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Chapter 9: Rigid Bodies and Rotational Motion Angular velocity an object which rotates about a fixed axis has an average angular velocity wav : wav 2 1 usually rad/s but sometime rpm, rps t 2 t1 t instantaneous angular velocity is given by: d w lim t 0 t dt since s r ds d r or v rw dt dt Phys 250 Ch9 p1 r s Angular Acceleration: the rate of change of angular speed ω ω dω d 2θ αav ; α lim 2 t 0 t t dt dt related to linear accelerati on in circular motion : dv dω atan r at rα dt dt v2 ac w 2 r r ac=w2r aT=ar total linear accelerati on a ac aT 2 2 Example: In a hammer throw, a 7.25 kg shot is swung in a circle 5 times and then released. The shot moves with an average radius of 2.1 m and an average angular acceleration of 2.3 rad/s2. What is the average tangential force and what is the maximum centripetal force on the hammer? Phys 250 Ch9 p2 Rotation with constant angular acceleration (just like linear 1-d) Angular Linear 1 2 0 w0t at 2 w0 w 0 t 2 1 2 x x0 v0t at 2 v0 v x x0 t 2 w w0 at v v0 at w2 w0 2a( 0 ) v 2 v0 2a ( x x0 ) 2 watch units consistency!!! Phys 250 Ch9 p3 2 Example: The wheel on a moving car slows uniformly from 70m rad/s to 42 rad/s in 4.2 s. What is the angular acceleration of the wheel? What angle does the wheel rotate in those 42 s? How far does the car go if the radius of the wheel is 0.32 m. Phys 250 Ch9 p4 Torque: the rotational analogue of force Torque = force x moment arm t = FL=F r sin moment arm = perpendicular distance through which the force acts L L F F L F Example: The bolts holding a head gasket are to be “torqued down” to 90 N-m. If a 45 cm wrench is used, what force should be applied perpendicular to the wrench handle? Phys 250 Ch9 p5 Example: The crank arm of a bicycle pedal is 16.5 cm long. If a 52.0 kg woman puts all her weicht on one pedal, how much torque is developed when the crank is horizontal? How much torque is developed when the pedal is 15º from the top? Phys 250 Ch9 p6 Equilibrium: stability, steadiness, balance etc. Mechanical Equilibrium: absence of change in motion => Net Force = 0 ! F 0 (usually, no motion) sum of x force components = sum of y force components = F F x y 0 0 With Rotational Equilibrium Rotational Equilibrium: absence of change in rotation (usually: no rotation) => net torque is zero Sti = 0 about any axis! t i 0 for all torques lying in the same plane Watch signs for torque F L Positive torque for counterclockwise rotation: t = F L Phys 250 Ch9 p7 L F Negative torque for clockwise rotation: t = F L Center of Gravity (CG) aka Center of Mass: the point of an object from which it could be suspended without tending to rotate. The point where all the mass of an object can be considered to be located. CG does not need to be located within the physical object! Horseshoe, for example usually easily identified from symmetry. Example: A 5 kg mass hangs from the 5 cm mark on a 1 meter long rod. An unknown mass hangs from the 85 cm mark. The rod has a mass of 2.0 kg and is balanced at the 35 cm mark. What is the unknown mass? Phys 250 Ch9 p8 Example: A sign weighing 400 N is suspended from the end of a 350 N horizontal uniform beam. What is the tension in the cable? 35o w Phys 250 Ch9 p9 Elasticity “stretchiness/springiness” -how materials respond to stress compression tension shear “Stretch-ability” = amount of stress (applied force) produces a strain (elongation/compression/shear) Hooke’s Law: the amount of stretching is proportional to the applied force. F=kx The details of such springiness depends upon the size and shape of the material as well as how the forces are applied 1 Ton x x 2 Tons Phys 250 Ch9 p10 Elastic Limit: the maximum stress (force) which can be applied to an object without resulting in permanent deformation. Plastic Deformation: the permanent deformation which results when a materials elastic limit has been exceeded. Ultimate strength: greatest tension (or compression or shear) the material can withstand. *snap* A malleable or ductile material has a large range of plastic deformation. Fatigue: small defects reduce materials strength well below original strength. Phys 250 Ch9 p11 Young’s Modulus: how things stretch (elastically) stress: force per area = F/A A L0 L0 compression L A tension A L strain: fractional change in length = change in length per original length = L/Lo Elastic modulus = stress/strain Young’s modulus (for stretching in one direction) Y Phys 250 Ch9 p12 F A L L0 A Example: A steel elevator cable supports a load of 900 kg. The cable has a diameter of 2.0 cm and an initial length of 24 m. Find the stress and the strain on the cable and the amount that it stretches under this load. Phys 250 Ch9 p13 Torque and Moment of Inertia For a single mass: FT= maT FTr =maTr t = mar r = mr2 a moment of inertia I = mr2 t= I a looks like F = ma for a system of objects ( a rigid object) I = Smiri2 Phys 250 Ch9 p14 ac=w2r L R2 L R2 R 1 I ML2 3 Thin Rod (axis at end) 1 ML2 12 Thin Rod I 1 MR 2 2 Solid Disk I a 1 2 2 M ( R1 R2 ) 2 Hollow Cylinder I a R b b 1 I M (a 2 b 2 ) 3 Thin Rectangula r Plate (about edge) 1 I M (a 2 b 2 ) 12 Rectangula r Plate (through center) R 2 MR 2 5 Solid Sphere I Phys 250 Ch9 p15 I MR 2 Thin Walle d Hollow Cylinder R 2 MR 2 3 Thin Walle d Hollow Sphere I Example: A cylindrical winch of radius R and moment of inertia I is free to rotate without friction. A cord of negligible mass is wrapped about the shaft and attached to a bucket of mass m. What is the acceleration of the bucket when it is released? Phys 250 Ch9 p16 Angular Momentum L=Iw (like p = m v ) + Angular momentum is conserved in the absence of external torques Lstart = Lend for a point mass moving in a circle L = mvr = mr2w conservation of angular momentum implies Kepler’s 3rd law! Example: Ann ice skater starts spinning at a rate of 1.5 rev/s with arms extended. He then pulls his arms close to his body, decreasing his moment of inertia to ¾ of its initial value. What is the skater’s final angular velocity? Phys 250 Ch9 p17 Rotational Kinetic Energy for a single point particle KE 1 2 1 2 2 mv mr w 2 2 for a solid rotating object 1 1 2 2 m1v1 m2 v2 2 2 1 1 2 2 m1r1 w 2 m2 r2 w 2 2 2 1 2 2 (m1r1 m2 r2 )w 2 2 1 1 mr 2 w 2 Iw 2 2 2 KE Phys 250 Ch9 p18 I mr KE 2 1 2 Iw 2 Combined Translation and Rotation KE = KEtranslation + KErotation KE 1 2 1 2 mv Iw 2 2 when rolling without slipping s= r v=wr a : angular acceleration The Great Race lost PE = gained KE same radius, object with the smallest I has most v => wins race a = a r, mgh KE 1 2 1 2 mv Iw 2 2 2 1 1 v 1 1 I 2 mv2 I mv2 v 2 2 2 r 2 2r 1 I m 2 v 2 2 r Phys 250 Ch9 p19