Physics 106 Lecture 1 Introduction to Rotation SJ 7th Ed.: Chap. 10.1 to 3 • Course Introduction • Course Rules & Assignment • TiTopics Overv iew • Rotation (rigid body) versus translation (point particle) • Rotation concepts and variables • Rotational kinematic quantities Angular position and displacement Angular velocity & acceleration • Rotation kinematics formulas for constant angular acceleration • Analogy with linear kinematics 1 Physics B Topics Overview PHYSICS A motion of point bodies COVERED: kinematics - translation dynamics ∑Fext = ma conservation laws: energy & momentum motion of “Rigid Bodies” (extended, finite size) PHYSICS B rotation + translation, more complex motions possible COVERS: rigid bodies: fixed size & shape, orientation matters kinematics of rotation dynamics ∑Fext = macm and ∑ Τext = Iα rotational modifications to energy conservation conservation laws: energy & angular momentum TOPICS: 3 weeks: rotation: ▪ angular versions of kinematics & second law ▪ angular momentum ▪ equilibrium 2 weeks: gravitation, oscillations, fluids 1 Angular variables: language for describing rotation Measure angles in radians simple rotation formulas Definition: 360o 180o • 2π radians = full circle 1 radian = = = 57.3o • 1 radian = angle that cuts off arc length s = radius r 2π π s arc length ≡ s = r θ (in radians) θ ≡ rad r r s θ’ θ Example: r = 10 cm, θ = 100 radians Æ s = 1000 cm = 10 m. Rigid body rotation: angular displacement and arc length Angular displacement is the angle an object (rigid body) rotates through during some time interval..... ...also the angle that a reference line fixed in a body sweeps out A rigid body rotates about some rotation axis – a line located somewhere in or near it, pointing in some y direction in space • One polar coordinate θ specifies position of the whole body about this rotation axis. θ • By convention, θ is measured CCW from the x-axis Reference x line rotates • It keeps increasing past 2π, can be negative, etc. with body • Each point of the body moves around the axis in a circle with some specific radius rotation axis “o” rigid body fixed to body parallel to z-axis Angular displacement: • Net change in the angular coordinate Reference line rotating with body y Δθ ≡ θfinal − θinital (an angle in rad.) Δs = r Δθ Arc length: Δs θf • Measures distance covered by a point as it moves r r through Δθ (constant r) θo Δs ≡ rΔθ (a distance along a circular arc) x 2 Rigid body rotation 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 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 3 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: 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) x 2 vT 2 F = ma a = = ω r (use v = ωr) centripetal c c r T So far: nothing about angular velocity changing 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 a and also to radius r dvT dω aT ≡ = r = rα • Units: length / time 2 dt dt 4 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. Rotation variables: angular and linear Angular motion variables Δθ, ω, α: • apply to the whole rotating body 2 dθ dω d θ v Δθ ω ≡ α ≡ = T 2 dt dt dt aT atotal Linear motion variables r, s, vT, aT, ac: r • applifiiily to a specific point on a rotating a body & are signed magnitudes c • values all proportional to r – distance from the rotation axis ω,α s ==rΔθ vT ωr 2 x ac = ω r aT = αr Total linear acceleration of a point (rotation): • vector sum of radial and tangential components (normal to each other) 2 2 atotal = ac + aT Some basic implications: • IF ω is constant, then α=0, aT=0. But ac = ar is not 0 • IF ω is not constant, then α and aT are not 0, ac and VT are varying with time • IF α is constant, then aT is constant for a particular point, different for different r Δω is proportional to Δt (angular kinematics) vT is not constant (so this is not UCM) } 5 Rotational variables are vectors, having direction The angular displacement, speed, and acceleration ( θ, ω, α ) are the magnitudes of rotational displacement, velocity, and acceleration vectors The directions are actually given by the right-hand rule Point thumb along + axis fingers curl CCW (positive) sense Example: Find the angular speed ω for the hour, minute, and second hands of an analog clock [rad/sec] Second Period T = 60 sec Hand f = Frequency = 1/T = 1/60 revolution/sec ω = #rev/sec x # rad/rev = 2πf = 2π/60 sec ω = 0.1 rad/sec Minute Period T = 1 hour = 3600 sec Hand ω = 2πf = 2π/T = π/1800 ω = 1.75 x 10-3 rad/sec Physicists’s Clock Hour Period T = 12 hours = 12 x 3600 sec = 43.200 sec Hand ω = 2πf = 2π/T = 2π/(3600 x 12) ω = 1.45 x 10-4 rad/sec 6 Example: Assume the displacement θ is given as the following function of time 2 where v θ(t) = θ0 + bt + st b = 10, s = 5, r = 2 t Actually, b = ω0 and s = α/2 r 1. Find ω(t) = angular speed (differentiate) d this is not UCM, since ω(t) = θ(t) = 0 + b + 2st [rad/s] dt ω is not constant v (r) = ω(t) r = br + 2srt = 20 + 20t [m/s] t vt is not constant either 2. Find α(t) = angular acceleration (differentiate again) d d2 α is constant because θ(t) α ≡ ω(t) = θ(t) = 2s = 10 [rad/s2 ] dt dt2 does not have t dependence higher than t2 3. Find tangential and radial accelerations 2 tangential acceleration is constant, at ≡ αr = 2sr = 20 [m/s ] but centripetal acceleration is not 2 2 2 2 ac ≡ ω r = (b + 2st) r [m/s ] unless s = 0 (no t term in θ(t)) 2 2 1/2 atot ≡ ( at + ac ) Angular kinematics: constant α Linear and Angular Kinematics Equations (Same mathematical forms) Linear motion Angular motion • constant acceleration a • constant angular acceleration α x(t), v(t), a(t) variables θ(t), ω(t), α(t) dv dx dθ dω a = v = Definitions ω = α = dt dt dt dt vf (t) = v0 + at Kinematic ωf (t) = ω0 + αt Equations x (t) = x + v t + 1 at 2 θ (t) = θ + ω t + 1 αt2 f 0 0 2 f 0 0 2 v2 (t) = v2 + 2a[x − x ] 2 2 f 0 f 0 ωf (t) = ω0 + 2α[θf − θ0 ] 1 1 xf (t) = x0 + (vo + vf )t θ (t) = θ + (ω + ω )t 2 f 0 2 o f Check by differentiating: dx dθ v = = v + 1 (2)at = v + at ω = =+ωαωα1 (2) tt =+ dt 0 2 0 dt 002 Both sets of kinematic equations follow from the definitions of velocity and acceleration by integrating 7 Hints for solving rotational kinematics problems Similar to the techniques used in linear motion problems With constant angular acceleration, the techniques are much like those with constant linear acceleration.