# Chapter 8. Rotational Equilibrium and Rotational Dynamics

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1 Chapter 8 Rotational Equilibrium and Rotational Dynamics 1

2 Force vs. Torque Forces cause accelerations Torques cause angular accelerations Force and torque are related 2

3 Torque The door is free to rotate about an axis through O There are three factors that determine the effectiveness of the force in opening the door: The magnitude of the force The position of the application of the force The angle at which the force is applied 3

4 Torque, cont Torque, τ, is the tendency of a force to rotate an object about some axis τ = r F τ is the torque F is the force symbol is the Greek tau r is the length of the position vector SI unit is N. m 4

5 QQ1 Using a screwdriver, you try to remove a screw from a piece of furniture, but can t get it to turn. To increase the chances of success, you should use a screwdriver that (a) is longer, (b) is shorter, (c) has a narrower handle (d) has a wider handle. 5

6 Direction of Torque Torque is a vector quantity The direction is perpendicular to the plane determined by the position vector and the force If the turning tendency of the force is counterclockwise, the torque will be positive If the turning tendency is clockwise, the torque will be negative 6

7 Right Hand Rule Point the fingers in the direction of the position vector (beginning from the axis of rotation) Curl the fingers toward the force vector The thumb points in the direction of the torque Out of the page for the figure 7

8 Multiple Torques When two or more torques are acting on an object, the torques are added As vectors. Take sign into account If the net torque is zero, the object s rate of rotation doesn t change 8

9 General Definition of Torque The applied force is not always perpendicular to the position vector When the force is parallel to the position vector, no rotation occurs When the force is at some angle, the perpendicular component causes the rotation 9

10 General Definition of Torque, final Taking the angle into account leads to a more general definition of torque: τ = r F sin θ F is the force r is the position vector θ is the angle between the force and the position vector 10

11 Lever Arm The lever arm, d, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force d = r sin θ Formula for torque may be rearranged to show that the torque is proportional to the force F times lever arm τ = r F sin θ = F (r sin θ) = F d 11

12 Net Torque The net torque is the sum of all the torques produced by all the forces Remember to account for the direction of the tendency for rotation which gives the sign: Counterclockwise torques are positive Clockwise torques are negative 12

13 Torque and Equilibrium First Condition of Equilibrium r Σ F = 0 or r r Σ F = 0 and Σ F = 0 The net external force must be zero x This is a statement of translational equilibrium (we know it from previous chapters) This is a necessary, but not sufficient, condition to ensure that an object is in complete mechanical equilibrium: the object may rotate! y 13

14 Torque and Equilibrium, cont To ensure mechanical equilibrium, you need to ensure rotational equilibrium as well as translational The Second Condition of Equilibrium (for rotation) states The net external torque must be zero r Σ τ = 0 14

15 Equilibrium Example (8.3) The woman, mass m, sits on the left end of the see-saw The man, mass M, sits where the seesaw will be balanced Apply the Second Condition of Equilibrium and solve for the unknown distance, x Free body diagram 15

16 Axis of Rotation If the object is in equilibrium, it does not matter where you put the axis of rotation for calculating the net torque The location of the axis of rotation is completely arbitrary Often the nature of the problem will suggest a convenient location for the axis When solving a problem, you must specify an axis of rotation Once you have chosen an axis, you must maintain that choice consistently throughout the problem 16

17 Example (8.7 page 235) Beam The free body diagram includes the directions of the forces The weights act through the centers of gravity of their objects Fig 8.12, p.228 Slide 17 17

18 Fig. P8-20, 18p.256

19 Fig. 8-2, 19p.228

20 Fig. P8-58, 20p.260

21 Center of Gravity The force of gravity acting on an object must be considered In finding the torque produced by the force of gravity, all of the weight of the object can be considered to be concentrated at a single point, called the center of gravity 21

22 Calculating the Center of Gravity The object is divided up into a large number of very small particles with small weight Each particle will have a set of coordinates indicating its location (x,y) 22

23 Calculating the Center of Gravity, cont. We assume the object is free to rotate about its center The torque produced by each particle about the axis of rotation is equal to its weight times its lever arm For example, for a particle 1 with mass m 1 τ 1 = m 1 g x 1 23

24 Calculating the Center of Gravity, cont. We wish to locate the point of application of the single force whose magnitude is equal to the total weight of the object, and whose effect on the rotation is the same as all the individual particles. This point is called the center of gravity of the object 24

25 Coordinates of the Center of Gravity The coordinates of the center of gravity can be found from the sum of the torques acting on the individual particles being set equal to the torque produced by the weight of the object Σmx i i Σmy i i xcg = and ycg = Σm Σm i i 25

26 Center of Gravity (C.G.)of a Uniform Object The center of gravity of a homogenous, symmetric body must lie on the axis of symmetry. Often, the center of gravity of such an object is the geometric center of the object. Example: for the ring, the c.g. is in the center of the ring 26

27 Experimentally Determining the Center of Gravity The wrench is hung freely from two different pivots The intersection of the lines indicates the center of gravity A rigid object can be balanced by a single force equal in magnitude to its weight as long as the force is acting upward through the object s center of gravity 27

28 28p.230

29 Fig. Q8-12,29p.253

30 Notes About Equilibrium A zero net torque does not mean the absence of rotational motion An object that rotates at uniform angular velocity can be under the influence of a zero net torque This is analogous to the translational situation where a zero net force does not mean the object is not in motion 30

31 Solving Equilibrium Problems Draw a diagram of the system Include coordinates and choose a rotation axis Isolate the object being analyzed and draw a free body diagram showing all the external forces acting on the object For systems containing more than one object, draw a separate free body diagram for each object 31

32 Problem Solving, cont. Apply the Second Condition of Equilibrium This will yield a single equation, often with one unknown which can be solved immediately Apply the First Condition of Equilibrium This will give you two more equations Solve the resulting simultaneous equations for all of the unknowns 32

33 Example of a Free Body Diagram (Ladder) The free body diagram shows the normal force and the force of static friction acting on the ladder at the ground The last diagram shows the lever arms for the forces 33

34 Torque and Angular Acceleration When a rigid object is subject to a net torque ( 0), it undergoes an angular acceleration The angular acceleration is directly proportional to the net torque 34

35 Moment of Inertia The analog for force for rotation is torque The mass analog for the rotation is called the moment of inertia, I, of the object I Σmr SI units are kg m

36 Newton s Second Law for a Rotating Object Σ τ = I α The angular acceleration is directly proportional to the net torque The relationship is similar to F = ma Newton s Second Law for translation 36

37 QQ2 A constant net torque is applied to an object. Which one of the following will not be constant? (a) angular acceleration (b) angular velocity (c) moment of inertia (d) center of gravity 37

38 More About Moment of Inertia There is a major difference between moment of inertia and mass: the moment of inertia depends on the quantity of matter and its distribution in the rigid object. The moment of inertia also depends upon the location of the axis of rotation Also the units are different 38

39 Moment of Inertia of a Uniform Ring Image the hoop is divided into a number of small segments, m 1 These segments are equidistant from the axis I =Σ mr = MR 2 2 i i 39

40 Other Moments of Inertia 40

41 QQ3 The two rigid objects shown in Figure have the same mass, radius, and angular speed. If the same braking torque is applied to each, which takes longer to stop? (a) A (b) B (c) more information is needed 41

42 Fig. 8-16, 42p.238

43 Example, Newton s Second Law for Rotation Draw free body diagrams of each object Only the cylinder is rotating, so apply Στ = I α The bucket is falling, but not rotating, so apply ΣF = m a Remember that a = α r and solve the resulting equations 43

44 Rotational Kinetic Energy An object rotating about some axis with an angular speed, ω, has rotational kinetic energy ½Iω 2 Similar= to the formula for a known formula for kinetic energy (for translational motion) Ek = ½mv 2 Energy concepts can be useful for simplifying the analysis of rotational motion 44

45 QQ4 Two spheres, one hollow and one solid, are rotating with the same angular speed around an axis through their centers. Both spheres have the same mass and radius. Which sphere, if either, has the higher rotational kinetic energy? (a) The hollow sphere. (b) The solid sphere. (c) They both have the same kinetic energy. 45

46 Total Energy of a System Conservation of Mechanical Energy ( KE + KE + PE ) = ( KE + KE + PE ) t r g i t r g f Remember, this is for conservative forces, no dissipative (non-conservative) forces such as friction can be present Potential energies of any other conservative forces could be added (e.g. spring) 46

47 QQ 5 (Modified) Which arrives at the bottom first: (a) a ball (hollow inside) rolling without sliding down a certain incline A, (b) a solid cylinder rolling without sliding down incline A, or (c) a box sliding down a frictionless incline B having the same dimensions as A? Assume that all objects have the same mass and each object is released from rest at the top of its incline. 47

48 Work-Energy in a Rotating System In the case where there are dissipative forces such as friction, use the generalized Work-Energy Theorem instead of Conservation of Energy W nc = ΔKE t + ΔKE R + ΔPE 48

49 General Problem Solving Hints The same basic techniques that were used in linear motion can be applied to rotational motion. τ α Analogies: F becomes, m becomes I and a becomes, v becomes ω and x becomes θ 49

50 Problem Solving Hints for Energy Methods Choose two points of interest One where all the necessary information is given The other where information is desired Identify the conservative and non-conservative forces 50

51 Problem Solving Hints for Energy Methods, cont Write the general equation for the Work-Energy theorem if there are non-conservative forces Alternatively use Conservation of Energy if there are no non-conservative forces Use v =rω if necessary to combine translational and rotational terms Solve for the unknown 51

52 Fig. 8-25, 52p.245

53 Fig. P8-36, 53p.258

54 Angular Momentum Similarly to the relationship between force and momentum in a linear system, we can show the relationship between torque and angular momentum Angular momentum is defined as L = I ω (similar to M=mv) r ΔL Δ p mv ( f v) r i and Σ τ = (similar to = = Fnet ) Δ t Δt Δt 54

55 Angular Momentum, cont Conservation of Angular Momentum states: The angular momentum of a system is conserved (remains constant) when the net external torque acting on the systems is zero. That is, when Σ τ = = ω = ω 0, Li Lf or Ii i If f 55

56 Conservation of Angular Momentum, Example With hands and feet drawn closer to the body, the skater s angular speed increases L is conserved, I decreases, ω increases 56

57 QQ (Example 8.15) A student walks toward the center of the rotating platform. The angular speed of the platform will: (a) increase (b) decrease (c) stay the same 57

58 Fig. 8-30, 58p.249

59 QQ7 Al Gore quiz If global warming continues, it s likely that some ice from the polar ice caps of the Earth will melt and the water will be distributed closer to the Equator. If this occurs, would the length of the day (one revolution) (a) increase, (b) decrease, or (c) remain the same? 59

60 Conservation Rules, Summary In an isolated system (with no external forces acting on the system, and no internal dissipation of energy), the following quantities are conserved: Mechanical energy Linear momentum Angular momentum 60

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