τ rf = Iα I point = mr 2 L35 F 11/14/14 a*er lecture 1

Transcription

1 A mass s attached to a long, massless rod. The mass s close to one end of the rod. Is t easer to balance the rod on end wth the mass near the top or near the bottom? Hnt: Small α means sluggsh behavor and α τ/i. A) Easer wth mass near top. B) Easer wth mass near bottom. C) No dfference. τ rf Iα I pont mr 2 L35 F 11/14/14 a*er lecture 1

2 A mass s attached to a long, massless rod. The mass s close to one end of the rod. Is t easer to balance the rod on end wth the mass near the top or near the bottom? Hnt: Small α means sluggsh behavor and α τ/i. θ L θ θ mg F mgsnθ τ Iα α τ I F L ml mgsnθ 2 ml gsnθ 1 L L L35 F 11/14/14 a*er lecture 2

3 A mass s attached to a long, massless rod. The mass s close to one end of the rod. Is t easer to balance the rod on end wth the mass near the top or near the bottom? Hnt: Small α means sluggsh behavor and α τ/i. A) Easer wth mass near top. B) Easer wth mass near bottom. C) No dfference. α 1 L Large L means a small α, so the bar s easer to balance. L35 F 11/14/14 a*er lecture 3

4 Assgnments Announcements: No HW due n rectaton next week. CAPA 12 s now lve. You should have read Ch. 10 and 11. Comng Up: Mdterm exam 3: NEXT week -- Thurs Nov 20, same tme, same place. Prmary focus wll be on Chs Old exam 3 has been placed on D2L. Today: Contnung wth rotatonal moton, found n Chs. 10 & 11: torque, moment of nerta, rotatonal knetc energy, conservaton of energy ncludng rotatonal moton. Then movng on to angular momentum and ts conservaton: re-express all quanttes as vectors. L35 F 11/14/14 a*er lecture 4

5 Rotatonal Knetc Energy and Conservaton of Energy M ω v v ω (v/r) 2 (same unts: J) L35 F 11/14/14 a*er lecture 5

6 Rotatonal Knetc Energy and Conservaton of Energy KE tot 1 2 Mv Iω 2 I sphere 2 5 MR2 I hoop MR 2 I dsk 1 2 MR2 Sphere: v sphere R 2MgH I + MR gh > v dsk > v hoop Hoop: v hoop Dsk: v dsk gh 4 3 gh L35 F 11/14/14 a*er lecture 6

7 Rotatonal Knetc Energy and Conservaton of Energy KE tot 1 2 Mv Iω 2 v sphere 10 7 gh Whch has the greater speed at the bottom of the ramp, the sphere that rolls down the ramp or a block of the same mass that sldes down the ramp? (Assume sldng frcton s neglgble.) A) Block B) Sphere C) Both the same. L35 F 11/14/14 a*er lecture 7

8 Rotatonal Knetc Energy and Conservaton of Energy KE tot 1 2 Mv Iω 2 v sphere 10 7 gh What s the speed of the block at the bottom of the frctonless ramp? A) 11 7 gh B) 2gH C) 19 7 gh D) 3gH MgH 1 2 Mv 2 block v block 2gH > v sphere L35 F 11/14/14 a*er lecture 8

9 Rotatonal Knetc Energy and Conservaton of Energy Whch object wll go furthest up the nclne? I hoop MR 2, I dsk 1 2 MR2 A) Puck B) Dsk C) Hoop D) Same heght. H Because the hoop has the largest moment of nerta and therefore the hghest total knetc energy. KE PE f KE trans + KE rot MgH 1 2 Mv I v r MgH L35 F 11/14/14 a*er lecture 9

10 Two Mscellaneous Subjects before gong on to angular momentum 1. Center of Mass y r rcm Example: m 1 2 kg m 2 1 kg y 1 (2,4) x (5,1) 2 x x (3,3) r L35 F 11/14/14 a*er lecture 10 x CM m 1 x 1 + m 2 x 2 m 1 + m 2 y CM m 1y 1 + m 2 y 2 m 1 + m 2 x CM y CM 2(2) +1(5) (4) +1(1) 2 +1 x y m 9 3 m 3m m 9 3 m 3m

11 Two Mscellaneous Subjects before gong on to angular momentum 1. Center of Mass y Remember: r rcm x r x CM y CM x y 1. You can choose to place your orgn anywhere, but your center of mass wll be relatve to your orgn. 2. It s often useful to place your center of mass at one of your masses. 3. If there s a contnuous dstrbuton of masses, then the center of mass formula s revsed nto an ntegral. L35 F 11/14/14 a*er lecture 11

12 Two Mscellaneous Subjects 1. Center of Mass x CM x A hyrogen (m 1u) and a chlorne (m 35u) atom are separated by about 100 pco-meter (.e., m) n a HCl molecule. How far from the chlorne atos the center of mass of the molecule? Cl y m H x A) m B) m C) m D) m x cm m H x H 1 m H + m Cl m m L35 F 11/14/14 a*er lecture 12

13 Two Mscellaneous Subjects before gong on to angular momentum 2. Parallel Axs Theorem cm L cm If you know the moment of nerta of an object of mass M about a partcular axs through the object s center-of-mass, then ts moment of nerta about any axs parallel to the orgnal axs s: I I cm + Md 2 d L/2 I cm 1 12 ML2 I I cm + M L ML ML2 1 3 ML2 L35 F 11/14/14 a*er lecture 13

14 Two Mscellaneous Subjects before gong on to angular momentum 2. Parallel Axs Theorem I I cm + Md 2 I cm 2 5 Mr 2 If the moment of nerta through the center of mass of a sphere of radus r and mass M s I cm as gven at left, what s the moment of nerta about a parallel axs tangent to the edge of the sphere as shown at rght? A) Mr 2 B) 2 5 Mr 2 C) 7 5 Mr 2 D) 12 5 Mr 2 I I cm + Mr Mr 2 + Mr Mr 2 L35 F 11/14/14 a*er lecture 14

15 Transton to Dscusson of Angular Momentum Prevously: consdered only rotatng systems wth a fxed axs angular poston (rad) angular velocty (rad/s) angular acceleraton (rad/s 2 ) moment of nerta (kg m 2 ) Torque (N m) θ ω α I τ All scalar quanttes but wth a sgn. L35 F 11/14/14 a*er lecture 15

16 Transton to Dscusson of Angular Momentum Now: consder rotatng systems wth a moveable axs angular poston (rad) angular velocty (rad/s) angular acceleraton (rad/s 2 ) moment of nerta (kg m 2 ) Torque (N m) θ ω α I τ Some quanttes become vectors. And we ntroduce another quantty, angular momentum, whch s a vector quantty: L (kg m 2 /s) L35 F 11/14/14 a*er lecture 16

17 Transton to Dscusson of Angular Momentum Angular velocty as a vector quantty: ω ω dθ dt Drecton of angular velocty s gven by the Rght Hand Rule : magntude of angular velocty L35 F 11/14/14 a*er lecture 17

18 Transton to Dscusson of Angular Momentum Angular acceleraton as a vector quantty: α d ω dt Δ ω Δt ω 2 ω 1 Δt L35 F 11/14/14 a*er lecture 18

19 Transton to Dscusson of Angular Momentum Angular acceleraton as a vector quantty: A dsk s spnnng as shown wth angular velocty ω. It begns to slow down. Whle t s slowng, what s the drecton of ts vector angular acceleraton α? α d ω dt Δ ω Δt ω 1 L35 F 11/14/14 a*er lecture 19 ω 2 Δω ω 2 ω 1 + Δω α ponts n the drecton Δω ω 2 ω 1 Δt