIONAL MOTION 2. A wheel rotates with constant angular acceleration of π rad/s 2. During the time interval from t 1 to t 2, its angular displacement is π rad. At time t 2

More information

Chapter 10 Practice Test

Chapter 10 Practice Test Chapter 10 Practice Test 1. At t = 0, a wheel rotating about a fixed axis at a constant angular acceleration of 0.40 rad/s 2 has an angular velocity of 1.5 rad/s and an angular position of 2.3 rad. What

More information

Physics 2210 Homework 18 Spring 2015

Physics 2210 Homework 18 Spring 2015 Physics 2210 Homework 18 Spring 2015 Charles Jui April 12, 2015 IE Sphere Incline Wording A solid sphere of uniform density starts from rest and rolls without slipping down an inclined plane with angle

More information

Summer Physics 41 Pretest. Shorty Shorts (2 pts ea): Circle the best answer. Show work if a calculation is required.

Summer Physics 41 Pretest. Shorty Shorts (2 pts ea): Circle the best answer. Show work if a calculation is required. Summer Physics 41 Pretest Name: Shorty Shorts (2 pts ea): Circle the best answer. Show work if a calculation is required. 1. An object hangs in equilibrium suspended by two identical ropes. Which rope

More information

FIGURE 2. Total Acceleration The direction of the total acceleration of a rotating object can be found using the inverse tangent function.

FIGURE 2. Total Acceleration The direction of the total acceleration of a rotating object can be found using the inverse tangent function. Take it Further Demonstration Versus Angular Speed Purpose Show that tangential speed depends on radius. Materials two tennis balls attached to different lengths of string (approx. 1.0 m and 1.5 m) Procedure

More information

General Physics (PHY 2130)

General Physics (PHY 2130) General Physics (PHY 130) Lecture 0 Rotational dynamics equilibrium nd Newton s Law for rotational motion rolling Exam II review http://www.physics.wayne.edu/~apetrov/phy130/ Lightning Review Last lecture:

More information

Rotational Motion. Rotational Motion. Rotational Motion

Rotational Motion. Rotational Motion. Rotational Motion I. Rotational Kinematics II. Rotational Dynamics (Netwton s Law for Rotation) III. Angular Momentum Conservation 1. Remember how Newton s Laws for translational motion were studied: 1. Kinematics (x =

More information

11-2 A General Method, and Rolling without Slipping

11-2 A General Method, and Rolling without Slipping 11-2 A General Method, and Rolling without Slipping Let s begin by summarizing a general method for analyzing situations involving Newton s Second Law for Rotation, such as the situation in Exploration

More information

Exercise Torque Magnitude Ranking Task. Part A

Exercise Torque Magnitude Ranking Task. Part A Exercise 10.2 Calculate the net torque about point O for the two forces applied as in the figure. The rod and both forces are in the plane of the page. Take positive torques to be counterclockwise. τ 28.0

More information

Physics 23 Exam 3 April 2, 2009

Physics 23 Exam 3 April 2, 2009 1. A string is tied to a doorknob 0.79 m from the hinge as shown in the figure. At the instant shown, the force applied to the string is 5.0 N. What is the torque on the door? A) 3.3 N m B) 2.2 N m C)

More information

Fall 2007 RED Barcode Here Physics 105, sections 1 and 2 Please write your CID Colton

Fall 2007 RED Barcode Here Physics 105, sections 1 and 2 Please write your CID Colton Fall 007 RED Barcode Here Physics 105, sections 1 and Exam 3 Please write your CID Colton -3669 3 hour time limit. One 3 5 handwritten note card permitted (both sides). Calculators permitted. No books.

More information

Chapter 12 Torque Pearson Education, Inc.

Chapter 12 Torque Pearson Education, Inc. Chapter 12 Torque Schedule 3-Nov torque 12.1-5 5-Nov torque 12.6-8 10-Nov fluids 18.1-8 12-Nov periodic motion 15.1-7 17-Nov waves in 1D 16.1-6 19-Nov waves in 2D, 3D 16.7-9, 17.1-3 24-Nov gravity 13.1-8

More information

Rotational N.2 nd Law

Rotational N.2 nd Law Lecture 0 Chapter 1 Physics I Rotational N. nd Law Torque Course website: http://faculty.uml.edu/andriy_danylov/teaching/physicsi IN THIS CHAPTER, you will continue discussing rotational dynamics Today

More information

FALL TERM EXAM, PHYS 1211, INTRODUCTORY PHYSICS I Saturday, 14 December 2013, 1PM to 4 PM, AT 1003

FALL TERM EXAM, PHYS 1211, INTRODUCTORY PHYSICS I Saturday, 14 December 2013, 1PM to 4 PM, AT 1003 FALL TERM EXAM, PHYS 1211, INTRODUCTORY PHYSICS I Saturday, 14 December 2013, 1PM to 4 PM, AT 1003 NAME: STUDENT ID: INSTRUCTION 1. This exam booklet has 14 pages. Make sure none are missing 2. There is

More information

Description: Using conservation of energy, find the final velocity of a "yo yo" as it unwinds under the influence of gravity.

Description: Using conservation of energy, find the final velocity of a yo yo as it unwinds under the influence of gravity. Chapter 10 [ Edit ] Overview Summary View Diagnostics View Print View with Answers Chapter 10 Due: 11:59pm on Sunday, November 6, 2016 To understand how points are awarded, read the Grading Policy for

More information

Chapter 12. Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx

Chapter 12. Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx Chapter 1 Lecture Notes Chapter 1 Oscillatory Motion Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx When the mass is released, the spring will pull

More information

Lecture 11 - Advanced Rotational Dynamics

Lecture 11 - Advanced Rotational Dynamics Lecture 11 - Advanced Rotational Dynamics A Puzzle... A moldable blob of matter of mass M and uniform density is to be situated between the planes z = 0 and z = 1 so that the moment of inertia around the

More information

Work and kinetic Energy

Work and kinetic Energy Work and kinetic Energy Problem 66. M=4.5kg r = 0.05m I = 0.003kgm 2 Q: What is the velocity of mass m after it dropped a distance h? (No friction) h m=0.6kg mg Work and kinetic Energy Problem 66. M=4.5kg

More information

Upthrust and Archimedes Principle

Upthrust and Archimedes Principle 1 Upthrust and Archimedes Principle Objects immersed in fluids, experience a force which tends to push them towards the surface of the liquid. This force is called upthrust and it depends on the density

More information

Phys 270 Final Exam. Figure 1: Question 1

Phys 270 Final Exam. Figure 1: Question 1 Phys 270 Final Exam Time limit: 120 minutes Each question worths 10 points. Constants: g = 9.8m/s 2, G = 6.67 10 11 Nm 2 kg 2. 1. (a) Figure 1 shows an object with moment of inertia I and mass m oscillating

More information

AAPT UNITED STATES PHYSICS TEAM AIP 2016

AAPT UNITED STATES PHYSICS TEAM AIP 2016 216 F = ma Exam 1 AAPT UNITED STATES PHYSICS TEAM AIP 216 216 F = ma Contest 25 QUESTIONS - 75 MINUTES INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN Use g = 1 N/kg throughout this contest.

More information

Revolve, Rotate & Roll:

Revolve, Rotate & Roll: I. Warm-UP. Revolve, Rotate & Roll: Physics 203, Yaverbaum John Jay College of Criminal Justice, the CUNY Given g, the rate of free-fall acceleration near Earth s surface, and r, the radius of a VERTICAL

More information

PS 11 GeneralPhysics I for the Life Sciences

PS 11 GeneralPhysics I for the Life Sciences PS 11 GeneralPhysics I for the Life Sciences ROTATIONAL MOTION D R. B E N J A M I N C H A N A S S O C I A T E P R O F E S S O R P H Y S I C S D E P A R T M E N T F E B R U A R Y 0 1 4 Questions and Problems

More information

UNIVERSITY OF TORONTO Faculty of Arts and Science

UNIVERSITY OF TORONTO Faculty of Arts and Science UNIVERSITY OF TORONTO Faculty of Arts and Science DECEMBER 2013 EXAMINATIONS PHY 151H1F Duration - 3 hours Attempt all questions. Each question is worth 10 points. Points for each part-question are shown

More information

Translational Motion Rotational Motion Equations Sheet

Translational Motion Rotational Motion Equations Sheet PHYSICS 01 Translational Motion Rotational Motion Equations Sheet LINEAR ANGULAR Time t t Displacement x; (x = rθ) θ Velocity v = Δx/Δt; (v = rω) ω = Δθ/Δt Acceleration a = Δv/Δt; (a = rα) α = Δω/Δt (

More information

Uniform Circular Motion:-Circular motion is said to the uniform if the speed of the particle (along the circular path) remains constant.

Uniform Circular Motion:-Circular motion is said to the uniform if the speed of the particle (along the circular path) remains constant. Circular Motion:- Uniform Circular Motion:-Circular motion is said to the uniform if the speed of the particle (along the circular path) remains constant. Angular Displacement:- Scalar form:-?s = r?θ Vector

More information

Phys 106 Practice Problems Common Quiz 1 Spring 2003

Phys 106 Practice Problems Common Quiz 1 Spring 2003 Phys 106 Practice Problems Common Quiz 1 Spring 2003 1. For a wheel spinning with constant angular acceleration on an axis through its center, the ratio of the speed of a point on the rim to the speed

More information

t = g = 10 m/s 2 = 2 s T = 2π g

t = g = 10 m/s 2 = 2 s T = 2π g Annotated Answers to the 1984 AP Physics C Mechanics Multiple Choice 1. D. Torque is the rotational analogue of force; F net = ma corresponds to τ net = Iα. 2. C. The horizontal speed does not affect the

More information

ω = ω 0 θ = θ + ω 0 t αt ( ) Rota%onal Kinema%cs: ( ONLY IF α = constant) v = ω r ω ω r s = θ r v = d θ dt r = ω r + a r = a a tot + a t = a r

ω = ω 0 θ = θ + ω 0 t αt ( ) Rota%onal Kinema%cs: ( ONLY IF α = constant) v = ω r ω ω r s = θ r v = d θ dt r = ω r + a r = a a tot + a t = a r θ (t) ( θ 1 ) Δ θ = θ 2 s = θ r ω (t) = d θ (t) dt v = d θ dt r = ω r v = ω r α (t) = d ω (t) dt = d 2 θ (t) dt 2 a tot 2 = a r 2 + a t 2 = ω 2 r 2 + αr 2 a tot = a t + a r = a r ω ω r a t = α r ( ) Rota%onal

More information

Version A (01) Question. Points

Version A (01) Question. Points Question Version A (01) Version B (02) 1 a a 3 2 a a 3 3 b a 3 4 a a 3 5 b b 3 6 b b 3 7 b b 3 8 a b 3 9 a a 3 10 b b 3 11 b b 8 12 e e 8 13 a a 4 14 c c 8 15 c c 8 16 a a 4 17 d d 8 18 d d 8 19 a a 4

More information

Rotational Motion. 1 Purpose. 2 Theory 2.1 Equation of Motion for a Rotating Rigid Body

Rotational Motion. 1 Purpose. 2 Theory 2.1 Equation of Motion for a Rotating Rigid Body Rotational Motion Equipment: Capstone, rotary motion sensor mounted on 80 cm rod and heavy duty bench clamp (PASCO ME-9472), string with loop at one end and small white bead at the other end (125 cm bead

More information

Rotational Motion. Lecture 17. Chapter 10. Physics I Department of Physics and Applied Physics

Rotational Motion. Lecture 17. Chapter 10. Physics I Department of Physics and Applied Physics Lecture 17 Chapter 10 Physics I 11.13.2013 otational Motion Torque Course website: http://faculty.uml.edu/andriy_danylov/teaching/physicsi Lecture Capture: http://echo360.uml.edu/danylov2013/physics1fall.html

More information

Quiz Number 4 PHYSICS April 17, 2009

Quiz Number 4 PHYSICS April 17, 2009 Instructions Write your name, student ID and name of your TA instructor clearly on all sheets and fill your name and student ID on the bubble sheet. Solve all multiple choice questions. No penalty is given

More information

Use the following to answer question 1:

Use the following to answer question 1: Use the following to answer question 1: On an amusement park ride, passengers are seated in a horizontal circle of radius 7.5 m. The seats begin from rest and are uniformly accelerated for 21 seconds to

More information

Physics 53 Exam 3 November 3, 2010 Dr. Alward

Physics 53 Exam 3 November 3, 2010 Dr. Alward 1. When the speed of a rear-drive car (a car that's driven forward by the rear wheels alone) is increasing on a horizontal road the direction of the frictional force on the tires is: A) forward for all

More information

4.0 m s 2. 2 A submarine descends vertically at constant velocity. The three forces acting on the submarine are viscous drag, upthrust and weight.

4.0 m s 2. 2 A submarine descends vertically at constant velocity. The three forces acting on the submarine are viscous drag, upthrust and weight. 1 1 wooden block of mass 0.60 kg is on a rough horizontal surface. force of 12 N is applied to the block and it accelerates at 4.0 m s 2. wooden block 4.0 m s 2 12 N hat is the magnitude of the frictional

More information

Rotational Inertia (approximately 2 hr) (11/23/15)

Rotational Inertia (approximately 2 hr) (11/23/15) Inertia (approximately 2 hr) (11/23/15) Introduction In the case of linear motion, a non-zero net force will result in linear acceleration in accordance with Newton s 2 nd Law, F=ma. The moving object

More information

AP Physics Multiple Choice Practice Torque

AP Physics Multiple Choice Practice Torque AP Physics Multiple Choice Practice Torque 1. A uniform meterstick of mass 0.20 kg is pivoted at the 40 cm mark. Where should one hang a mass of 0.50 kg to balance the stick? (A) 16 cm (B) 36 cm (C) 44

More information

Planet X has the same mass as the Earth, but 1/2 the radius. (Planet X is more dense than Earth). What is the acceleration of gravity on Planet X?

Planet X has the same mass as the Earth, but 1/2 the radius. (Planet X is more dense than Earth). What is the acceleration of gravity on Planet X? Planet X has the same mass as the Earth, but 1/2 the radius. (Planet X is more dense than Earth). What is the acceleration of gravity on Planet X? A) g (same as Earth) B) 2g C) 4g D) 8g E) None of these.

More information

Distance travelled time taken and if the particle is a distance s(t) along the x-axis, then its instantaneous speed is:

Distance travelled time taken and if the particle is a distance s(t) along the x-axis, then its instantaneous speed is: Chapter 1 Kinematics 1.1 Basic ideas r(t) is the position of a particle; r = r is the distance to the origin. If r = x i + y j + z k = (x, y, z), then r = r = x 2 + y 2 + z 2. v(t) is the velocity; v =

More information

Practice Test 3. Multiple Choice Identify the choice that best completes the statement or answers the question.

Practice Test 3. Multiple Choice Identify the choice that best completes the statement or answers the question. Practice Test 3 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A wheel rotates about a fixed axis with an initial angular velocity of 20 rad/s. During

More information

Chapter 17 Two Dimensional Rotational Dynamics

Chapter 17 Two Dimensional Rotational Dynamics Chapter 17 Two Dimensional Rotational Dynamics 17.1 Introduction... 1 17.2 Vector Product (Cross Product)... 2 17.2.1 Right-hand Rule for the Direction of Vector Product... 3 17.2.2 Properties of the Vector

More information

A uniform rod of length L and Mass M is attached at one end to a frictionless pivot. If the rod is released from rest from the horizontal position,

A uniform rod of length L and Mass M is attached at one end to a frictionless pivot. If the rod is released from rest from the horizontal position, A dentist s drill starts from rest. After 3.20 s of constant angular acceleration, it turns at a rate of 2.51 10 4 rev/min. (a) Find the drill s angular acceleration. (b) Determine the angle (in radians)

More information

Rotation Work and Power of Rotation Rolling Motion Examples and Review

Rotation Work and Power of Rotation Rolling Motion Examples and Review Rotation Work and Power of Rotation Rolling Motion Examples and Review Lana Sheridan De Anza College Nov 22, 2017 Last time applications of moments of inertia Atwood machine with massive pulley kinetic

More information

Chapter 21 Rigid Body Dynamics: Rotation and Translation about a Fixed Axis

Chapter 21 Rigid Body Dynamics: Rotation and Translation about a Fixed Axis Chapter 21 Rigid Body Dynamics: Rotation and Translation about a Fixed Axis Chapter 21 Rigid Body Dynamics: Rotation and Translation about a Fixed Axis... 2 21.1 Introduction... 2 21.2 Translational Equation

More information

Y of radius 4R is made from an iron plate of thickness t. Then the relation between the

Y of radius 4R is made from an iron plate of thickness t. Then the relation between the Q. No. The moment of inertia of a body does not depend on : Option The mass of the body Option The angular velocity of the body Option The axis of rotation of the body Option The distribution of the mass

More information

Name: AP Physics C: Kinematics Exam Date:

Name: AP Physics C: Kinematics Exam Date: Name: AP Physics C: Kinematics Exam Date: 1. An object slides off a roof 10 meters above the ground with an initial horizontal speed of 5 meters per second as shown above. The time between the object's

More information

Advanced Higher Physics. Rotational motion

Advanced Higher Physics. Rotational motion Wallace Hall Academy Physics Department Advanced Higher Physics Rotational motion Problems AH Physics: Rotational Motion 1 2013 Data Common Physical Quantities QUANTITY SYMBOL VALUE Gravitational acceleration

More information

The Arctic is Melting

The Arctic is Melting The Arctic is Melting 1 Arctic sea has shown a large drop in area over the last thirty years. Japanese and US satellite data Jerry Gilfoyle The Arctic and the Length of Day 1 / 42 The Arctic is Melting

More information

Physics 201 Lab 9: Torque and the Center of Mass Dr. Timothy C. Black

Physics 201 Lab 9: Torque and the Center of Mass Dr. Timothy C. Black Theoretical Discussion Physics 201 Lab 9: Torque and the Center of Mass Dr. Timothy C. Black For each of the linear kinematic variables; displacement r, velocity v and acceleration a; there is a corresponding

More information

Practice Problems for Exam 2 Solutions

Practice Problems for Exam 2 Solutions MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01 Fall Term 008 Practice Problems for Exam Solutions Part I Concept Questions: Circle your answer. 1) A spring-loaded toy dart gun

More information

Lecture PowerPoints. Chapter 10 Physics for Scientists and Engineers, with Modern Physics, 4 th edition Giancoli

Lecture PowerPoints. Chapter 10 Physics for Scientists and Engineers, with Modern Physics, 4 th edition Giancoli Lecture PowerPoints Chapter 10 Physics for Scientists and Engineers, with Modern Physics, 4 th edition Giancoli 2009 Pearson Education, Inc. This work is protected by United States copyright laws and is

More information

Circular motion, Center of Gravity, and Rotational Mechanics

Circular motion, Center of Gravity, and Rotational Mechanics Circular motion, Center of Gravity, and Rotational Mechanics Rotation and Revolution Every object moving in a circle turns around an axis. If the axis is internal to the object (inside) then it is called

More information

Angular Speed and Angular Acceleration Relations between Angular and Linear Quantities

Angular Speed and Angular Acceleration Relations between Angular and Linear Quantities Angular Speed and Angular Acceleration Relations between Angular and Linear Quantities 1. The tires on a new compact car have a diameter of 2.0 ft and are warranted for 60 000 miles. (a) Determine the

More information

Lab 2: Equilibrium. Note: the Vector Review from the beginning of this book should be read and understood prior to coming to class!

Lab 2: Equilibrium. Note: the Vector Review from the beginning of this book should be read and understood prior to coming to class! Lab 2: Equilibrium Note: This lab will be conducted over 2 weeks, with half the class working with forces while the other half works with torques the first week, and then switching the second week. Description

More information

PES Physics 1 Practice Questions Exam 2. Name: Score: /...

PES Physics 1 Practice Questions Exam 2. Name: Score: /... Practice Questions Exam /page PES 0 003 - Physics Practice Questions Exam Name: Score: /... Instructions Time allowed for this is exam is hour 5 minutes... multiple choice (... points)... written problems

More information

The content contained in all sections of chapter 6 of the textbook is included on the AP Physics B exam.

The content contained in all sections of chapter 6 of the textbook is included on the AP Physics B exam. WORK AND ENERGY PREVIEW Work is the scalar product of the force acting on an object and the displacement through which it acts. When work is done on or by a system, the energy of that system is always

More information

Normal Force. W = mg cos(θ) Normal force F N = mg cos(θ) F N

Normal Force. W = mg cos(θ) Normal force F N = mg cos(θ) F N Normal Force W = mg cos(θ) Normal force F N = mg cos(θ) Note there is no weight force parallel/down the include. The car is not pressing on anything causing a force in that direction. If there were a person

More information

Written Homework problems. Spring (taken from Giancoli, 4 th edition)

Written Homework problems. Spring (taken from Giancoli, 4 th edition) Written Homework problems. Spring 014. (taken from Giancoli, 4 th edition) HW1. Ch1. 19, 47 19. Determine the conversion factor between (a) km / h and mi / h, (b) m / s and ft / s, and (c) km / h and m

More information

Practice Test 3. Name: Date: ID: A. Multiple Choice Identify the choice that best completes the statement or answers the question.

Practice Test 3. Name: Date: ID: A. Multiple Choice Identify the choice that best completes the statement or answers the question. Name: Date: _ Practice Test 3 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A wheel rotates about a fixed axis with an initial angular velocity of 20

More information

Solution Derivations for Capa #12

Solution Derivations for Capa #12 Solution Derivations for Capa #12 1) A hoop of radius 0.200 m and mass 0.460 kg, is suspended by a point on it s perimeter as shown in the figure. If the hoop is allowed to oscillate side to side as a

More information

Chapter 12. Rotation of a Rigid Body

Chapter 12. Rotation of a Rigid Body Chapter 12. Rotation of a Rigid Body Not all motion can be described as that of a particle. Rotation requires the idea of an extended object. This diver is moving toward the water along a parabolic trajectory,

More information

Equilibrium & Elasticity

Equilibrium & Elasticity PHYS 101 Previous Exam Problems CHAPTER 12 Equilibrium & Elasticity Static equilibrium Elasticity 1. A uniform steel bar of length 3.0 m and weight 20 N rests on two supports (A and B) at its ends. A block

More information

Dynamics of Rotational Motion: Rotational Inertia

Dynamics of Rotational Motion: Rotational Inertia Connexions module: m42179 1 Dynamics of Rotational Motion: Rotational Inertia OpenStax College This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License

More information

Conservation of Angular Momentum

Conservation of Angular Momentum Physics 101 Section 3 March 3 rd : Ch. 10 Announcements: Monday s Review Posted (in Plummer s section (4) Today start Ch. 10. Next Quiz will be next week Test# (Ch. 7-9) will be at 6 PM, March 3, Lockett-6

More information

Motion Of An Extended Object. Physics 201, Lecture 17. Translational Motion And Rotational Motion. Motion of Rigid Object: Translation + Rotation

Motion Of An Extended Object. Physics 201, Lecture 17. Translational Motion And Rotational Motion. Motion of Rigid Object: Translation + Rotation Physics 01, Lecture 17 Today s Topics q Rotation of Rigid Object About A Fixed Axis (Chap. 10.1-10.4) n Motion of Extend Object n Rotational Kinematics: n Angular Velocity n Angular Acceleration q Kinetic

More information

PHY218 SPRING 2016 Review for Final Exam: Week 14 Final Review: Chapters 1-11, 13-14

PHY218 SPRING 2016 Review for Final Exam: Week 14 Final Review: Chapters 1-11, 13-14 Final Review: Chapters 1-11, 13-14 These are selected problems that you are to solve independently or in a team of 2-3 in order to better prepare for your Final Exam 1 Problem 1: Chasing a motorist This

More information

General Physics I. Lecture 8: Rotation of a Rigid Object About a Fixed Axis. Prof. WAN, Xin ( 万歆 )

General Physics I. Lecture 8: Rotation of a Rigid Object About a Fixed Axis. Prof. WAN, Xin ( 万歆 ) General Physics I Lecture 8: Rotation of a Rigid Object About a Fixed Axis Prof. WAN, Xin ( 万歆 ) xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ New Territory Object In the past, point particle (no rotation,

More information

Axis. Axis. Axis. Hoop about. Annular cylinder (or ring) about central axis. Solid cylinder (or disk) about. central axis. central axis I = MR 2 1

Axis. Axis. Axis. Hoop about. Annular cylinder (or ring) about central axis. Solid cylinder (or disk) about. central axis. central axis I = MR 2 1 Instructor(s): Matchea/Yelton PHYSICS DEPATMENT PHY 2048 Exam 2 Noember 7th, 207 Name (print, last first): Signature: On my honor, I hae neither gien nor receied unauthorized aid on this examination. YOU

More information

Rotational Motion Test

Rotational Motion Test Rotational Motion Test Multiple Choice: Write the letter that best answers the question. Each question is worth 2pts. 1. Angular momentum is: A.) The sum of moment of inertia and angular velocity B.) The

More information

Momentum Circular Motion and Gravitation Rotational Motion Fluid Mechanics

Momentum Circular Motion and Gravitation Rotational Motion Fluid Mechanics Momentum Circular Motion and Gravitation Rotational Motion Fluid Mechanics Momentum Momentum Collisions between objects can be evaluated using the laws of conservation of energy and of momentum. Momentum

More information

Torque and Static Equilibrium

Torque and Static Equilibrium Torque and Static Equilibrium Ch. 7.3, 8.1 (KJF 3 rd ed.) Phys 114 Eyres Torque 1 Causing rotation: Torque Three factors affect the how a force can affect the rotation: The magnitude of the force The distance,

More information

Circular Motion and Universal Law of Gravitation. 8.01t Oct 4, 2004

Circular Motion and Universal Law of Gravitation. 8.01t Oct 4, 2004 Circular Motion and Universal Law of Gravitation 8.01t Oct 4, 2004 Summary: Circular Motion arc length s= Rθ tangential velocity ds v = = dt dθ R = Rω dt 2 d θ 2 dt tangential acceleration a θ = dv θ =

More information

CHAPTER 10 ROTATION OF A RIGID OBJECT ABOUT A FIXED AXIS WEN-BIN JIAN ( 簡紋濱 ) DEPARTMENT OF ELECTROPHYSICS NATIONAL CHIAO TUNG UNIVERSITY

CHAPTER 10 ROTATION OF A RIGID OBJECT ABOUT A FIXED AXIS WEN-BIN JIAN ( 簡紋濱 ) DEPARTMENT OF ELECTROPHYSICS NATIONAL CHIAO TUNG UNIVERSITY CHAPTER 10 ROTATION OF A RIGID OBJECT ABOUT A FIXED AXIS WEN-BIN JIAN ( 簡紋濱 ) DEPARTMENT OF ELECTROPHYSICS NATIONAL CHIAO TUNG UNIVERSITY OUTLINE 1. Angular Position, Velocity, and Acceleration 2. Rotational

More information

University of Houston Mathematics Contest: Physics Exam 2017

University of Houston Mathematics Contest: Physics Exam 2017 Unless otherwise specified, please use g as the acceleration due to gravity at the surface of the earth. Vectors x, y, and z are unit vectors along x, y, and z, respectively. Let G be the universal gravitational

More information

8.012 Physics I: Classical Mechanics Fall 2008

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

More information

Friction is always opposite to the direction of motion.

Friction is always opposite to the direction of motion. 6. Forces and Motion-II Friction: The resistance between two surfaces when attempting to slide one object across the other. Friction is due to interactions at molecular level where rough edges bond together:

More information

Equilibrium and Torque

Equilibrium and Torque Equilibrium and Torque Equilibrium An object is in Equilibrium when: 1. There is no net force acting on the object 2. There is no net Torque (we ll get to this later) In other words, the object is NOT

More information