# Chapter 10. Rotation

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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

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