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Physics B Topics Overview ∑ Physics B Topics Overview PHYSICS A motion of point bodies COVERED: kinematics - translation dynamics ∑Fext = ma conservation laws: energy & momentum 8 weeks: rotation PHYSICS B 2 weeks: gravitation COVERS: 2 weeks: oscillations 1 week: fluids 1 week: review Physics 106 Week 1 Introduction to Rotation SJ 8th Ed.: Chap. 10.1 to 3 • Rotation (rigid body) versus translation (point particle) • Rotation concepts and variables • Rotational kinematic quantities Angular position and displacement Today Angular velocity & acceleration • Rotation kinematics formulas for constant angular acceleration • Analogy with linear kinematics 2 1 “Radian” “radian” : more convenient unit for angle than degree Definition: 360o 180o • 2π radians = 360 degree 1 radian = = = 57.3o 2π π s arc length ≡ s = r θ (in radians) θ ≡ rad r θ (in rad) s =×2π rr =θ 2π r s θ’ θ Example: r = 10 cm, θ = 100 radians Æ s = 1000 cm = 10 m. Rigid body Rigid body: A “rigid” object, for which the position of each point relative to all other points in the body does not change. Example: Solid: Rigid body Liquid: Not rigid body Rigid body can still have translational and rotational motion. 2 Rotation of Rigid body in 3D Choose Z - axis as rotation axis (marks a constant direction in space) Use reference line perpendicular to rotation axis to measure rotation angles for the body EACH POINT ON A BODY SWEEPS OUT A CIRCLE PARALLEL TO X-Y PLANE 3D VIEW – Z axis up TOP VIEW – Z axis out of paper Reference line rotates around it REFERENCE LINE ROTATES WITH BODY Measure θ CCW from x axis THE ROTATION AXIS DIRECTION TAKES 2 ANGLES TO SPECIFY, e.g. (LATITUDE, LONGITUDE) For any point in the body, r is the perpendicular distance to the rotation axis Angular position of rotating rigid body A rigid body rotates about some rotation axis – a line located somewhere in or outside it, pointing in some direction y in space • One polar coordinate θ specifies θ position of the whole body about this Reference x line rotates rotation axis. with body • By convention, θ is measured CCW rotation axis “o” rigid body from the x-axis fixed to body • It keeps increasing past 2π, can be parallel to z-axis negg,ative, etc. • Each point of the body moves around the axis in a circle with some specific radius 3 Angular displacement of rotating rigid body Angular displacement: the angle an object (rigid body) rotates y through during some time interval. Equivalently, the angle that a reference θ line fixed in a body sweeps out Reference x line rota tes with body Angular displacement: rotation axis “o” rigid body • Net change in the angular coordinate fixed to body parallel to z-axis Δθ ≡ θfinal − θinital (an angle in rad.) Arc length: Δs • Measures distance covered by a point as it moves Reference line rotating with body y through Δθ (constant r) Δs = r Δθ Δs ≡ rΔθ (a distance along a circular arc) θf r r θo x Rigid body rotation: angular & tangential velocity Angular velocity ω: For any point, r is the perpendicular • Rate of change of the angular displacement distance to the rotation axis Δθ Δθ dθ ωave ≡ ωinst ≡ Lim ≡ Δt Δt → 0 Δt dt vT • Units: radians/sec. Positive in CCW sense • If is CONSTANT (uniform circular motion) ω r θf = θ0 + ωΔt θ = ωΔτ x • Frequency f = # of complete revolutions/unit time • f = 1/T T = period (time for 1 complete revolution ω = 2πf = 2π/T f = ω/2π Δs = rΔθ ds rd (r constant) Tangential velocity v = θ T: ds dθ • Units: distance / time v ≡ = r = rω T dt dt • Rate at which a point sweeps out arc length along circular path • Proportional to r, same ω vT = ωr 4 Angular velocity, period, and linear velocity 1.1. The period of a rotating wheel is 12.57 seconds. The radius of the wheel is 3 meters. It’s angular speed is closest to: A. 79 rpm B. 0.5 rad/s C. 2.0 rad/s D. .08 rev/s E. 6.28 rev/s 1.2. A point on the rim of the same wheel has a tangential speed closest to: A. 12.57 rev/s B. 0.8 rev/s C. 0.24 m/s Δs ≡ rΔθ D. 1.5 m/s E. 6.28 m/s vT = ωr ω = 2πf = 2π/T Rigid body rotation: angular and tangential acceleration Centripetal (radial) acceleration ac or ar (Phys105) vT • Body rotates at rate ω. r • Radial acceleration component, points toward rotation axis ac • Constant magnitude if ω is constant (UCM). Units: length/time2 ωτ • Changing if ω not constant (angular acceleration not zero) 2 x vT 2 ar== ω (use v= ω r) Fcentripetal = mac c r T Angular acceleration α: Δω Δω αave ≡ αinst ≡ Lim • Rate of change of the angular velocity Δt Δt→0 Δt aT • Units: radians/sec/sec. 2 dω d θ vT • CCW considered positive α ≡ = 2 r dt dt ac • for CONSTANT α: ωf = ω0 + αΔt ω,α x Tangential acceleration a : T aT = αr • Tangential acceleration component • Proportional to angular acceleration α and also to radius r dvT dω aT ≡ = r = rα • Units: length / time 2 dt dt 5 Note: for displacement, speed, and acceleration • The tangential quantities depend on r • r varies for different points on the object • All points on a rotating rigid body have the same angular displacement, but do not move through the same path length. • All points on a rigid body have the same angular speed, but not the same tangential speed. • All points on a rigid object have the same angular acceleration, but not the same tangential acceleration. Rotational Motion Example • For a compact disc player, the tangential speed must be constant to readdd da ta proper lly • The angular speed must vary: (vt = ωr) • For inner tracks, the angular speed must be larger than at the outer edge • The player has to vary the angular speed (rotation rate) accordingly. Linear and Angular Kinematics Equations (Same mathematical forms!!) Linear motion Angular motion (Phys105) (Phys106) x(t), v(t), a(t) variables θ(t), ω(t), α(t) dx dv dθ dω v = a = Definitions ω = α = dt dt dt dt Motion with constant velocity Rotation with constant angular velocity x f ()txvt=+0 θ f ()tt= θω0 + Motion with constant acceleration Rotation with constant angular acceleration vf (t) = v0 + at ωf (t) = ω0 + αt More similarity to come…. 6.
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