Lectue 13 Rotational motion Moment of inetia
EXAM 2 Tuesday Mach 6, 2018 8:15 PM 9:45 PM
Today s Topics: Rotational Motion and Angula Displacement Angula Velocity and Acceleation Rotational Kinematics Angula and Tangential Vaiables Centipetal and Tangential Acceleation Rolling Moments of Inetia
Get Oiented Eveything e ve done so fa elates to tanslational (linea) motion. Geneal motion involves both tanslation and otation! Think about putting a mak on the edge of at tie and then olling it don the oad
Let s fist deal ith pue otation The angle though hich the object otates is called the angula displacement. Dq q - q o By convention, the angula displacement is positive if it is counteclockise and negative if it is clockise. SI Unit of Angula Displacement: adian (ad)
Rotational Motion and Angula Displacement Fo a full evolution: q (in adians) Ac length Radius s 2p q 2p ad 2 p ad! 360
A Total Eclipse of the Sun The diamete of the sun is about 400 times geate than that of the moon. By coincidence, the sun is also about 400 times fathe fom the eath than is the moon. Fo an obseve on the eath, the angle subtended by the moon and the angle subtended by the sun is the same and explains hy this can esult in a total sola eclipse. Ac length q (in adians) Radius q (Sun) q (moon) s
Angula Velocity DEFINITION OF AVERAGE ANGULAR VELOCITY Aveage angula velocity q -qo t - t o Angula displacement Elapsed time Dq Dt SI Unit of Angula Velocity: adian pe second (ad/s) Look familia? INSTANTANEOUS ANGULAR VELOCITY lim Dt 0 lim Dt 0 Dq Dt
Angula Acceleation DEFINITION OF AVERAGE ANGULAR ACCELERATION Aveage angula acceleation Change in angula velocity Elapsed time a - t - t o o D Dt SI Unit of Angula acceleation: adian pe second squaed (ad/s 2 ) Ho about no????
Kinematics of Rotation
Example Duing the spin-dy cycle of a ashing machine, the moto slos fom 95 ad/s to 30 ad/s hile tuning the dum though an angle of 402 adians. What is the magnitude of the angula acceleation of the moto? (a) 64 ad/s 2 (b) 32 ad/s 2 (c) 10 ad/s 2 (d) 20 ad/s 2 (e) 1.0 ad/s 2 0 F 95 ad/s 30 ad/s Δθ 402 ad a? ω F 2 ω 0 2 + 2αΔθ α ω 2 2 F ω 0 2Δθ 10ad/s 2 ( 30 ad/s) 2 95 ad/s 2 402 ad ( ) 2 ( ) Note that the magnitude is 10 ad/s 2 hile the diection (-) is opposite to the angula velocity!
Tangential Velocity and Speed 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 beteen angula velocity, ω, and tangential velocity, o speed v T.
D q Dt v T s Dt Dq Dt æ ç è Dq ö Dt ø
Angula and tangential acceleation ( ) ( ) t t t v v a o o To T T D - D - D - t o D - a ) in ad/s ( 2 a a T a
Example On an amusement pak ide, passenges ae seated in a hoizontal cicle of adius 7.5 m. The seats begin fom est and ae unifomly acceleated fo 21 seconds to a maximum angula speed of 1.4 ad/s. What is the tangential acceleation of the passenges duing the fist 21 s of the ide? (a) 0.067 m/s 2 (b) 0.50 m/s 2 (c)1.4 m/s 2 (d)7.5 m/s 2 (e)11 m/s 2 What is the instantaneous tangential speed of the passenges 15 s afte the acceleation begins? (a) 0.067 m/s (b) 0.50 m/s (c) 1.4 m/s (d) 7.5 m/s (e) 11 m/s 7.5 m Dt 0 ad/s 0 F 21s 1.4 ad/s
Reiting centipetal acceleation and foce a v ( 2 2 T c ) 2 F mv m( 2 2 T c ) 2 m
Conceptual Poblem A igid body otates about a fixed axis ith a constant angula acceleation. Which one of the folloing statements is tue concening the tangential acceleation of any point on the body? (a) The tangential acceleation is zeo m/s 2. (b) The tangential acceleation depends on the angula velocity. (c) The tangential acceleation is equal to the centipetal acceleation. (d) The tangential acceleation is constant in both magnitude and diection. (e) The tangential acceleation depends on the change in the angula velocity. a T a D Dt a c 2
Rolling (ithout slipping) Put otational and tanslational motion togethe The tangential speed of a point on the oute edge of the tie is equal to the speed of the ca ove the gound. Fo an object that is olling ithout slipping, the tanslational and otational motions ae coupled! v a a
ACT: Tuck Speedomete The speedomete of a tuck is set to ead the linea speed of the tuck but uses a device that actually measues the angula speed of the ties. If lage diamete ties ae mounted on the tuck instead, ho ill that affect the speedomete eading as compaed to the tue linea speed of the tuck? a) speedomete eads a highe speed than the tue linea speed b) speedomete eads a loe speed than the tue linea speed c) speedomete still eads the tue linea speed The linea speed is v ωr. So hen the speedomete measues the same angula speed ω as befoe, the linea speed v is actually highe, because the tie adius is lage than befoe.
Rotational Kinetic Enegy and the Moment of Inetia Imagine a mass at the end of a light od, KE 1 2 mv2 1 2 m ( ω ) 2 1 ( 2 m 2 )ω 2 We can also ite the kinetic enegy as KE 1 2 Iω 2 Whee I, the moment of inetia, is given by I m 2