Lectue 13 Rotational motion Moment of inetia EXAM uesday Mach 8, 16 8:15 PM 9:45 PM oday s opics: Rotational Motion and Angula Displacement Angula Velocity and Acceleation Rotational Kinematics Angula and angential Vaiables Centipetal and angential Acceleation Rolling Moments of Inetia 1
Get Oiented Eveything we ve done so fa elates to tanslational (linea) motion. Geneal motion involves both tanslation and otation! hink about putting a mak on the edge of at tie and then olling it down the oad Let s fist deal with pue otation he angle though which the object otates is called the angula displacement. = θ θ o By convention, the angula displacement is positive if it is counteclockwise and negative if it is clockwise. SI Unit of Angula Displacement: adian (ad) Rotational Motion and Angula Displacement o a full evolution: π θ = = π ad Ac length s θ (in adians) = = Radius! π ad = 36
A otal Eclipse of the Sun he diamete of the sun is about 4 times geate than that of the moon. By coincidence, the sun is also about 4 times fathe fom the eath than is the moon. o an obseve on the eath, the angle subtended by the moon and the angle subtended by the sun is the same and explains why this can esult in a total sola eclipse. Ac length s θ (in adians) = = Radius θ (Sun) = θ (moon) Angula Velocity DEINIION O AVERAGE ANGULAR VELOCIY Angula displacement Aveage angula velocity = Elapsed time θ θo ω = = t t o SI Unit of Angula Velocity: adian pe second (ad/s) INSANANEOUS ANGULAR VELOCIY ω = lim ω = lim Look familia? Angula Acceleation DEINIION O AVERAGE ANGULAR ACCELERAION Change in angula velocity Aveage angula acceleation = Elapsed time ω ωo Δω α = = t t o SI Unit of Angula acceleation: adian pe second squaed (ad/s ) Can you hea me now???? 3
Kinematics of Rotation Example Duing the spin-dy cycle of a washing machine, the moto slows fom 95 ad/s to 3 ad/s while the tuning the dum though an angle of 4 adians. What is the magnitude of the angula acceleation of the moto? (a) 64 ad/s (b) 3 ad/s (c)1 ad/s (d) ad/s (e)1. ad/s ω = 95 ad/s ω = 3 ad/s = 4 ad α =? ω = ω + α α = ω ω = ( 3 ad/s ) 95 ad/s 4 ad = 1ad/s ( ) ( ) Note that the magnitude is 1 ad/s while the diection (-) is opposite to the angula velocity! angential Velocity and Speed We ve aleady seen that an angula displacement of θ coesponds to a tangential displacement of s fo a point a distance fom the axis of otation We have a simila elationship between angula velocity, ω, and tangential velocity, o speed v. 4
Δ ω = θ v s = = = v = ω (ω in ad/s) What about angula and tangential acceleation? v vo a = = ( ω) ( ω ) o ω ωo = ω ωo α = a = α ( α in ad/s ) Example On an amusement pak ide, passenges ae seated in a hoizontal cicle of adius 7.5 m. he seats begin fom est and ae unifomly acceleated fo 1 seconds to a maximum otational speed of 1.4 ad/s. What is the tangential acceleation of the passenges duing the fist 1 s of the ide? (a).67 m/s (b).5 m/s (c)1.4 m/s (d)7.5 m/s (e)11 m/s What is the instantaneous tangential speed of the passenges 15 s afte the acceleation begins? (a).67 m/s (b).5 m/s (c) 1.4 m/s (d) 7.5 m/s (e) 11 m/s = 7.5 m = 1s ω = ad/s ω = 1.4 ad/s a = α ω = ω + α ω 1.4 ad/s α = = =.67 ad/s 1s a = α = (7.5 m)(.67 ad/s ) a =.5 m/s v = ωt= 15s ω = ω + α ω = (.67 ad/s )(15 s) = 1ad/s v = (7.5 m)(1 ad/s) = 7.5 m/s 5
Rewiting centipetal acceleation a v ( ω) c = = = ω mv m( ω) c = = = mω Conceptual Poblem A igid body otates about a fixed axis with a constant angula acceleation. Which one of the following statements is tue concening the tangential acceleation of any point on the body? (a) he tangential acceleation is zeo m/s. (b) he tangential acceleation depends on the angula velocity. (c) he tangential acceleation is equal to the centipetal acceleation. (d) he tangential acceleation is constant in both magnitude and diection. (e) he tangential acceleation depends on the change in the angula velocity. Δω a = α = a = ω c Rolling (without slipping) Put otational and tanslational motion togethe he tangential speed of a point on the oute edge of the tie is equal to the speed of the ca ove the gound. o an object that is olling without slipping, the tanslational and otational motions ae coupled! v = ω a = α 6
Rotational Kinetic Enegy and the Moment of Inetia Imagine a mass on a sting, We can also wite the kinetic enegy as Whee I, the moment of inetia, is given by I = m 7