Exm II Physics 101: Lecture 15 olling Objects Tody s lecture will cover Textbook Chpter 8.5-8.7 Physics 101: Lecture 15, Pg 1
Overview eview K rottion = ½ I w Torque = Force tht cuses rottion t = F r sin q Equilibrium F Net = 0 t Net = 0 Tody t Net = I (rottionl F = m) Energy conservtion revisited Physics 101: Lecture 15, Pg
Liner nd Angulr Liner Angulr Displcement x q Velocity v w Accelertion Inerti m I KE ½ m v ½ I w NL F=m t = I Momentum p = mv L = Iw Tody Physics 101: Lecture 15, Pg 3
ottionl Form Newton s nd Lw t Net = I Torque is mount of twist provide by force» Signs: positive = CCW Moment of Inerti like mss. Lrge I mens hrd to strt or stop from spinning. Problems Solved Like Newton s nd Drw FBD Write Newton s nd Lw Physics 101: Lecture 15, Pg 4
The Hmmer! 3% 6% 6% You wnt to blnce hmmer on the tip of your finger, which wy is esier A) Hed up B) Hed down C) Sme I just tried it in my home nd I guess it is esier to blnce the hmmer with the hed up. Angulr ccelertion is smller the lrger the rdius the lrger the moment of inerti. Physics 101: Lecture 15, Pg 5
9% 63% 8% The Hmmer! You wnt to blnce hmmer on the tip of your finger, which wy is esier A) Hed up B) Hed down C) Sme t = I Key ide: higher ngulr m g sin(q) ccelertion = m mens more difficult Inerti to blnce. Torque increses with increses s Angulr ccelertion decreses with! So lrge is esier g sin(q) / = to blnce. Wht is ngulr ccelertion? mg Physics 101: Lecture 15, Pg 6
Flling weight & pulley A mss m is hung by string tht is wrpped round pulley of rdius ttched to hevy flywheel. The moment of inerti of the pulley + flywheel is I. The string does not slip on the pulley. Strting t rest, how long does it tke for the mss to fll distnce L. I T m Wht method should we use to solve this problem? A) Conservtion of Energy (including rottionl) B) t Net = I nd then use kinemtics Either would work, but since it sks for time, we will use B. mg L Physics 101: Lecture 15, Pg 7
Flling weight & pulley... For the hnging mss use F Net = m mg - T = m For the flywheel use t Net = I I T sin(90) = I elize tht = T T = I Now solve for, eliminte T: m mg L = m m g I Physics 101: Lecture 15, Pg 8
Flling weight & pulley... Using 1-D kinemtics we cn solve for the time required for the weight to fll distnce L: L 1 y = y0 v0t t = 1 t t L = I T m mg where = m m g I L Physics 101: Lecture 15, Pg 9
Torque ACT Which pulley will mke it drop fstest? 1) Smll pulley ) Lrge pulley 3) Sme = m m g I Lrger, gives lrger ccelertion. Physics 101: Lecture 15, Pg 10 5
Tension m1 T 1 T m m3 F Compre the tensions T 1 nd T s the blocks re ccelerted to the right by the force F. A) T 1 < T B) T 1 = T C) T 1 > T T 1 < T since T T 1 = m. It tkes force to ccelerte block. T 1 m m 1 Compre the tensions T 1 nd T s block 3 flls A) T 1 < T B) T 1 = T C) T 1 > T T m 3 T > T 1 since T T 1 = I. It tkes force (torque) to ccelerte the pulley. Physics 101: Lecture 15, Pg 11
olling A wheel is spinning clockwise such tht the speed of the outer rim is m/s. Wht is the velocity of the top of the wheel reltive to the ground? + m/s Wht is the velocity of the bottom of the wheel reltive to the ground? - m/s y x m/s m/s You now crry the spinning wheel to the right t m/s. Wht is the velocity of the top of the wheel reltive to the ground? A) -4 m/s B) - m/s C) 0 m/s D) +m/s E) +4 m/s Wht is the velocity of the bottom of the wheel reltive to the ground? A) -4 m/s B) - m/s C) 0 m/s D) +m/s E) +4 m/s Physics 101: Lecture 15, Pg 1
olling An object with mss M, rdius, nd moment of inerti I rolls without slipping down plne inclined t n ngle q with respect to horizontl. Wht is its ccelertion? Consider CM motion nd rottion bout the CM seprtely when solving this problem I M q Physics 101: Lecture 15, Pg 13
olling... Sttic friction f cuses rolling. It is n unknown, so we must solve for it. First consider the free body digrm of the object nd use F NET = M cm : In the x direction Mg sin q - f = M cm Now consider rottion bout the CM nd use t Net = I relizing tht t = f nd = q M Mg f f = I f = I q Physics 101: Lecture 15, Pg 14
olling... We hve two equtions: Mg sin q - f = M f = I We cn combine these to eliminte f: For sphere: M sin q = g M I A M I M sin q 5 = g gsin q = M M 7 5 q Physics 101: Lecture 15, Pg 15
Energy Conservtion! Friction cuses object to roll, but if it rolls w/o slipping friction does NO work! W = F d cos q d is zero for point in contct No dissipted work, energy is conserved Need to include both trnsltionl nd rottionl kinetic energy. K = ½ m v + ½ I w Physics 101: Lecture 15, Pg 16
Trnsltionl + ottionl KE Consider cylinder with rdius nd mss M, rolling w/o slipping down rmp. Determine the rtio of the trnsltionl to rottionl KE. Trnsltionl: K T = ½ M v ottionl: K = ½ I w use I = 1 M nd w = V ottionl: K = ½ (½ M ) (V/) = ¼ M v H = ½ K T Physics 101: Lecture 15, Pg 17
olling Act Two uniform cylinders re mchined out of solid luminum. One hs twice the rdius of the other. If both re plced t the top of the sme rmp nd relesed, which is moving fster t the bottom? () bigger one (b) smller one (c) sme K i + U i = K f + U f 1 1 MgH = I w MV 1 MgH = M V 1 1 MV V = 4 gh 3 Physics 101: Lecture 15, Pg 18
Summry t = I Energy is Conserved Need to include trnsltionl nd rottionl Physics 101: Lecture 15, Pg 19