Homework: 49, 56, 67, 60, 64, 74 (p )

Transcription

1 Hoework: 49, 56, 67, 60, 64, 74 (p )

2 49. bullet o ass 0g strkes a ballstc pendulu o ass kg. The center o ass o the pendulu rses a ertcal dstance o c. ssung that the bullet reans ebedded n the pendulu, calculate the bullet s ntal speed. The collson here s a perectly nelastc collson, the lnear oentu o the syste bullet pendulu s consered because the external pulse J on the syste s zero: ( M ) V V M ter the collson, the echancal energy o the syste bullet-block-earth s consered: ( M ) V M ( gh M ) gh V Read also Saple Proble 9-8 (page 8) gh

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4 56. In the beore part o the gure below, car (ass 00 kg) s stopped at a trac lght when t s rear-ended by car (ass 400 kg). oth cars then slde wth locked wheels untl the rctonal orce ro the slck road (wth a low µ k o 0.0) stops the, at dstance d 8. and d 6.. What are the speeds o (a) car and (b) car at the start o the sldng, just ater the collson? (c) ssung that lnear oentu s consered durng the collson, nd the speed o car just beore the collson. (d) Explan why ths assupton ay be nald.

5 0 ad 0 ad µ k gd The agntude o the acceleraton o each car s deterned by: (c) I p consered: (d) k a F Veloctes o car and ater the collson: µ kg k µ k µ gd ; µ 0, Δp 0, Δp J J F F Δt Δt 0 0 k gd Howeer, ag Thereore, the assupton that p s consered (or Δp 0) ay be nald ag p s consered the rctonal orce exerted on the cars ro the road s neglgble durng the collson. g

6 60. lock (ass.6 kg) sldes nto block (ass.4 kg), along a rctonless surace. The drectons o three eloctes beore () and ater () the collson are ndcated; the correspondng speeds are 5.5 /s,.5 /s, and 4.9 /s. What are the (a) speed and (b) drecton (let or rght) o elocty? (c) Is the collson elastc? We choose the poste drecton s rghtward (a) The lnear oentu o the syste () s consered (no rcton): ( ) (.5 4.9).9(/s).6 (b) > 0, so the drecton s to the rght (c) ΔK K K the collson s elastc ( ) ( ) 0 (J)

7 64. steel ball o ass.5 kg s astened to a cord that s 70c long and xed at the ar end. The ball s then released when the cord s horzontal. t the botto o ts path, the ball strkes a.8 kg steel block ntally at rest on a rctonless surace. The collson s elastc. Fnd (a) the speed o the ball and (b) the speed o the block, both just ater the collson. Conseraton o echancal energy: gh gl gh U g gh h l Conseraton o lnear oentu (no rcton): gl () U g 0 () ( ) ( ) The collson s elastc: ΔK 0

8 () () ) ( ) ( ) ( ;

9 74. Two.0kg bodes, and, collde. The eloctes beore the collson are /s and /s.ter the collson, /s. What are (a) the nal elocty o and (b) the change n the total knetc energy (ncludng sgn)? j ˆ 30 ˆ 5 j ˆ 5 ˆ 0 j ˆ 0 5ˆ ' We assue that the total lnear oentu o the two bodes s consered: ' ' : ' ' j ˆ 5 0ˆ ' ' ' K K (b) 500 (J) Δ K K K è The collson here s an nelastc collson snce KE s not a constant.

10 Part C Dynacs and Statcs o Rgd ody Chapter 5 Rotaton o a Rgd ody bout a Fxed xs 5.. Rotatonal Varables 5.. Rotaton wth Constant ngular cceleraton 5.3. Knetc Energy o Rotaton, Rotatonal Inerta 5.4. Torque, and Newton s Second Law or Rotaton 5.5. Work and Rotatonal Knetc Energy 5.6. Rollng Moton o a Rgd ody 5.7. ngular Moentu o a Rotatng Rgd ody 5.8. Conseraton o ngular Moentu

11 Oerew We hae studed the oton o translaton, n whch objects oe along a straght or cured lne. In ths chapter, we wll exane the oton o rotaton, n whch objects turn about an axs.

12 5.. Rotatonal arables: We study the rotaton o a rgd body about a xed axs. Rgd bodes: odes can rotate wth all ts parts locked together and wthout any change n ts shape. Fxed axs: xed axs eans the rotatonal axs does not oe. ngular Poston: Reerence lne: To deterne the angular poston, we ust dene a reerence lne, whch s xed n the body, perpendcular to the rotaton axs and rotatng wth the body. The angular poston o ths lne s the angle o the lne relate to a xed drecton. s θ r θ : radans (rad) re π rad

13 ngular Dsplaceent Δθ θ θ Conenton: Δθ > 0 n the counterclockwse drecton. Δθ < 0 n the clockwse drecton. ngular Velocty θ θ Δθ erage angular elocty: ωag t t Δt Δθ dθ Instantaneous angular elocty: ω l Δt 0 Δt dt Unt: rad/s or re/s or rp; (re: reoluton) ngular cceleraton ω ω Δω erage angular acceleraton: αag t t Δt Δω dω Instantaneous angular acceleraton: α l Δt 0 Δt dt Unt: rad/s or re/s Note: ngular dsplaceent, elocty, and acceleraton can be treated as ectors (see page 46).

14 5.. Rotaton wth Constant ngular cceleraton For one densonal oton: dx d ; a dt dt Let s change arable naes: x θ, ω, a α ω ω αt ω 0 0 θ θ ω t ω 0 0 αt α ( θ θ ) Checkpont (p. 48): In our stuatons, a rotatng body has angular poston θ(t) gen by (a) θ3t-4, (b) θ-5t 3 4t 6, (c) θ/t -4/t, and (d) θ5t -3. To whch stuatons do the angular equatons aboe apply? (d) 0

15 5.3. Knetc Energy o Rotaton a. Lnear and ngular Varable Relatonshp The poston: s θ where angle θ easured n rad; s: dstance along a crcular arc; r: radus o the crcle The speed: ds dt r dθ r dt ω r where ω n radan easure The perod o reoluton: The cceleraton: d dt Tangental acceleraton: Radal acceleraton: πr π T dω ω r dt r a t a r ω α r r

16 b. Knetc Energy o Rotaton: The KE o a rotatng rgd body s calculated by addng up the knetc energes o all the partcles: K I r K : Rotatonal Unt or I: kg 3 3 ( ωr) ω Inerta (or oent o... r nerta), K Iω

17 c. Calculatng the Rotatonal Inerta: I the rgd body conssts o a ew partcles: For contnuous bodes: I I r r d Parallel-xs Theore: The theore allows us to calculate I o a body o ass M about a gen axs we already know I co : I I co Mh h: the perpendcular dstance between the gen axs and the axs through the center o ass o the body.

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