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Chapter 10: Dynamics of rotational motion • Torque: Is it a force? • torques rotational motion (just as forces linear accelerations) • combination of translation and rotation: rolling objects • Calculation of work done by a torque • Angular momentum conservation • rotational dynamics and angular momentum: they are related The race of objects with different moments • This problem is a “classic” and appears on most professors’ exams on this material. (example 10.5) • Which objects reaches the bottom first? Torque: How to rotate an object more effectively • A force applied at a right angle to a lever will generate a torque. • The distance from the pivot to the point of force application will be linearly proportional to the torque produced. Lines of force and calculations of torques • The directions of torques: use the righthand rule (RHR). τ = r × F = rF sin φ = Ftan r There are many ways to understand the above formula; What would you choose? (Level arm) φ: angle between the two vectors r and F How to turn an object more effectively? Larger force! Longer distance from the rotating axis Apply that force perpendicular to the radius! Calculate an applied torque: Loosening a pipe fitting • To be a good plumber (they make more money than you do!), what do you need? • Strength; Knowledge; and good tools. A plumber pushes straight down on the end of a long wrench as shown. What is the magnitude of the torque he applies about the pipe at lower right? A. (0.80 m)(900 N)sin 19° B. (0.80 m)(900 N)cos 19° C. (0.80 m)(900 N)tan 19° D. none of the above Calculate an applied torque: Loosening a pipe fitting • To be a good plumber (they make more money than you do!), what do you need? • Strength; Knowledge; and good tools. A plumber pushes straight down on the end of a long wrench as shown. What is the magnitude of the torque he applies about the pipe at lower right? A. (0.80 m)(900 N)sin 19° B. (0.80 m)(900 N)cos 19° C. (0.80 m)(900 N)tan 19° D. none of the above Q10.1 The four forces shown all have the same magnitude: F1 = F2 = F3 = F4. Which force produces the greatest torque about the point O (marked by the blue dot)? F1 F3 O F2 F4 A. F1 B. F2 C. F3 D. F4 E. not enough information given to decide Q10.2 Which of the four forces shown here produces a torque about O that is directed out of the plane of the drawing? A. F1 B. F2 C. F3 D. F4 E. more than one of these F1 F3 O F2 F4 Q10.4 A force acts an object at a point located at the position What is the torque that this force applies about the origin? A. zero B. C. D. E. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Newton’s laws still valid: Τ = Iα is just like F = ma • Refer to Figures 10.6 and 10.8 while considering the application of Figure 10.7. What will the newton’s laws about motions become in rotational dynamics? Unwinding cable: Example 10.3 • Example 9.9 used energy methods. What is speed of the block when it hits the floor? vy 1 1 1 mgh = mv y2 + ( MR 2 )ω 2 ,ω = 2 2 2 R v y2 2gh g ⇒ vy = ⇒ ay = = M 2h 1+ M 1+ 2m 2m • Use rotational dynamics: ∑τ = RT = Iα = (0.5MR )α ∑ F = mg + (−T) = ma 2 z € y z y • Block and pulley’s accelerations are related v y = Rω ⇒ ay = Rα z € dv y dω =R ⇒ dt dt z € A rigid body in motion about a moving axis • What is the total kinetic energy of a rigid body rotating about a moving axis 1 1 2 K = Mv cm + Icmω 2 2 2 Translational rotational € How to derive this: 1 K i = mi (v cm + v i' ) 2 2 1 1 2 K = ∑ K i = ∑ ( miv cm ) +∑ (miv cm • v i' ) +∑ ( miv 'i2 ) 2 2 Translational 0 rotational (∑ miv i' ) • v cm ) = 0 Rolling with and without slipping • Rolling with slipping may be calculated. Slipping makes things worse (for driving and calculations). • What does “without slipping mean”? The race of objects with different moments • This problem is a “classic” and appears on most professors’ exams on this material. (example 10.5) • Which objects reaches the bottom first? Icm = cMR 2 1 1 2 Mgh = Mv cm + Icmω 2 2 2 1 1 v 2 = Mv cm + (cMR 2 )( cm ) 2 2 2 R ⇒ v cm = € 2gh 1+ c What does c=0 mean? Compact objects wins the race! Consider the acceleration of a rolling sphere • Refer to Example 10.7 and Figure 10.19. Consider the acceleration of a rolling sphere II • Figure 10.20 will introduce an effect of friction. Work and power in rotational motion • Figure 10.21 is a game I loved as a child. • Examples 10.8 and 10.9 allow us to follow calculations of power and torque. The race of objects with different moments • We learned that the most compact object rolls down fastest: smallest moment of inertia: “I” • What caused these objects to roll in the first place? Q10.6 A lightweight string is wrapped several times around the rim of a small hoop. If the free end of the string is held in place and the hoop is released from rest, the string unwinds and the hoop descends. How does the tension in the string (T) compare to the weight of the hoop (w)? A. T = w B. T > w C. T < w D. not enough information given to decide Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Consider the acceleration of a rolling sphere • Friction is the reason the ball rolls without slipping • Friction also is the reason the ball rotates! • Does the friction do work? ∑F x = Mgsin β + (− f ) = Macm −x 2 = Rf = I α = ( MR 2 )α z z z 5 acm −x = Rα z ∑τ ⇒ acm −x = ?g f = ? Mg µs−min = ? € Work and angular momentum in rotational motion • We have already learnt “work” in linear motion. • Just change the symbols! W = τ (θ 2 − θ1 ) W = F(x 2 − x1 ) W = ∫ x2 x1 1 1 F(x)dx = mv 22 − mv12 2 2 p = mv dp Ftot = dt € € θ2 1 1 W = ∫θ τ (θ )dθ = Iω 22 − Iω12 1 2 2 L = Iω New concepts dL τ tot = dt Spinning professor: Not me (Example 10.11) I1prof = 3.0kg • m 2 ω1 = 0.5rev /s I2prof = 2.2kg • m 2 r2dumbbell = 0.20m ω 2 = ?rev /s r1dumbbell = 1.0m mdumbbell = 5.0kg € total total Angular momentum conservation I1 ω1 = I2 ω 2 (Is the kinetic energy conserved? Decreased? Or increase? Remember the pair skaters pushing each other?: € € m1 v1 = m2v 2 How does a car’s clutch works? • Engine flywheel (A) will run against the clutch plate until they have the same angular velocities • Different clutch plates have different moments of inertia • Is the total kinetic energy conserved? Assume Ftot-external=0 Gyroscopic precession • The precession of a gyroscope shows up in many “common” situations. If we release the flywheel axis without the wheel spinning, the system will drop. What will happen if the flywheel was spinning fast at the beginning? Consider a similar situation: Gyroscope: A device to measure/maintain orientation Precession:Change of the rotation axis v0=0 v0 not 0 F not 0 F not 0 A rotating flywheel dL τ tot = =r×F dt € • The shifting self-rotation axis of the earth is a special case of precession! (The reason?) Since the torque τ is always perpendicular to the angular Momentum L, |L| remains the same The only thing changing is the L direction, hence the rotation axis Summary τ = r × F = rF sin φ = Ftan r • Torque is not a force; Cross products • Torque changes the angular momentum/angular velocity dω dL τ= = Iα = I dt dt Newton’s 2nd law • Work-energy theorem in rotational dynamics W = • Angular momentum ∫ θ2 θ1 1 1 τdθ = Iω 22 − Iω12 2 2 L = r × p = r × mv L = Iω Particle Rigid body • Angular momentum is conserved when total external torque is 0 • Total kinetic energy of a rotating rigid body K= 1 1 2 Mv cm + Icmω 2 2 2 • Code word rolling without slipping: zero velocity at contact point • Precession: changing rotation axis €