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dynamics practice problems including kinematics and rigid body motion
Typology: Exercises
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Ho mew or k Ass i gn m en t 5 P ro bl e m 1 Illu strat io n
Giv e n: ① d, = d a o r r , = L h = 150 mm = 0 - 1 5 m C o e ffi c i e nt o f R es ti tut i o n:
T o fin d: Ang u l ar Vel oc ity o f th e b a r 'w ' im media te ly aft e r imp a ct.
As su mp t i on s : T h e co nn e cti n g ba r is o f neg li g ib le ma ss.
Sol u ti o n:
St ef : Re f e ren c e fram es.
( Th i s a ls o t ells us M, = M z)
6 0 0 m m
150 m m ↓
r : 1 22 =^8
h=
M 2 - m MI MI
r : I
s t e el
6 00 mm ai
1 50 m m ↓
h =
M 2 - m M I MI
i
i
' N 'f r ame
a t ' a 'f ram e
Ste p 2 : Go ve rn i ng E quat i ons of m o tio n I W I- 2 = A KE Wo r k d o ne by grav i ty bet wee n ① - In it i a l s t ate 2 -^ ju^ s^ t^ b^ efo^ r^ e^ h^ it^ ting th e Bra ss an d st e el p la te s.
So, Wg'^ "^ =^ f^ E^ g.^ d-s^ (^ o^ nl^ y^ c^ ons^ ide^ r^ i^ ng^ Bal^ l^1 ) SI = § - m g I. - * ↑ = mg ( is ,) =m g h 5 1
S ince th e s yst em st ar t s fr o m re s t ; A KE = ± m " #. "V 7 @ O (^0) V 7 = F gh
Du e t o symm etr y ; V E = I gn
Th e n e x t g o ve rn i ng e qua ti on we use i s: e = I V 23 - V 11 = R el a ti ve vet. E v il
a- te r c ol l is ion R ela ti ve ve l - b e fo re co ll i si on
OR , For Ba ll 1 - BRA SS : e , = I v 1 - ( 0 7 + 0 9 ) [ No te : t h e b r ass pl at e E vi + COE + 0 51 a^ ss^ um^ ed^ t^ o^ b^ eat r e st be fo r e & a fte r ] coll i s ion O R, (^) e , =
¥.
(V ii =- ug h
9 ) Ve loc i ty o f Ba l l 1 j us t b e fo re c oll i si on.
O R , 1 1 % 7 11 = e , V i = e ,E gh Sim i l arl y; 11 % 11 1 = e s Ig ht
Po st c oll isi on
s tee l
v
N i si N ui t " v 1
Nii = "v e
→ i B U T (^) "i t ≠ " it Th is ca us es r otat ion of the b ar Co mpu ti n g "t A r eq uir e s th e re la tiv e ve lo cit y equ a t ion →
☑ it
Wn a i ' A' 2
0 A ñ O i
' μ'
NJ 2 / (^0) - -^ "^ ✓^ ◦^ A^ +^ AT^4 0 A^ +^ N^ @A^ ✗^ Fo^ r
N ow , w e sup er im p ose the ' A' a n d ' O'f r am es suc h th at the y sh a re ori gi n s.
T hi s w e u se th e si mi lar R e lati v e v elo c it y E OM
Re l - r e l - of 2 wo t. O i n' N ' basi s
f or 1. N fl / (^0) = NTOA + A I^4 0 A^ +^ NIA^ ✗^ F^ OI
Equa ti o n 2 -^1 g^ iv^ es^ - NJ 2 10 - N J % = NW A ×@ 02 - 8 0 )
O R, (^) e z T g h ↑ e,^ Th^ i^ =^ '^ wk^ ×^ (^4 27 -^ (-^1 /^2 7 ))
O R, (^) (ez - e ) Fg h ↑ lw^ j^ →^ %^ ; I ← Com p a ring t he c om p o ne n ts ;
(e x- e) Tgh = to
OR , (^) W = (^) ea - e. DE #
N o w, s ubs ti tute n um eri ca l val u es ; W = ( 0. 6 - 0. 4 ) % 2 × 9. 8 1 40. 15
W = 0. 5 7 2 ra d /s
An y [t^ he^ s^ ig^ n^ co^ n^ fi^ r^ ms^ tha^ t ou r a s sume d di rec tio n of r ot at io n is COR R EC T
"A ss um e c o w i s + ve r ota tio n"
P ro b le m 2 I ll us trat ion :
Giv e n: ① Id en ti ca l b a l ls i. e; PA = B I M A = MB c oef fici ent of rest i tu ti on , e = 0. 7
To f ind : ① Ve l oci ty of e ach b a ll j us t a f te r im pa c t % a ge loss o f KE d u e to co lli s io n
No p a rt ic ul a r a s su m pt i on s to H I GHLI GH T! Solu ti on : ^
Ste ph: R ef er en c e fr a me s *
0 , r
= 6 ft/S r VA
, r
' N' f ram e
S te p 2 : Go v er ni n g Eq uatio ns of M o t io n. A (^) C ons e rvat io n o f Li n ea r Mo m e nt um →
☒" J:O + ✗ " 5 : 1 0 = m " 5 7 10 + m Ñ f%
W e k n ow N J A lo i n itia l ve locities } Ny B l o i
i = VA Si n 30 ° ↑ - V A 05 30 05 an d,
= (^) ↳ ↑
s o; N^ #A^ lo^ +^ N^ J?/^0 =^ V^ A^ si^ n^30 °^ ↑^ +^ (^ VB^ -^ V^ a^ c^ o^ s^30 )^ ↑
"D ur i ng Col li s ion a t an an g le ; t he ta ng e n t ial c om po ne n t o f ve lo c ity do es no t cha nge. "
He r e ↑ i s t he t ang e nt i al d ir e c ti o n a t t he c o nta ct po i nt-
A f + V 4 5 + ✓ ¥? ↑ + U p? ↑ = V a s in 30 ° i + ( VB - Va c os 30 °
fr om ①.
Vf t
i
OR, V^ Af^ t^ +^ Uft^ =^ V^ A^ s^ i^ n^3 0 °
a nd ;
¥? + U p? = VB - V a c os 3 0 °
B (^) Co e ff ic i en t o f R e s titu ti on ( By de fi nit io n app l ie s a lon g t he ] l oc a l no rm a l dir e c t io n
@ = (^) V fn ✓ I n
Vi? - V i n
O R , (^) e (v , - (Va co s 30 °) =^ ✓^ E^ n^ ✓^ fu
= 8 ft /s
= 8 - 6 c os 3 0 ° = 8 - 35 = 2. 8 0 4 ft /s -
O R , Tfn^ -^ ✓^ E^ n^ =^0.^7 (^8 - (^ -^5 -^1 9 6 ))=^9.^23
2 b
f ro m (^2) b (^) a n d 2 C i^ w^ e^ so^ lve^ s^ i^ mu^ lta^ n^ eo^ u^ sly^ to^ ob^ ta^ in;
Vfn = 6 - 0 2 f t/s (^) i
✓f Ba = - 3. 2 2 ft/ s
But w e a ls o kn o w by de fin iti on of 2 D c ol li s i on s that ; in the ta nge nti al di r e c ti o n th e (^) ↓v eloc i tie s o f th e pa r ti cle s re s p ec tiv e d o no t c han g e - So, (^) V ff = ✓ A it
✓♀ = V I.
@^ f - " T aft = V A si n 30 ° ↑ i "T ff = 0
So , co mbin i ng r es ① & r e s
N T A f = 3 ↑
An d , N JB f =. 0 ↑
N ow , fo r s eco nd p ar t ; we ne ed to c o m pu te I K E i a nd E KE f.
IM V I. T: + ± m i?. T?
=
Res 1
(^2) i
O R, 1 17 71 1 = 6 - 72 6 f t/s
IM [ 3 6 + 6 4 ] = 5 0 m u n its.
I KE F = E M Tf.^ If^ +^ Im^ v^73.^ I^ "
Im [ " FA IR + 11 V 71 ]
± M ( ( 67 26 5 + 4 25 )
2 7. 80 37 m un it s.
/ d ec re ase £ L OS S
IKE : - IK E f
= 22. 1 96 2 m u n i t s.
S o, (^) % a ge L O SS = (^2 2). 1 96 2 M
(^50) m
OR , (^) % LOS S = 44. 39 25 % A y