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Engineering Dynamics Practice Problems, Study Guides, Projects, Research of Mechanical Engineering

Rensselaer Polytechnic Institute (RPI)Mechanical Engineering

Engineering Dynamics practice problems

Typology: Study Guides, Projects, Research

2022/2023

Uploaded on 02/05/2025

hanlin-wang
hanlin-wang 🇺🇸

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pf2

Partial preview of the text

Download Engineering Dynamics Practice Problems and more Study Guides, Projects, Research Mechanical Engineering in PDF only on Docsity!

Ex am 2 ; Pr a ct i c e P r ob le m 1

G iv e n: M y = M 2 =M I, = 0. 6 ; 0 2 = 0 - 9

Sol uti on: Th er e are 2 c o lli si o n s. T h ey are pa rti al ly in el a s tic. S o , w e can o nly con se rv e l i nea r mom ent u m f o r th i s pro b l em! Co ll isi on 1 : (A ID co llis i o n) A l o ng ↑

M I Vi , i t M z (O ) = M, If + My Ta b

OR , m yria = m v ioa it m u .t t , O R,

Thi s i s one eq ua ti on w ith m or e th a n o ne un k n o wns!

So , w e u se the ef fi c i ency o f co l lis i on to f in d a sec on d eq uat i o n! I, = V 20 - up) ( V i i_o g O R, e^ vi^ :(^ V^2 6 - vi^ d^ )

N ow , C ol l is io n 2 (^ A^2 D^ col^ l^ isi^ on^ )

The c ol li sion i s "P AR TIA LL Y " in el as ti c. BUT L I NE AR M OM EN T UM i s c on se rv e d!

M V a, i - X V 2 6 + 215 , 6 o r, 4 h , = v qc os oit v y si. n o - 5 + 21 6 7

O R , 2 S^ CA^ L^ AR^ E^ QU^ ATION^ S-

A. 2.^8 28 =^ V^2 ,^ f^ C^ os^0 2 ,^6 +^1 3 ,^6

  1. 8 28 =

A t t his st age , I ha v e - V 4 6 , V 3 , 6 , 0

B

all as unk n ow ns ( 3 ) and onl y 2 eq uati ons! W e wi ll us e th e effi ci ency re lat io n t o ob ta in th e 3 ʳ ᵈ e q ua ti on a nd ma ke t he s yst e m d e te r mi ni st ic.

ez =^ V^3 6 n^ Vyn

Li n - Y in

0 % 0. 9 = 1 % 0. i - V 20. :)

w he re n is th e n or m al d i r ec t ion!

J i. ↑i^ i 3 i^.^ i

O R , 0. 9 =

O R ,

OR ,^0 -^6 ×^5 =^ V^26 - V^ i^ t^ (A^ s^ s^ um^ e^ V^ I^ >^ V^1 6 ) 0 1 4 3 = V 26 - V 10 -

i

S o, f rom ① I (^2) V 2 6 = 4 m 1 5 i v # 1 m /s (A s s um pti o n i s cor r ec t) s o , V at = 4 9 ; it = I ai

S ol ve syst em of E qu a tion s!

Fr om A^2 ⑥ i

5 37 4 - -^3 1 °

Pl u ggi n g this in to C^ f

A n d fr om B

E l

T h er ef o r e i AKE^ ac^ ro^ ss^ the^2 "P^ ar^ tia^ l^ ly^ "^ i^ n^ e^ la^ s^ t^ i^ c^ co^ l^ l^ is^ ion^ s^ is^ -

≤m ( 5 ) 70 + 0

I ni tia l K E of t he s ys t em

5 m 1 5

" 2

± m 1 2. 92 9 97 £ 2 m ( 1 - 79 1 ) "

  • (^2) mW "

Ba ll 3

Ba ll 2

Ba ll ①

n •. K E; = 1 2. 5 m J

a nd, KEf = 4. 2 83 3 + 3. 2 07 8 + 0 - 5 m

=

M y = 2 m

i

r ñ

V 2 ,b s in o

'^ ^

l

N

v

o

Reb o u nd a ng le!

q

ez a i y i on )

↑ (E)

ai ↑ 95 °

y i = V i t + U!

v i i. (^) O

lL 1 02 e

na, %

i

ñ

O R , 4 c os 0 ↑ + Y s i O j = (V 26 c os 0 + 24 6 ) i + V 2 6 t rio ↑

V 36 - V 20 cos 0

( 42 - 0 )

  1. 546 = V 36 - V} Lo so^ c

OR , Us^ t^ =^1.^79 1 m^ /s

V 2 8 Cos 0 = - 0 - 754 66

and E^ l^ i

t om O = 2. 82 1

  • 0 - 7 546 =

and (^) V 2 , 6 = C^.^8 28 /^2 +^ (-^0 -^75 46 )^ -^ =^2.^926 944 m/s

> 0 =^ -^7 5 -^0 5 79

7 - 99 0 9 81 m J

Th er e for e ;^ %^ ag^ e^ c^ ha^ n^ ge^ =^

1 2. 5 - 7 2 , 9 1 × 10 0

= 36 %^ Lo^ ss^ in^ K^ E

C oll is io nI →

Lot s i n KE = 12 -^5 - E.f^ i#^ ttm--^4 D^ =^32 %^

LO SS

c olli sio n 2

F ine =

f. 5

= 5 -^989 %^ Lo^ ss.

P ra ct ic e P robl e m? E xa m 2

Q^ s

V ☒ = 5 m /s Ap : 2 m 1 6 "

r 4 = (^5 0) b (^0) V mm 1 0 2 = 5 0 0

12 = 3 0 0 m m

1 •^ fi^ n^ d

p

gov er ni ng E qua tio n- N J P = N - O °

i

i (^) 'N'

t

ñ

i s

A

Oc r ig i d ) JP / 0 + N WA ✗ po p

OR, N IP _ NW^ A^ ✗^ T^ OP Le t NW A = Wa k o r W A 9 3 N IP = Wa a s ✗ 49 = l, wa ai

i % :

O R,

Tr a n sfo rm at io n

i %

a s = c oso , I + sin o , ↑ ai = co s o, i - s in o, ↑

so , N^ IP^ =^ Awa^ co^ s^0 ,^5 + 1 , WA si n o,?

2 F i nd IP? G ov erni n g E qua t io n NIP = N T^ O^ +^ Ag^ PO^ +^ NI^ A^ ✗^ TO^ P^ +^ N^ ot/^ NIA^ ✗^ TO^ P^ )

  • 28 A^ ✗AT^ Pl^ o ri gi d

W aa i x (w a a i x e, a ;)

( r i g id)

OR , NI P = (^) ✗ A a s ✗ l, a i +

OR, NI ' = 4 h a a s + Wa a s ✗ Gw a n

OR , N^ IP^ =^ l,^ da^ ai^1 ,^ w^ a'^ ai

(^3) Fi nd Wp a and Wo p? N o w, f irs t e xp r ess in g v e lo ci t y of 0 w o t- Pi

N J P + B IN D + NI B × IPQ Np? ri gi d

O R, - 5 j = di wa su it, it Gw a lo se, I + (W B % ✗ - lab ,)

  • 5 5 = l , W A s ui t, i t G W A c os 015 - la ws %

T r a ns fo rm a tio n % ↑

c onve r t to { 1 , 5 }.

OR,

A

^ 02

b- = _s in Oi + c os 0 2 J b) = c ol ori + si no, J O R, - 5 5 - l , Wa s in o, it liw a c o so , ;- l aw, s in g? 12 W B co s 0 2 J OR , TWO^ S^ CAL^ AR^ E^ H

O =L , WA Se rio , + l y WB s i n Or

A nd, ' -^5 =^ G^ WA^ co^ so,^ -^ l^ a^ w,^ c^ o^ s^ on

A

B

Un kn own s - (WA , W B )!! A (^) ✗ lo se , - B^ ✗ s it/

  • 5 si n o, = 1 2 WB ( sin O zl o so , + c os O rs ino ,)

OR, WB^ =^5 Si^ n^01 la si n ( 02 + 9 )

A (^) I cos o ut B^ sin^ Or

  • 5 sin 0 2 = l , w a (i n o, co s out c os o, si n ce a )

O R , WA^ =^ -^55 in^ O^ r

1 , S ui ( 01 1 - 0 2 )

    • W pa

(^4) f in d ✗ p a O R; 41 1.

For thi s p rob lem ; E xpr es s rel a t ive a cc e le rat i on of Q w ot. P

N = "I " + Ba ra^ -^ N^ IB^ ✗^ T^ PQ^ +^ No^ B^ ✗ (N^ iB^ xFP^ Q^ )

  • C^ or^ io^ l^ is^0 OR , (^2 5) = l i r a a i - l, w a i t 12 h , b i - l a W bbl

U se t rans fo r ma ti on s h er e-

or , 25 = e r a (S ino , i t cos o,j ) - l ,W a~ ( c ost , i- si no,; )

  • 1 2 x p C - A r io , i + co so ,; ) -

la WB" (co s O ri + si n o r; )

2 Sc ala r Eq uat io ns -

D = I/✗^ A^ s^ in^ o,^ - e,^ w^ ar^ c^ oso^ ,^ -^ y^ a,^ sin^ g-^ la^ wg^ oso,

2 = (^) I/ ✗ A Co so, + Wa r s i no , + l ax, co s or - law s si n o r

U NK N OWNS - ✗ A /✗ B)

e qu at i ons ar e l i ne ar i n t he un kno w ns.

S OLV E.. _.