Lecture Outlines Chapter 11 Physics, 3rd Edition James S. Walker © 2007 Pearson Prentice Hall This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted. The work and materials from it should never be made available to students except by instructors using the accompanying text in their classes. All recipients of this work are expected to abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors who rely on these materials.

Rotational Dynamics and Static Equilibrium Chapter 11 Rotational Dynamics and Static Equilibrium

Units of Chapter 11 Torque (vridmoment) Torque and Angular Acceleration Zero Torque and Static Equilibrium Center of Mass and Balance Dynamic Applications of Torque Angular Momentum (rörelsemängdsmoment)

Units of Chapter 11 Conservation of Angular Momentum Rotational Work and Power The Vector Nature of Rotational Motion*

11-1 Torque From experience, we know that the same force will be much more effective at rotating an object such as a nut (mutter) or a door if our hand is not too close to the axis. This is why we have long-handled wrenches (shiftnyckel), and why doorknobs are not next to hinges (gångjärn).

11-1 Torque We define a quantity called torque: The torque increases as the force increases, and also as the distance increases.

Exercise 11-1 För att öppna dörren i figur 11-1b krävs ett vridmoment på 3,1 Nm. Vilken (tangentiell) kraft krävs om hävarmen är 0,94 m ? b) 0,35 m ? Enligt formeln nedan så behövs i fallet F = 3,1 Nm/0,94 m = 3,3 N F = 3,1 Nm/0,35 m = 8,9 N

11-1 Torque Only the tangential component of force causes a torque:

11-1 Torque This leads to a more general definition of torque:

11-1 Torque If the torque causes a counterclockwise angular acceleration, it is positive; if it causes a clockwise angular acceleration, it is negative.

Example 11-1 Torques to the Left and Torques to the Right Two helmsmen, in disagreement about which way to turn a ship, exert the forces shown below on a ship’s wheel. The wheel has a radius of 0.74 m, and the two forces have the magnitudes F1 = 72 N and F2 = 58 N. Find (a) the torque caused by F1 and (b) the torque caused by F2. (c) In which direction does the wheel turn as a result of these two forces?

Example 11-1 Torque to the left and Torque to the Right Två styrmän är oense om hur man skall styra båten och applicerar krafterna F1 = 72 N och F2 = 58 N enligt figur. Vad är vridmomentet för respektive kraft då r = 0,74 m? Hur kommer ratten att vrida sig? τ1 = r F1 sin 50,0° = 40,8 Nm τ2 = - r F2 = - 42,9 Nm τ = 40,8 Nm – 42,9 Nm = - 2,1 Nm (dvs medurs)

Figure 11-4 Torque and angular acceleration A tangential force F applied to a mass m gives it a linear acceleration of magnitude a = F/m. The corresponding angular acceleration is α = τ/I, where τ = rF and I = mr2.

11-2 Torque and Angular Acceleration Newtons andra lag lyder: Från kapitel 10 vet vi att för en massa m som roterar på avståndet r från en axel så gäller att om vinkelhastighen ändras att Δvt = rΔω så att den tangentiella accelerationen at kan skrivas at = rΔω/Δt = r α (ekvation 10-14) Vinkelacceleration blir (I = tröghetsmomentet) α = at/r = Ft /mr = Ft r/mr2 = τ/I

11-2 Torque and Angular Acceleration Once again, we have analogies between linear and angular motion:

Example 11-2 A Fish Takes the Line Spänningen i linan blir då T Example 11-2 A Fish Takes the Line Spänningen i linan blir då T. Rullen roterar utan friktion under tiden t. Vad blir a) Δθ ? b) Hur mycket av linan dras ut? c) ωf =? A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

Example 11-2 A Fish Takes the Line Spänningen i linan blir då T Example 11-2 A Fish Takes the Line Spänningen i linan blir då T. Rullen roterar utan friktion under tiden t. Vad blir a) Δθ ? b) Hur mycket av linan dras ut? c) ωf =? τ = R T τ = I α α = RT / I a) Δθ = θ – θ0 = ω0 t + αt2/2 = {ω0 = 0} = RTt2/2I b) Δx = R Δθ = R2Tt2/2I c) ωf = ω0 + α t = RTt / I [vilket vi naturligtvis också hade fått från ekvationen ωf2 = ωi2+ 2α Δθ ] A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

Conceptual Checkpoint 11–1 Which Block Lands First? (α = RT / I) The rotating systems shown below differ only in that the two spherical movable masses are positioned either far from the axis of rotation (left), or near the axis of rotation (right). If the hanging blocks are released simultaneously from rest, is it observed that (a) the block at left lands first, (b) the block at right lands first, or (c) both blocks land at the same time?

Example 11-3 Drop It A person holds his outstretched arm at rest in a horizontal position. The mass of the arm is m and its length is 0.740 m. When the person releases his arm, allowing it to drop freely, it begins to rotate about the shoulder joint. Find (a) the initial angular acceleration of the arm, and (b) the initial linear acceleration of the man’s hand. (Hint: In calculating the torque, assume the mass of the arm is concentrated at its midpoint. In calculating the angular acceleration, use the moment of inertia of a uniform rod of length L about one end; I = 1/3 mL2.)

Example 11-3 Drop it Armens massa är m och dess längd 0,740 m Example 11-3 Drop it Armens massa är m och dess längd 0,740 m. a) Vad är begynnelse(vinkel)accelerationen? b) Vad är den linjära (tangentiella) begynnelseaccelerationen för handen? Hint: Armen kan betraktas som en homogen stång (som har tröghetsmomentet I = mL2/3). a) τ = I α α = mgL/2I = 3g/2L = 19,9/s2 Enligt 10-14 b) at = r α = L α = 14,7 m/s2 (för r = 2L/3 är at = g) A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

11-3 Zero Torque and Static Equilibrium Static equilibrium occurs when an object is at rest – neither rotating nor translating.

11-3 Zero Torque and Static Equilibrium If the net torque is zero, it doesn’t matter which axis we consider rotation to be around; we are free to choose the one that makes our calculations easiest. (F1 = ? , F2 = ?)

Example 11-4a Taking the Plunge (L = 5,00 m, d = 1,50 m, m = 90 kg Example 11-4a Taking the Plunge (L = 5,00 m, d = 1,50 m, m = 90 kg. Hur stora är F1 och F2?) A 5.00-m long diving board of negligible mass is supported by two pillars. One pillar is at the left end of the diving board, as shown below; the other is 1.50m away. Find the forces exerted by the pillars when a 90.0-kg diver stands at the far end of the board.

Example 11-4b Taking the Plunge (d•F1 = (L- d)•mg, d•F2 = mg•L) F1 = - 2060 kN, F2 = 2943 N

Active Example 11-2 Walking the Plank: Find the Mass A cat walks along a uniform plank that is 4.00 m long and has a mass of 7.00 kg. The plank is supported by two sawhorses, one 0.440 m from the left end of the board and the other 1.50 m from its right end. When the cat reaches the right end, the plank just begins to tip. What is the mass of the cat?

Active Example 11-2 Walking the Plank Find the Mass M = 7,00 kg, plankan är 4,00 m och står på två sågbockar, 0,440 m överhäng till vänster och 1,50 m till höger. När katten står längs ut till höger på plankan är normalkraften på vänstra bocken = 0. Beräkna kattens massa. Vridmomentet kring högra bocken ger direkt att mg(1,50 m) = Mg(0,50 m) (CM på mitten) m = M/3 = 2,33 kg A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

11-3 Zero Torque and Static Equilibrium When forces have both vertical and horizontal components, in order to be in equilibrium an object must have no net torque, and no net force in either the x- or y-direction.

Vi vill beräkna T och f (Standard y/x) Beräkna momenten i f:s angreppspunkt. Vridmomentet = 0 ger direkt T eftersom mgH = TV Kraftjämvikt i y-led ger fy = mg Kraftjämvik i x-led ger fx = T Om m = 2,00 kg V = 12,0 cm och H = 15,0 cm blir T = mgH/V = 1,25 mg f = ((1,25)2 + 12)1/2 mg ≈ 1,6 mg Kanske värt att lägga märke till krafterna på stöden är större än vad föremålet väger, som de skall hålla upp! A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

Example 11-5 Arm in a Sling Betrakta underarmen och handen som homogen med massan 1,31 kg och längden 0,300 m. a) Hur stor är spänningen T? b) fx = ?, fy = ? A hiker who has broken his forearm rigs a temporary sling using a cord stretching from his shoulder to his hand. The cord holds the forearm level and makes an angle of 40.0° with the horizontal where it attaches to the hand. Considering the forearm and hand to be uniform, with a total mass of 1.31 kg and a length of 0.300 m, find (a) the tension in the cord and (b) the horizontal and vertical components of the force, f, exerted by the humerus (the bone of the upper arm) on the radius and ulna (the bones of the forearm).

Example 11-5 Arm in a Sling Vi vill beräkna T, fx och fy (Standard y/x) Beräkna momentet i f:s angreppspunkt. T sin(40,0°) L = mgL/2 T = 1,31•9,81/(2•0,643) = 10,0 (N) Kraftjämvikt i y-led ger fy + T sin(40,0°) = mg fy = 1,31•9,81 – 6,43 = 6,42 (N) Kraftjämvik i x-led ger fx = Tcos(40,0°) = 7,66 (N) A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

Active Example 11-3 Don’t Walk Under the Ladder: Find the Forces An 85-kg person stands on a lightweight ladder, as shown. The floor is rough; hence, it exerts both a normal force, f1, and a frictional force, f2, on the ladder. The wall, on the other hand, is frictionless; it exerts only a normal force, f3. Using the dimensions given in the figure, find the magnitudes of f1, f2, and f3.

Active Example 11-3 Don’t Walk under the Ladder: Find the Forces (m = 85,0 kg) (Standard y/x) Vridmomentet kring stegens nedre punkt ger att mg(0,70 m) = f3(3,8 m) f3 = 150 N Kraftjämvikt i x-led ger f2 = f3 = 150 N Kraftjämvikt i y-led ger f1 = mg = 830 N A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

11-4 Center of Mass and Balance If an extended object is to be balanced, it must be supported through its center of mass.

11-4 Centre of Mass Balance (Standard y/x) Vridmomentet kring upphängningspunkten ger direkt att m1x1 - m2x2 = 0 dvs m1x1= m2x2 Använd nu definitionen på masscentrum enligt 9-14 Xcm = ∑mixi /M = (m1{-x1} + m2x2 )/(m1+ m2) med hjälp av relationen ovan inses direkt att Xcm = 0 så upphängningpunkten går genom systemets tyngdpunkt A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

Example 11-6 A Well-Balanced Meal Beräkna m1, m2 och m3 Example 11-6 A Well-Balanced Meal Beräkna m1, m2 och m3. Stavar och trådar kan anses masslösa. As a grade-school project, students construct a mobile representing some of the major food groups. Their completed artwork is shown below. Find the masses m1, m2, and m3 that are required for a perfectly balanced mobile. Assume the strings and the horizontal rods have negligible mass.

11-4 Centre of Mass Balance (Standard y/x) Vridmomentet kring understa upphängningspunkten ger direkt att m1•g•12 cm = m2•g•18 cm dvs m1 = 1,5 m2 Vridmomentet kring näst understa upphängningspunkten ger direkt att 2,5• m2 •g •6 cm = 0,30 kg•g•24 cm dvs m2 = 0,48 kg m1 = 0,72 kg Vridmomentet kring näst understa upphängningspunkten ger direkt att (0,72 + 0,48 + 0,30 + 0,20)kg•g•18 cm = m3•g•31 cm m3 = 0,99 kg A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

11-4 Center of Mass and Balance This fact can be used to find the center of mass of an object – suspend it from different axes and trace a vertical line. The center of mass is where the lines meet.

Figure 11-8 Equilibrium of a suspended (upphängt) object (a) If an object’s center of mass is directly below the suspension point, its weight creates zero torque and the object is in equilibrium. (b) When an object is rotated, so that the center of mass is no longer directly below the suspension point, the object’s weight creates a torque. The torque tends to rotate the object to bring the center of mass under the suspension point.

Conceptual Checkpoint 11–2 Compare the Masses mhuvud > = < mskaft? A croquet mallet balances when suspended from its center of mass, as indicated in the drawing at left. If you cut the mallet in two at its center of mass, as in the drawing at right, how do the masses of the two pieces compare? (a) The masses are equal; (b) the piece with the head of the mallet has the greater mass; or (c) the piece with the head of the mallet has the smaller mass.

11-5 Dynamic Applications of Torque When dealing with systems that have both rotating parts and translating parts, we must be careful to account for all forces and torques correctly.

11-5 Dynamic Applications of Torque (Standard y/x) Newtons andra lag ger för tyngden T – mg = ma Newtons andra lag ger för blocket -T R = I α Om repet inte glider gäller att relationen mellan vinkel accelerationen och (den linjära) acceleration är (10-14) a = R • α som insatt ger att – T = a • I/R2 som slutligen ger a = - g/(1 + I/mR2) som är ekvation 11-10 A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

Example 11-7 The Pulley Matters Bestäm T2 och vagnens acceleration a A 0.31-kg cart on a horizontal air track is attached to a string. The string passes over a disk-shaped pulley of mass 0.080 kg and radius 0.012 m and is pulled vertically downward with a constant force of 1.1 N. Find (a) the tension in the string between the pulley and the cart and (b) the acceleration of the cart.

Example 11-7 The Pulley Matters (Om m = 0 gäller T1 = T2) M = 0,31 kg, m = 0,080 kg, r = 0,012 m, T1 = 1,1 N Newtons andra lag ger vagnen T2 = Ma Newtons andra lag ger för blocket rT1 - r T2 = I α Om repet inte glider gäller att relationen mellan vinkel accelerationen och (den linjära) acceleration är (10-14) a = r • α eliminera α och ”kom ihåg” att Idisk = mr2/2 rT1 – rT2 = mra/2 = mrT2/2M A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

Example 11-7 The Pulley Matters (fortsättning) rT1 – rT2 = mra/2 = mrT2/2M som omstuvat blir T2 = T1/(1+ m/2M) = 1,1/(1 + 0,080/(2 • 0,31)) = {0,129} = 0,97N a = T2/M = 0,97 N/0,31 kg = 3,1 m/s2 Märk att T1 > T2 så att blocket kommer att rotera moturs (som väntat) A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

11-6 Angular Momentum Using a bit of algebra (I = mr2 och ω = v/r), we find for a particle moving in a circle of radius r,

Figure 11-11 The angular momentum of circular motion A particle of mass m, moving in a circle of radius r with a speed v. This particle has an angular momentum of magnitude L = rmv.

11-6 Angular Momentum For more general motion,

Exercise 11-3 Beräkna rörelsemängdsmomentet för a) en frisbee som väger 0,13 kg och som kan anses vara en homogen skiva med r = 7,5 cm och ω = 1,15 rad/s L = I ω = mr2/2 • ω = 4,2 10-4 kgm2/s (Js) b) en person (m = 95,0 kg) som springer med v = 5,1 m/s i en cirkel med r = 25 m L = r p = 25 m • 95,0 kg • 5,1 m/s = 12 kJs A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

Conceptual Checkpoint 11–3 Angular Momentum Conceptual Checkpoint 11–3 Angular Momentum? Har ett föremål som rör sig längs en rät linje alltid/ibland/aldrig ett rörelsemängdsmoment? Does an object moving in a straight line have nonzero angular momentum (a) always, (b) sometimes, or (c) never?

Figure 11-13 Angular momentum in linear and circular motion An object moving in (a) a straight line and (b) a circular path. In both cases, the angular position increases with time; hence, the angular momentum is positive.

Example 11-8 Jump On Running with a speed of 4.10 m/s, a 21.2-kg child heads toward the rim of a merry-go-round, as shown. If the radius of the merry-go-round is 2.00 m, and the child moves in the direction shown, what is the child’s angular momentum with respect to the center of the merry-go-round?

Example 11-8 Ett barn (m = 21,2 kg) springer med hastigheten 4,10 m/s mot en karusell enligt figuren. Beräkna barnets rörelsemängdsmoment, relativt karusellens centrum om r = 2,00 m. L = m v r cos45° = 123 kgm2/s (Js) A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

11-6 Angular Momentum Looking at the rate at which angular momentum changes,

Exercise 11- 4 I en lätt vind känner en väderkvarn ett konstant vridmoment på 255 Nm. Om vingarna först var i vila (det vill säga Li = 0), vad är deras rörelsemängdsmoment (Lf), 2,00 sekunder senare? ΔL = Lf – Li = τ Δt = 255 Nm • 2,00 s = 510 kgm2/s = Lf A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

11-7 Conservation of Angular Momentum If the net external torque on a system is zero, the angular momentum is conserved. The most interesting consequences occur in systems that are able to change shape:

Example 11-9 Going for a Spin For a classroom demonstration, a student sits on a piano stool holding a sizable mass in each hand. Initially, the student holds his arms outstretched and spins about the axis of the stool with an angular speed of 3.74 rad/s. The moment of inertia in this case is 5.33 kg-m2. While still spinning, the student pulls his arms in to his chest, reducing the moment of inertia to 1.60 kg-m2. What is the student’s angular speed now?

Example 11-9 Beräkna vinkelhastigheten, som ursprungligen var 3,74 rad/s, för en student på en pianostol (med tröghetsmomentet 5,33 kgm2) som sedan ändrar sitt tröghetsmomentet till 1,60 kgm2? Rörelsemändsmomentet bevaras så det gäller Li = Lf dvs Li = Ii ωi = If ωf = Lf så att ωf = 5,33 kgm2 •3,74 rad/s / 1,60 kgm2 = 12,5 rad/s A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

Photo 11-8 Hurricane Andrew This 1992 satellite photo of Hurricane Andrew (left), one of the most powerful hurricanes of recent decades, clearly suggests the rotating structure of the storm. The violence of the hurricane winds can be attributed in large part to conservation of angular momentum: as air is pushed inward toward the low pressure near the eye of the storm, its rotational velocity increases.

Photo 11-9 Crab Nebula Pulsar Among the fastest rotating objects known in nature are pulsars: stars that have collapsed to a tiny fraction of their original size. Since all the angular momentum of a star must be conserved when it collapses, the dramatic decrease in radius is accompanied by a correspondingly great increase in rotational speed. The Crab nebula pulsar, the remains of a star whose explosion was observed on Earth nearly 1000 years ago, spins at about 30 rev/s. This X-ray photograph shows rings and jets of high-energy particles flying outward from the whirling neutron star at the center.

Active Example 11-4 A Stellar Performance Find the Angular Speed Beräkna vinkelhastigheten, för en stjärna (pulsar) med radien R = 2,3•108 m, som ursprungligen var 2,4•10-6 rad/s, om stjärnan kollapsar till ett klot med radien 20,0 km. (Behandla stjärnan som ett homogent klot och anta att ingen massförlust sker vid kollapsen.) Rörelsemängdsmomentet bevaras så det gäller Li = Lf dvs Ii ωi = If ωf Tröghetsmomentet för en homogen sfär är 2mR2/5 så ωf = ωi Ii / If = ωi (Ri /Rf)2 = 2,4 10-6 rad/s (1,15)2 108 = = 320 rad/s (50,5 varv/s) → T = 2π/ω ≈ 20 ms A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

Conceptual Checkpoint 11-4 Compaire Kinetic Energies En isprinsessa halverar sitt tröghetsmoment och dubblerar på detta sätt sin vinkelhastighet. Är då hennes slutliga rörelseenergi a) > b) = c) < jämfört med hennes ursprungliga rörelseenergi? [Ki = Ii ωi2/2 Kf = If ωf2/2] A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

11-7 Conservation of Angular Momentum As the moment of inertia decreases, the angular speed increases, so the angular momentum does not change. Angular momentum is also conserved in rotational collisions:

Rotational Collisions (p Rotational Collisions (p.341) Givet: It ω0 Ir Rörelsemängdsmomentet bevaras (men inte energin, det är en inelastisk kollision) så det gäller Li = Lf dvs It ω0 = If ωf men If = It + Ir ωf = It ω0 /(It + Ir) A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

Active Example 11-5 Conserve Angular Momentum: Find the Angular Speed A 34.0-kg child runs with a speed of 2.80 m/s tangential to the rim of a stationary merry-go-round. The merry-go-round has a moment of inertia of 512 kg-m2 and a radius of 2.31 m. When the child jumps onto the merry-go-round, the entire system begins to rotate. What is the angular speed of the system?

Active Example 11-5 Conserve Angular Momentum: Find the Angular Speed m = 34,0 kg, Ik(arusell) = 512 kg m2, r = 2,31 m När barnet hoppar upp på karusellen med v = 2,80 m/s börjar hela systemet rotera. Hur stort blir ωf? Rörelsemängdsmomentet bevaras (men inte energin, det är en inelastisk kollision) så det gäller Li = rmv Lf = (Ik + mr2) ωf ωf = mvr/(Ik + mr2) = 34•2,8•2,31/(512 + 34•2,31•2,31) = 0,317/s A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

11-8 Rotational Work and Power A torque acting through an angular displacement does work, just as a force acting through a distance does. The work-energy theorem applies as usual.

Figure 11-15 Rotational work A force F pulling a length of line from a fishing reel does the work W = FΔx. In terms of torque and angular displacement, the work can be expressed as W = τ Δθ.

11-8 Rotational Work and Power Power is the rate at which work is done, for rotational motion as well as for translational motion. Again, note the analogy to the linear form:

Exercise 11-5 (p.343) Vridmomentet för handtaget till en glassmaskin är 5,7 Nm. Hur mycket arbete uträttas i varje varv? Hur stor är effekten om varje varv tar 1,5 s? W = τΔθ = 5,7 Nm • 2π (rad) = 36 J P = τ • Δθ/Δt = 35,8 J/1,5 s = 24 W A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find (a) the angular displacement of the spool, (b) the length of line pulled from the spool, and (c) the final angular speed of the spool.

11-9* The Vector Nature of Rotational Motion The direction of the angular velocity vector is along the axis of rotation. A right-hand rule gives the sign.

11-9* The Vector Nature of Rotational Motion A similar right-hand rule gives the direction of the torque.

11-9* The Vector Nature of Rotational Motion Conservation of angular momentum means that the total angular momentum around any axis must be constant. This is why gyroscopes are so stable.

Summary of Chapter 11 A force applied so as to cause an angular acceleration is said to exert a torque. Torque due to a tangential force: Torque in general: Newton’s second law for rotation: In order for an object to be in static equilibrium, the total force and the total torque acting on the object must be zero. An object balances when it is supported at its center of mass.

Summary of Chapter 11 In systems with both rotational and linear motion, Newton’s second law must be applied separately to each. Angular momentum: For tangential motion, In general, Newton’s second law: In systems with no external torque, angular momentum is conserved.

Summary of Chapter 11 Work done by a torque: Power: Rotational quantities are vectors that point along the axis of rotation, with the direction given by the right-hand rule.