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Chapter 8

Chapter 8 . Rotational Motion. Hello! From: Jon. Forms of Motion . Translational Body moving such that a line drawn between any two of its points remains parallel to itself Ex – car riding down a road Rotational

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Chapter 8

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  1. Chapter 8 Rotational Motion

  2. Hello! From: Jon Forms of Motion • Translational • Body moving such that a line drawn between any two of its points remains parallel to itself • Ex – car riding down a road • Rotational • Body moving such that a line drawn between any two of its points does NOT stay parallel to itself • Ex – galaxy spinning through space

  3. Kinematics of Rotation • Most things moving in a circle are measure in DEGREES • Based on Babylonian number system • Consider a set of beads swinging in a circle • Each bead sails through a different ARC • Each bead sweeps through the same ANGLE

  4. Types of Motion • Rectilinear • Rotation about a point outside the body • Curvilinear • Rotation about a point in the body • Rotation and translation

  5. Angular displacement • Θ = l/r • Θ = angle (radians) • l = arc-length (meters) • r = radius (meters) • Circumference • 2π(radians) = 6.28(radians) = 360˚

  6. Problem 8.1 • Andromeda galaxy subtends an angle of 4.1˚ with a diameter of 163 x 10³light-years. How far away is it in light-years? Θ = l/r Θ = 4.1º = 0.072 radians l = 163000 light-years r = l/Θ = 163000 ly/0.072 radians = 2.3 million light-years

  7. Angular speed • l /t = rΘ/t • Average angular speed = distance on arc time • ω(lower case –omega) • ω = Θ / t • Units – radians/second -(revolutions/second) • v = rω(only when angular speed is in radians)

  8. Example 8.2 • A horse (Dr. Faber) finished 1.00 mile race around a circular track in 1.00 min 32.2 seconds, averaging 62.8 km/hr. What was the horse’s average angular speed? v = 62.8 km/hr =17.4 m/s l= 1 mile t = 92.2 seconds r = ? r = 1.00 mile/2π = 0.159 mi = 0.256 km ω = ? ω = v/r = .00174 km/s / 0.256 km = 0.0679 rad/s

  9. Difference between angular speed and linear speed (on an angle) • Crack-the-Whip • CD • Marching band

  10. Angular Acceleration • Variations in ω occur as often as variations in v • Same formula – different symbols ά = Δ ω/ Δt = Δv/r Δt = l a/r ά - angular acceleration (rad/s²) ω – angular velocity ( rad/s) t – time (sec) v – tangential velocity (m/s) r – radius (m) l – length on arc (m) a – tangential acceleration (m/s²)

  11. Example 8.3 • How fast is the angular velocity 2.3 cm from the center of a CD if the linear velocity is 1.2 m/s? v = 1.2 m/s r = 2.3 cm = 0.023 m ω = ? ω = v/r = 1.2 m/s / 0.023 m = 52.2 rad

  12. Example 8.4 r = 50 m Find : v = ? ω = 0.60 rad/s ac = ? ά = 0.20 rad/s² aT = ? total a = v = ωr = 50 m (0.60 rad/s) = 30 m/s ac = v²/r = (30m/s)²/ 50 m = 18 m/s² aT = rά = 50 m (0.20 rad/s²) = 10 m/s²

  13. Example 8.4 • A car moving around a track with a radius of 50 m has an angular velocity of 0.60rad/s and an angular acceleration of 0.20 rad/s². What is the linear speed? Centripetal acceleration? Tangential acceleration? Total linear acceleration at that moment? r = 50 m Find : v = ? ω = 0.60 rad/s ac = ? ά = 0.20 rad/s² aT = ? total a = ?

  14. Example 8.4 • a²total = ac² + at² = (18m/s²)² + (10m/s²)² = 21m/s² Φ = tan-¹ ac/at = 61°

  15. Equations of Constant Angular acceleration • Big Five from Ch 2 – for velocity and acceleration • Remember that: l = rΘ v = rω aT = rα • Big Five of Angular Acceleration

  16. Free Rolling Wheel • Remember that: l = rΘ v = rω aT = rα • These formulas convert from angular to tangential (linear)

  17. Rotational Equlibrium • What do balances, levers, and seesaws have in common? • Pivoted beam in balance • Rotational equilibrium • Aristotle's Law of Levers – Unequal forces, acting perpendicular on a pivoted bar, balance each other, provided that F1r1= F2r2

  18. Example 8.7 • A boy (30kg)want to balance across from his dog(10kg) on a seesaw. If the dog is 3.0 m from the pivot, where must the boy sit to have the board balance? • F1= mg = 30 x 9.81 • F2= mg = 10 x 9.81 • r2= 3.0 m r1= F2 r2/ F1= 1.0 m

  19. Torque • Steelyard – measure heavy items by balancing it with a lighter item on a long arm • Using a wrench – • Axis of rotation • Moment-arm – position vector r • From the axis of rotation to point of application of the force • Force – Perpendicular to the moment arm

  20. Torque • Torque – measure of the twist produced by a force around an axis • τ = Fr = Fr sinΘ • Direction • Clockwise – cw • Counterclockwise – ccw • Units for torque • Newton-meter • Nm

  21. Example 8.8 • A weight-lifter extends his legs to lift a weight strapped to his feet. What is the torque when his 50 cm legs are extended 30º from straight down while holding 1.0 kg ? τ= Fr sinΘ F = mg τ= mg r sinΘ = 1.0 kg (9.81 m/s²)(.5 m) sin 30 = 2.5 Nm Θ Fw

  22. Sum of torque – Example 8.9 • A hand exerts a 200N force at the end of a 1.0 m lever. A return spring that pulls the lever back from the midpoint has a horizontal force of 80 N. What is the torque when the lever is pulled out 60º from the wall?

  23. Second Condition of Equilibrium • Remember the first condition of equilibrium? • Σ F = 0 • Second Condition: • Σ τ = 0

  24. Center of Gravity and Extended Bodies • All of the atoms of an object “pool” together to make a spot where the total weight reacts • CENTER OF GRAVITY – point where total weight can be imagined to act • When object sits on the ground – c.g. acts in-line with normal force • When object is suspended by one string – c.g. acts in-line with supporting string

  25. Torque and C.g. • Use torque to calculate c.g. xcg = Σ Fw x /Σ Fw Torque

  26. Example 8.11 • A physical therapist needs to know the cg of child’s leg. If the cg of the lower leg(59N) is at 19 cm and the cg of the upper leg (97N) is at 42 cm from the bottom of the foot, where is the cg of the entire leg? • xcg = Σ Fw x /Σ Fw = 59 N (.19 m) + 97 N (.42 m) 59 N + 97 N = .33 m

  27. Example 8.12 • The bicep attaches to the radius 5.0 cm from elbow. The mass of a hand and forearm is 2.0 kg with a cg at 16 cm from elbow. What is the force of the biceps if the hand is holding a 2.0 kg ball 34 cm from the elbow? • Σ τ = 0 0 = marmg(.16m) + mballg(.34m) – Fbicep(.05m) Fbicep = 250 N

  28. Stability and Balance • Stable – will return to original place after a minor displacement • Rule of thumb – an object supported at a point above the cg will be stable, while one supported below the cg will be unstable. • Humans – weight must be over hips • Why do wrestlers, football players crouch? • Buildings – • Why hasn’t the Leaning Tower of Pisa fallen? • Why is the Empire State building’s base cover an entire city block?

  29. Rotational Inertia • Law of Rotation – An object rotating about an axis tends to keep rotating about that axis • Rotational Inertia – resistance to change in rotation • Depends on mass • Depends on distance (torque) • The greater the distance from the axis, the greater the inertia

  30. Varying Rotational inertia • Baseball bat • Full swing • “Chocked-up” • Swing leg • At the hip • With knee bent – reduces inertia • Flipping pencil • At the middle • From one end • Rotate between fingers on long axis

  31. Formulas for Rotational Inertia

  32. Rolling • Which will roll down an incline with greater acceleration, a hollow cylinder or a solid one of same mass? • The one with the smaller rotational inertia • Think of inertia as “Laziness” • The more inertia, the longer it takes to roll

  33. Rotational Inertia and Gymnastics • Symmetrical lines • Pass through cg • Rotational inertia is least along transverse • Skaters outstretched arms –slow • Tuck them in – 6x faster • R.I. is greatest along longitudinal axis • Summersault • Vaulting – Change axis from hands to cg – more spin • Medial axis - cartwheels

  34. Angular momentum • Inertia of rotating – an object in rotation stays in motion unless acted upon unbalanced force • Angular momentum – vector quantity that has direction and magnitude = Inertia x Velocity • Gyroscope • Swivel turns any direction without exerting a torque on the whirling gyroscope • Stays pointed in same direction • Bicycle • Spinning wheels have angular momentum – stay balanced • No spin – fall over

  35. Conservation of Angular Momentum • If no unbalanced external torque acts on a rotating system, the angular momentum of that system is constant • When a rotating body contracts, the speed increases • Uses • Galaxies – Flat and narrow • Ice skaters • Cats – landing on feet • Astronauts – twist without spinning

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