EN221 - Fall2008 - HW # 7 Soluions Pof. Vivek Shenoy 1.) Show ha he fomulae φ v ( φ + φ L)v (1) u v ( u + u L)v (2) can be pu ino he alenaive foms φ φ v v + φv na (3) u u v v + u(v n)a (4) (a) Using v JV, an changing he inegal o he efeence fame. bu, φ v φ v (φjv ) ( φj + φj)v ( φj + φj L)V ( φ + φ L)JV ( φ + φ ivv)v (5) (gaφ) v + φ ivv iv(φv) φ φ + (gaφ) v ( φ + φ ivv)v 1
( φ ( φ ) + (gaφ) v + φ ivv v ) + iv (φv) v (6) Using ivegence heoem φ φ v + φv na (7) (b) Using v JV, an changing he inegal o he efeence fame. (uj) uv V ( uj + uj)v ( uj + uj L)V ( u + u L)JV ( u + u ivv)v (8) bu, uv (gau) v + u ivv iv(v u) u u + (gau) v ( u + u ivv)v ( ) u + (gau) v + u ivv v ( ) u + iv(v u) v (9) Using ivegence heoem u u v + u (v n)a (10) 2
2.) Consie he cuing configuaion shown in he figue below. The wokpiece is assume o be an incompessible maeial of mass ensiy ρ. The opening angle of he saionay cuing ool is α. The wok piece is pushe owas he ool by a foce of magniue F, a a consan spee V 0. The hickness (lengh in he x 3 iecion) of he wokpiece is b (no shown in he figue). The conac beween he chip an ool obeys Coulomb ficion wih coefficien of ficion µ, so ha µn, whee an N ae he ficional an nomal-eacion foces, especively, as shown in he figue. Noe ha he chip emeges a a spee V an makes an angle β o he hoizonal. (a) Using consevaion of mass, elae he chip spee V an he cuing spee V 0. (b) Using linea momenum balance, compue he cuing foce F an he ool eacions N an in ems of ρ, b, h, V 0, α, β an µ. fom he figue DAB + α + β π DAB π (α + β) also i can be seen ha fom he Eqns(11 an 12) sin DAB sin [π (α + β)] c BA h sin β BA sin (α + β) c sinβ h sin (α + β) c h sin β (11) (12) (13) le (BE) x an (BF) l so he oal mass of he wokpiece is (which is incompessible) Noe: ρ is consan M ρxhb + ρlcb + mass of small egion in beween(consan) M 0 0 ρẋhb + ρ lch v l ẋh V oh c c (ẋ V o ) (14) The momenum is P ρv l c b cosα e 1 + ρv l c b sin α e 2 + ρv o x c b e 1 + momenum of smallegion(consan) P P Foce [F Nµ cosα N sin α]e 1 + [N cosα + Nµ sinα]e 2 () (ρv 2 c b cosα + ρv 2 0 h b)e 1 + (ρv 2 c b sin α)e 2 (16) 3
on compaing he Eqns( an 16) we obian N ρv 2 c b sin α (17) µ sinα cosα Nµ (18) [ V 2 ] c sinα(µ cos α + sin α) F ρb + V0 2 V 2 c (19) µ sinα cosα 4
3).If enoes he acion pe uni aea on a suface whose nomal is in iecion n, show ha he squae of he magniue of he shea sess on ha suface is (n.) 2 If n i (i 1, 2, 3) ae he componens of n elaive o he pincipal axes of he Cauchy sess enso, show ha he above expession may be wien ( 2 3 ) 2 n 2 2n 2 3 + ( 3 1 ) 2 n 2 3n 2 1 + ( 1 2 ) 2 n 2 1n 2 2 whee n i (i 1, 2, 3) ae he pincipal Cauchy sesses. Show ha he aveage of his is expessible as an euce ha his expessible as 1 { (2 3 ) 2 + ( 3 1 ) 2 + ( 1 2 ) 2} 2 { [I1 (T)] 2 2I 2 (T) } whee I 1 (T) an I 2 (T) ae he fis wo pincipal invaians of T now, σ T n s s s ( n)n [ ( n)n] [ ( n)n] 2( n)(n ) + ( n)( n) ( n) 2 (20) If σ 1, σ 2 an σ 3 ae he pincipal sesses 3 σ σ i p i p i p 1,p 2,p 3 ae pincipal iecions (21) i1 n n i p i i σ i n i (no epeae summaion) s s (σ1n 2 2 1 + σ2n 2 2 2 + σ3n 2 2 3) (σ 1 n 2 1 + σ 3 n 2 3 + σ 3 n 2 3) (22) n 2 1 + n 2 2 + n 2 3 1 hence, s s (σ 2 1n 2 1 + σ 2 2n 2 2 + σ 2 3n 2 3)(n 2 1 + n 2 2 + n 2 3) (σ 1 n 2 1 + σ 3 n 2 3 + σ 3 n 2 3) (σ 2 σ 3 ) 2 n 2 2n 2 3 + (σ 3 σ 1 ) 2 n 2 3n 2 1 + (σ 1 σ 2 ) 2 n 2 1n 2 2 (23) 5
now; < n 2 2n 2 3 > is obaine as follows Le, n 1 cosθ (24) n 2 sinθ cosφ (25) n 3 sinθ sin φ (26) 2π π < n 2 2 n2 3 > 0 0 (sin2 θ sinφ cos φ) 2 sin θθφ 2π π 0 0 sin θθφ 1 Similaly, < n 2 3 n2 1 > 1 an < n2 1 n2 2 > 1 (27) I 1 (T) σ 1 + σ 2 + σ 3 (28) I 2 (T) σ 1 σ 2 + σ 2 σ 3 + σ 3 σ 1 (29) upon subsuion 2 ( I1 (T) 2 3I 2 (T) ) 1 ( (σ2 σ 3 ) 2 + (σ 3 σ 1 ) 2 + (σ 1 σ 2 ) 2) (30) 6
4). Fomulae he balance of angula momenum fo a maeial boy ace on by a boy oque c pe uni mass in aiion o he boy foce b, an a conac oque u (n) pe uni aea in aiion o he conac foce (n). Esablish, by a eaheon agumen, he exisence of a couple sess enso µ such ha u n µ T n. The angula momenum L is L ρx v (31) L since mass is conseve ρ ρ hus, ρ (x v) (32) (x v) v v + x a x a (33) L T oque ρx a (34) T ( ρb x + c) + (x + u)a (35) ρx a [ρ(x a x b) c] bu, σ T n ( ρb x + c) + (x + u)a (x + u)a (36) (x σ T n)a (x ivσ τ) (37) (P-20, Pg44, Chawick) τ is axial veco of (σ σ T ) [x (ρa ρb ivσ) ρc] τ + u A (38) using he Foce balance ρa ρb ivσ 0 also using agumens simila o hose on 146-147 of Ogen we can show, u µ T n an µ T na ivµ hus, (ρc + τ + ivµ) 0, (39) ρc + τ + ivµ 0 (40) 7