# 10/9/2003 PHY Lecture 11 1

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1 Announcements 1. Physc Colloquum today --The Physcs and Analyss of Non-nvasve Optcal Imagng. Today s lecture Bref revew of momentum & collsons Example HW problems Introducton to rotatons Defnton of angular varables Moment of nerta Energy assocated wth rotatons 10/9/003 PHY Lecture 11 1

2 From HW 9 6. HW6 9.P.031. A space vehcle s travelng at v 0 = 5700 km/h relatve to the Earth when the exhausted rocket motor s dsengaged and sent backward wth a speed of v = 85 km/h relatve to the command module. The mass of the motor s four tmes the mass of the module. What s the speed of the command module relatve to Earth after the separaton? [ ] km/h 4M 4M v 0 M Mv = 4M ( v v ) v M v M v M 5 0 M solve for v M + Mv M 10/9/003 PHY Lecture 11

3 From HW 9 7. HW6 9.P.035. A certan radoactve nucleus can transform to another nucleus by emttng an electron and a neutrno. (The neutrno s one of the fundamental partcles of physcs.) Suppose that n such a transformaton, the ntal nucleus s statonary, the electron and neutrno are emtted along perpendcular paths, and the magntudes of the lnear momenta are kg m/s for the electron and kg m/s for the neutrno. As a result of the emssons, the new nucleus moves (recols). (a) What s the magntude of ts lnear momentum? [ ] kg m/s (b) What s the angle between ts path and the path of the electron? [ ] (c) What s the angle between ts path and the path of the neutrno? [ ] (d) What s ts knetc energy f ts mass s kg? [ ] J p e p N θ e p ν θ ν 10/9/003 PHY Lecture 11 3

4 From HW HW6 10.P.046. Two.0 kg masses, A and B, collde. The veloctes before the collson are v A = j and v B = j. After the collson, v' A = j. All speeds are gven n meters per second. (a) What s the fnal velocty of B? [ ] m/s + [ ] m/s j (b) How much knetc energy was ganed or lost n the collson? [ ] J m( v A + v B ) = m( v A + v B ) ( j ) + ( j) = ( j) E = 1 1 ( + v ) m( v + v ) m v A B A B + v B 10/9/003 PHY Lecture 11 4

5 From HW HW6 10.P.047. An alpha partcle colldes wth an oxygen nucleus, ntally at rest. The alpha partcle s scattered at an angle of 64.0 above ts ntal drecton of moton, and the oxygen nucleus recols at an angle of 47.0 on the opposte sde of that ntal drecton. The fnal speed of the nucleus s m/s. In atomc mass unts, the mass of an alpha partcle s 4.0 u. The mass of an oxygen nucleus s 16 u. (a) Fnd the fnal speed of the alpha partcle. [ ] m/s (b) Fnd the ntal speed of the alpha partcle. [ ] m/s v v f v O 10/9/003 PHY Lecture 11 5

6 Angular moton s angular dsplacement θ(t) dθ angular velocty ω (t) = dt angular acceleraton dω α (t) = dt natural unt == 1 radan elaton to lnear varables: s θ = r (θ f -θ ) v θ = r ω a θ = r α 10/9/003 PHY Lecture 11 6

7 v 1 =r 1 ω r 1 ω r v =r ω Specal case of constant angular acceleraton: α = α 0 : ω(t) = ω + α 0 t θ(t) = θ + ω t + ½ α 0 t ( ω(t)) = ω + α 0 (θ(t) - θ ) 10/9/003 PHY Lecture 11 7

8 Example: Compact dsc moton ω 1 In a compact dsk, each spot on the dsk passes the laser-lens system at a constant lnear speed of v θ = 1.3 m/s. ω 1 =v θ /r 1 =56.5 rad/s ω =v θ /r =.4 rad/s What s the average angular deceleraton of the CD over the tme nterval t=4473 s? α = (ω -ω 1 )/ t = rad/s 10/9/003 PHY Lecture 11 8 ω

9 Object rotatng wth constant angular velocty (α = 0) ω v=ω v=0 Knetc energy assocated wth rotaton: K = 1 where : I m v = m r m r moment of nerta 10/9/003 PHY Lecture ω 1 Iω ;

10 10/9/003 PHY Lecture 11 10

11 Peer nstructon queston: Suppose each of the followng objects each has the same total mass M and outer radus and each s rotatng counterclockwse at an constant angular velocty of ω=3 rad/s. Whch object has the greater knetc energy? (a) (Sold dsk) (b) (crcular rng) 10/9/003 PHY Lecture 11 11

12 Varous moments of nerta: sold cylnder: I=1/ M sold sphere: I=/5 M sold rod: I=1/3 M 10/9/003 PHY Lecture 11 1

13 Calculaton of moment of nerta: Example -- moment of nerta of sold rod through an axs perpendcular rod and passng through center: I = m r M M = dr r = r dr = 1 3 M Extra credt: Wrte out the evaluaton of I for another shape. 10/9/003 PHY Lecture 11 13

14 How to make objects rotate. θ r Defne torque: τ = r x F τ = rf sn θ r snθ F sn θ F θ F = ma r F τ = r ma = Iα 10/9/003 PHY Lecture 11 14

15 From HW /9/003 PHY Lecture 11 15

16 Newton s second law appled to center-of-mass moton dv F = m Ftotal = dt M dv dt CM Newton s second law appled to rotatonal moton τ v dv F = m r F = r m dt = τ τ r total F m = ω r = = r I d dω dt ( ω r ) dt = Iα dm 10/9/003 PHY Lecture dv dt (for rotatng about prncpal axs) r I F m d

17 Another example: A horzontal 800 N merry-go-round s a sold dsc of radus 1.50 m and s started from rest by a constant horzontal force of 50 N appled tangentally to the cylnder. Fnd the knetc energy of sold cylnder after 3 s. F K = ½ I ω τ = Ι α ω = ω + αt = αt In ths case I = ½ m and τ = F ( 50N ) F K = g t = 9.8m/s (3s) = J mg 800N 10/9/003 PHY Lecture 11 17

18 10/9/003 PHY Lecture e-examnaton of Atwood s machne T 1 T T T 1 I T 1 -m 1 g = m 1 a T -m g = -m a τ =T T 1 = I α = I a/ + + = + + = / Ig τ / g a I m m m m I m m m m

19 Another example: Conservaton of energy: K f + U f = K + U h m 1 m v 1 h/ v 10/9/003 PHY Lecture 11 19

20 Peer nstructon queston Three objects of unform densty a sold sphere (a), a sold cylnder (b), and a hollow cylnder (c) -- are placed at the top of an nclne. If they all are released from rest at the same elevaton and roll wthout slppng, whch object reaches the bottom frst? (a) sold sphere (b)sold cylnder (c)hollow cylnder 10/9/003 PHY Lecture 11 0

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