Topic X: Dynamics, Kinematcs, and Vibrations
A pjeie i r r a non ,h a iniia eliy I 000 m/s n a au anl o 30° o h horizona Appxmaely what distnce he annn wil h pocl s r he rod i e poin o pin of eleas eleas? ? impa i 500 m blw e pin
A 00 g bl i ued ong a s
h , a surfa:e y an extea 00 N fr L oin of bctwe1 he l and the sfc is 0 appoxmatey wa hoinal aeeraion i xpeiened by h bok de he exea rc?
- - . '
A 3 m/s B) 38 m/s C 3 m/s (D) 50 /s2
x
A) 800 () 67 000
Il
Il
C 78000 D) !H 000 A heman hs a's engine as is enerng a hab Th a ms o a dd so wih i s fon fon nd uhng h ock The heman a i 80 kg. He mvs m om hs sa in h ak o h ron o h boa in , png b able ah ah he he k f h mpy a has mss of 300 kg, ad dgadng al ion appximaly hw wil he sheman hv o jp o rah h duck A m B) 1.3 m (C) 9 m D 0 m 2
The eevaor ry uiding has a ns 000 g. maximm veoiy and mxm alion e 1/ nd m/2 rpecivey A paeng wih a mass o f 7 g sands o a baoom sa i h eevaor h eva aen a is m"im1m rain Wha s mo ny he sae readig jus a he eevar chs s maxmm alaon A 7 N L50 N 3.
A 000 g ar pulls a 500 g il The car and ir i r el o 50 m/h o 75 m/h a a rae of m/ The linea impse ha h ar mpar o e !aie is s neay (A) 3500 ·s () 8700 Ns C 3 000 N· (D 7 000 N 5.
A wh wi dis 08 m os alo a fa sur a m/s I a AB n h wl pimeer meas 0 wh a is mo nay h veliy o po A whn poi nt B n ac te gund 6
3
(C)) (C
810 N
D) 80 N
A 30 /s (B) 3 m/s C 3.8 m/ D m/s
P P I
•
pcom
DE X2
F
M A I A
R E v l E
A u A L
A kg blok is sig a horioLal irclr dik eg ale) a a rad o 0. o he c. he oeiien o iio b h blok and dsk s 2. Th dsk oas ih a niorm aglar vocy 'ha s mo aly he minmum agar vloiy o he di k ha ill a as s h bock bock o slip (A) 1.4 rd/s (B) 20 rad/s (C 39 rd/ (D .l ad/ pere sphr s pojc pojc p a icio icio l l clin c lin 8. A pere by a pring \Vhih propery inas? A) agla velocy (B oa eergy (C poal enegy D lneRr monm 9 A bal bal i doppd o o es a a poin po in 2 m above he gomd ino a mooh, iioless ch h al xis he h 2 m above h god a a an agle ° om he horizona. Air esisa is gigil Approma o il he bal ave i he hor zal diecion bee hiing he gond 7.
-0
SOLUOS - 00 m sie i is belo belo he lah plae
98
- 000
: 90
- 0 + ( 1 10 0 ) ) O
he ime o impa /.
=
-
2
00
4ac
[quarat
frmul]
� s
±
00
2
() .90
(-100 m)
2) 490
+104.8 s, -2!LGG s
:
he horizoa disanc is
vo 00 08 s)co s)co 30 30
90803 ( ! I 000 m)
50
/
2
T w w s (DJ.
'
(A) 2 m B) 20 m (C 22 m (D 24 m sphere movng a 3 m/ olide ih i h a 10 kg 1 O A 6 kg sphere phere aveig a 2 m/s in he same drecio The 6 kg sphr comes comes o a comple op afer he olliso \ha is mos eary he e vlocy o he 0 kg spher immedialy afr h colision? (A) 00 m/ ) 28 m/s (C) 43 m/s (D m/ •
2
P P
v s i { - 2 t2 v v i + 0
w w i p a s s . m
2.
Th vloiy o he sherma sherma elaive o o he boa
m t s 1 m/s
he boa move a a he isherman i sherman ovs h vlocy o h shr shran an rlave o he ock i
+ v" I
Ue he oervaio o momum fshenaVl1r0 +mi t Vb
71lfshCna
vfi.s.shhen:)
+ o v;>
DE X2
F
M A I A
R E v l E
A u A L
A kg blok is sig a horioLal irclr dik eg ale) a a rad o 0. o he c. he oeiien o iio b h blok and dsk s 2. Th dsk oas ih a niorm aglar vocy 'ha s mo aly he minmum agar vloiy o he di k ha ill a as s h bock bock o slip (A) 1.4 rd/s (B) 20 rad/s (C 39 rd/ (D .l ad/ pere sphr s pojc pojc p a icio icio l l clin c lin 8. A pere by a pring \Vhih propery inas? A) agla velocy (B oa eergy (C poal enegy D lneRr monm 9 A bal bal i doppd o o es a a poin po in 2 m above he gomd ino a mooh, iioless ch h al xis he h 2 m above h god a a an agle ° om he horizona. Air esisa is gigil Approma o il he bal ave i he hor zal diecion bee hiing he gond 7.
-0
SOLUOS - 00 m sie i is belo belo he lah plae
98
- 000
: 90
- 0 + ( 1 10 0 ) ) O
he ime o impa /.
=
-
2
00
4ac
[quarat
frmul]
� s
±
00
2
() .90
(-100 m)
2) 490
+104.8 s, -2!LGG s
:
he horizoa disanc is
vo 00 08 s)co s)co 30 30
90803 ( ! I 000 m)
50
/
2
T w w s (DJ.
'
(A) 2 m B) 20 m (C 22 m (D 24 m sphere movng a 3 m/ olide ih i h a 10 kg 1 O A 6 kg sphere phere aveig a 2 m/s in he same drecio The 6 kg sphr comes comes o a comple op afer he olliso \ha is mos eary he e vlocy o he 0 kg spher immedialy afr h colision? (A) 00 m/ ) 28 m/s (C) 43 m/s (D m/ •
2
P P
v s i { - 2 t2 v v i + 0
w w i p a s s . m
2.
Th vloiy o he sherma sherma elaive o o he boa
m t s 1 m/s
he boa move a a he isherman i sherman ovs h vlocy o h shr shran an rlave o he ock i
+ v" I
Ue he oervaio o momum fshenaVl1r0 +mi t Vb
71lfshCna
vfi.s.shhen:)
+ o v;>
I G 0 s c
I c s J
E x :
7 (
vx
=
ilf.henu
\ bo boaa t
(
= ( v v )
11boat
km km 5 h 36 h = 3472 N·s (35 N·) (5 kg) 75
= .2 /s
t /.
(
=
5
2
( m)
k
e answe is (A.
The dt t ihrm wll v o jmp i
n
DE X
Im Im ! ! !v = !v)
8 kg + 30 kg
S v
v I B I 0 s
5.
l -
- l ji ,! 1 e <
A D
Th mpul lvr o a s i ual o h g 1 i o1m (t ipl-om1tu pripl)
Hev, Vf
K I E I c s J
h whl's aiu is 8 Po. bos h is1ou t of rottio w it ont wt h grond 6
(5 )
[backward]
Tlw sw s s A A 3. h a drc rc picatio of Nwo cond law h aceleaon of Le eevao add o .h grava ional acelertio
F = .(9 +
=
(
= 75 kg) 98 8 N
r2 r2 ) 2
s
=
= (8
( 8 N)
T swe s ) 4 he weg i recd te vrtca componet o
h appid r T icioal orc i
F1 = p N = L g Fy =
(
(5) kg) 8
= N
(5 (5 N) 3°
[agor t.r]
= 28 m2 = 2 T vlociy of point
=
) 2
3 m
A i
(
( 3 m 3
vo
(8
8 = 2 m/ (2 m/) T sw s w s (D
Ue Newo' eond law C C = L = F F =
F - Ff
7 o te lock o gin to l the centriugal oc qal t riconal ce
,. =
Fe F
(5 N)co 3 3 N k g
2
323 ms
(32 m/)
2 = 1 N = 2
W=
T sw s (A
(2) =
n.s1
5 1
= r
P P I
•
w w p s s o
DE X-4
F
M
c
A
N I
cA L v
w
M A N u f L
S sst sst i iinlss, iin lss, i n mn ing i ng p rt rr stp stp rtng rt ng r g vl vlit it i nnt nnt n t t t i itin lss tl nrg s cns. linic r} is c vt pnl nrg. As linr vlci dcss, ss <s n n et
lve sig g udri udr i ul ul.. =
9. lng in lvin wn pins A nd prsns dcs n ls ptntil r T kint ki nt n n in i nss rrpningl rrpningl
= 22 s, 02 s l l t dtn v vld ld :net :net f vlci vv cs B 4 (22 s)s 45Q 28 (22 m
8.
e swe is (
B
2a : ± b±
\
:
( 1:) + ( I O g 2.5 1:) (6 kg) 0 vlc kg spr kg)) 3 6 kg
k
3 kg v 0 kg 3 s
=
Tfe swe is (C).
l l llws ws p p p pjci jci wn wn pns pn s B nd
T
y -qt + vsin(B) � v ssnn B)B) + y =
Bs lndng pin s w c xi,
y=2
2 - 1414 : (sisinn 45° ) t. + ( 2 m) 0 4 + (2 ) 0 4 =
w w w a s s
m
s t w ff nsrv nsrvin in n n
v g6h = m.g6h= 2 v = 296h V= (2(2)) 8 811 (lo (lo ) = 40 s
•
Il
Tle Tl e swe swe is (C)
u (2
(2) 4
m
P P I
=
+ (1 kg)v
. troduion o (inematis 2 atils ad Rig Bodis . . . . 3 Distance an Speed . . . . . . . . 4. Retangar Coodnates . . . 5 Rtilar Moio . . . . . . 6 onstant Aelerato1 . . . 7 NonConstat eleration . . . . . . . . . 8 rvilear otio . . . . . . Curvlar \oto: lane Cirua Moto . . . . . . . . . 1. Cuvlnear Moto Transverse an Raial ompoents r Plaar Motio . 1 1 . Cuvlnea Motion: ormal an angetal ompoents . . . . . . . . . . 12 elative �otion 13 Liear a Rotatona Variabes . . . 1 Projectle Moto . . . . . . . . . . . Nomcatue
f
g
v : y
c.ion offin o iion frn gviion cin, 9.81 poiion ri pmn n pi im vci orizonl ditnc i
/ Hz /2
l
m/ m m
37 3-1 37-2 3-2 37-3 373 37 375
. IRODUCIO O EMCS
Dyn is t study of moving obes. e sbjec is ivied into inematis an kints. Knmti s t sty of a body's moto ipeent of te rces te body is a stuy of te geometry of moton witout consiertio of t cases of motio Kinematis deals oy wit elatiosips amog positio, veloity, aeleaton ad time
375 37- 37-7 37-7 378 37-
?
AICl'�1.t..<� -·
A boy in otio a b onsiere a pt if roation of te oy is asent or sigii cant A partcle os ot possess otatioal kiti egy l parts of a partile ave the same istantaeous isplaemet veocity an accelertio iid bdy o not derm wen loaded and an be onsiere a ombination of two or mor partils tat rmai a t a ed nte istane rom ea oter At any give instat te pars partics) of a rgid body can hav diffent isplaements veloities ad aeleatios i te bod as rotatioal as wl as tansla tioa motio
Equato 3 1 ad Eq 3 2 Istataos oty ad ato
v /
mbo
Q B
p
nu clrion ngur poiion rdi of crur ng vcy
rd/ rd
rd/
. 32
Vaao
Subrip
0
y
B
inil onn fn nr rdi l ngnl hr zon l il rvr
Do
Fo te positio vctor o a partcle r, te instantaeous veoity ad aceleratio are give by Eq. 37 ad Eq 372 respetvey
xaml
T positio o a partcle movig aong te axs s given by r t) t2 t + 8 were r is i ts o meters
=
w w w 2 a s s .
372
F
E C A N A
n is n snd L
p l we
\a
?
n E V W
M A / L
s ot nar h o f
ge 37 Rnul ooaes
(A) /s
B 0 (C) 1 1 /s
(D
12 /s
Soltion
loty a on i th is ri i o l i quan w rp
v) d / d 8 = =
l
v5 (2)(5) /s
I
h vt m of h pa ti ' poton s h to h a oth mantd and d on h Ca
dt ym J tgua dt fm) y
e swe s A).
3 DSTNCE ND SPEED s "isp la n t" al isn a dn ang n na Dpacm o r d pm) is h n han n a pa's poson a dnnd th poston to t). Dit
Equao 33 Catesa Ut eto om
:r+ !U
ed h amatd lnth of th path tad dn al dton sas and an b nd b addn h pah nths ov dun pos n wh h lot y gn ds h a Th d i an is lys g an o qa dsacnt
zk
37.3
h ut. r frm a pos in or n q 373 Exame
Sa, "o and "s ha irn nns: ly a to h aig bh a n u and iion; pd s a alar antt, qa o t ad o c y Wn sp1 pd, icn s not onsdd 4
RECTNGULR COORDINTES
poton of a p a t i pci
wth n to a oodna sst. h oodnats a na o dn iy si.in in insion c in w nson s, two oord na ar ay . A rd in at an sn a lna iion, as n th rcglar dna sys o a psnt an an g a poson as n h pola y onsd th pat e shon n F g 371 s poston as wl as ts oct and aton an pd in h y oms cr r, anua i-
n , and n c r I
•
J p p s s .
h poson of a p n Caran cdna t. s 51 n th drton y = 6 h cto n and z 5 n h zirecton. Wa h o pals poso r?
5li 6j 5/k (B) r 5 6j + 6tk 6f + 5j 5 A
(D
+
r=
61 + 5j 6c
Suin Uing Eq 373 iion s
e
t r m o h i's
+ yj zk =5+6j+5 T aswe (A
I N E M T I C S
37-3
5. ECTILINE MOTON
6 CONSTNT CCELEION
Equat 34 ug Eq 39: Pal tla Mt
Equat 3 1 4 Tug Eq 31 lty ad Dsplamt wt Cstat La lat
v s v=1 = -
[l]
37.4
[Pr
37.5
= \ v [] v = v0 + Ocl
s
·' +
7 37.7
+
= no
v) = ao(t lu) +
= lo /2 + I - o) + v2 = v� 2ns - s
79
j jj
v(
A ei ystm i oe in wic paticl ove only g ie Ano ae i i sytem.) T elaonip aog poo, veocy, a accato a liea ye ae gven y Eq. 374 oug Eq. 37.9 Eqao 37.4 oug Eq 37.6 ow raop geneal (inclug vara acceeao o patcl 1 Eqao 37 7 tog Eq 37.9 ow rlatonps gven coant acclrao, 2 We va o e are uu o e equa ton, e oon, vlocy, a acceeao a ow a aa as Equat 3 10 Tug Eq 31 3 Catsa lty ad lat
=
! + Y + / V :' + +
/ =
71 7 17
Vo
Dco
x
715
78
I
v v + 2 s - s11
714
710 7 1 1
= o
Dcto
Accleaon i a coa n ay ca, c a a eefalling y wt coat acceeao I e acclraion coa e accao r can ake out o gral ow n Sec 375 T ial tac o te ogn 0; e ial velocy a coat, 0; a a contan acceleaon not 0 Exm
In aar gavity ock A eer a fre o 0000 an ock B exer a rce o 700 Bot ock ae iialy l aionary Tere i o frcon, an t puley ave o a Pey A a a acceerao of 14 /2 once te ock are reeae.
71 71
Decto
Te veocy an acclraion ae e wo va v o t poso vco a own n q 370 a Eq. 3 7
8quatio 376 c e deid fom = d• and v i s y li1nint ng . One scnai we the cceraion depnd poiin a aice g ccltd (or dcrad b a comprion spr T ng ce dnd n he sprg xtsion s he accleati do al 2Te Rfc ak (NCS Hbk) csit i ht it scipts t dsgae Fr aple, in Dynamc in i us subscrts dinate te lcain of te accratin int (e c r acelato f he cntid) te dition r latd ax ( in a accelration i he dirctin), th ty o acclation (.g., in a r normal accra tn), and th momnt i tim (. 0 in r inial acclatio In q 37. thro q 3.9, th NC Hbook uss subcipt t dgnat th natu of t acratio (i., te ubcrip indica contant celratn Eler in th NC Hbook, he ubsipt c d t dignate centrod an ctr.
x
7500 N
0000 N
Wa i o nay e velociy of lok A 2 after e lock ae rla? (A) 0 / (B 35 / () 4 / D 49 / P P
wwwpp2psom
37-4
FE MCHAICAL REVE MAUA
St
7 NON°CONSTN CCELERTION
Us q 3715 lv f vcy blk
V A = aA ( l lu) + v = m/s
= )
1 i.4 ( 2 s s
A.
+ ms
Eqato 3722 a Eq 3723 eloty a Dsplaemet o NoCostat eleato
v{I he aswr s ()
s(t) Eqato 3 7 1 8 Tog Eq 3721 : eoty a Dspaemet wth Costat ga eleato
o(t)
w t) O()
=
=
a
+
= a0{1 - 2/ + 0( l) 1 w� ( - Bo
2
v() .,
2
(t)dt
eio
19
Th vliy nd dipceme rpcvly cns1 ccri, a(t), lcld si Eq. 722 d Eq 72.
2
Exmle
2
A pc ily vlig m/s xpiencs
8
u(t - lo)
J J
eiio
Equi 18 ug q 72 giv L qn cs nlr cceri
ine ncs i cclei dircn m s shw pric rces an cclrin f 20 m/s n secds.
6
xmle
A l ots 200 rpm whn h pwr s
m/s2
sddy c he ywl dcrs cs re 2 d/s nd cmes res 6 mn er \' s m rly gl dispcm h lywl?
(A 4 () 9
x
3 d
6
"s nry w s dsc rvd by he p i dg hs 6 scnds?
0 rd C 40 x 0 d (D) 270 x 103 d
(A
60
( 0
(C) 2 m D) 18
S
m Eq 2, ngr dispcm i Sutn
O() = · ( - )2 / 2 w( - ) O
)6 ) )( 2
-2 rd s
+ 2 +o
=
� 6 mi - 0 mi 2 2
rev
_
2r
d (6 mn - mi) v
d
x
103 <
•
20 6
w w w .
= / 6 2
t
s3
Frm q 7.22 velcy i
0 x 0 d
aswe is (C)
I
Te expss r cci ncn m
Sic v(O
=
0, C
=
10.
IEMATIS
Fm q :n. 23, e p c s ( =
1ime 32 Plne Cicular Mt
J J
2
36
y
+ O d
(d
375
+ / + 2
a cakula.io1 of ce e he sc (psitin s(O) 0, s C2 T e ce l g h firt 6 s (6 =
(.
2 3 + LO 3G
0 = 180 = 18
G
Equa 326 ugh Eq 331 Cdaes
Te nswe s
J'
3
v, = !
3
3.8
v,
J =
)d + w"
3
39 33
w (d f
35
cn ng ccet ( h g y w g peme 0, n e c m 32 Eq 325
Eq 326 gh 33 i e Lite reltonshp bewee ey n te Ces c nt f pce n pne cc m Equa 332 hug Eq. 3.3 Pla Cdaes
·
8. CULINE MOION
iline moion b he mtn ptc lg p h sgt Spi l xmpl cl mt e pl e cc m pe m. ptls lg lg p the p elcy ccl my e pecf ng hey wee cl mt my e me cnnt exp e nm em f e yem eg p cnts
Pln oion w s roio nl ic moion, ngul oion ciul moion m pe x pth (See Fig 372 )
33 37.33
33
v,
3.35
a,
B 2;(
33
37.37
=
Dscri
9 CULINE MOION PLNE CICUL MOION
33
Dcrit
e l
= z
!
Dscrii
=
Uy =
Equa 3.24 ad Eq. 325 aable gula elea
z
n ol coodine he ps p s c y s ge 0. q 332 thgh q 337 g h lp betwee el y ee pc p c m polar te ytm PPI
•
w w w p p s s . o
37-6
FE EAIA REVIEW NU
Solt on
Equao 3738 Troug Eq 3741 Reiliear Forms o Curviliea Moion
itio
[
1 =
374
( dyd:1/J32 374
�
2
I.
3739
ds
dO = - 4 . - 4 ( ) = d. ) (3(3 - ( 4 ) (3 ) - 4
3738
dv =v=
+
e aula vliy is
swr (.
= 1 1 al/s
10 CURILINER MOTON TRNSERSE ND RDIL COMPONENTS FOR
....F�...9I.1�.... ... ...... ....... ......... ..
Te relaios bewee aelerio veloiy a di a t is iv by poso i a nib q 8 hou E 41
·
37.45
374
+
3747
= r d
i· =
37.42
a = dwc aO = wd
re
i·e, 0 " - O )e ·Oen
Equao 3742 Troug Eq 3744 Parie gular Moio
W = dBd
Equaio 3745 Troug Eq 3749 Polar Coordae Forms o Cuvliear Moio
37.43
Vritio
3748
dt
37.49
3744
Vritio
c = d tio
ton
e
Te beavior o a roa parile is ee by is alar poio B, aa veloy a alar aelero, ese vaiabes are aaoous o v a a vaiabls r iear syses Alar variabls a be subsue oer-oe r lear variabls i os eqaos
he posiio of a paile i a pola cooiae sysem ay also b xpr as a veor o a1ue a reio sii by veor Se he veoiy of a picl s o usaly i_ee raially ou ro e er of e ooriae syse i ca be ive o wo copos cale dial a tve wc are paralle a perpelar especively o he ui ia vecor. Fire llusra e raia a avere ompoe o velocy i a pola cooriae syse a e raial a ui ravese veor a use i he vecor rs o e oio qaios
Examl
T posiio o a car rveli ao a rve is sribe by e llowi ucio o i (i scos .
e0,
re 33 Rd n Tsv n y
B(t.) = - 2t - 4 + 0 Va is os arly e aula veloiy afer 3 s o rvl?
A -16 /s ( 40 a /s (C) r /s (D) 5 ra /s
•
a
x
l· l M S
. CUILINE OTO: OL D GEL COPOES ···· Equt 3.50 d Eq 3.51 Vlty d Rsultt t
v = (le1
37
ot)e + (/en
37
Deiton
Fo plan ccla t, th vc 11 f oiion, vocy, nd cclrai a ivn y q. 3752 Eq 3753 and Eq 37.< Th aL o h ng1a voy lo nd y q. 375 nd E 375G 1 2 RELTIVE MOTION
h t e/e mn d whn n f pl dc wth c o ohng n oto. Th ptc' pton, vty, cc n y pcd wih pt to nt1P vig pt wh t t movig n1 of nc, nwn Ne1J1a a m e rerne
Vrition v ae e
37-7
v2
Derio
A ti vig i a c vna h will hv nan-
a li1 voy la ccla Th li vai wl ic ngnly o th ah a a kw s a19e1al e/iy v" a19al e/e a vly Th c tha otn J p o th vd pt w gnly d ow h n oton, th pat wl xpn nwa cclt ppnd o th tgnl vloy lrton, own a h nrm /a1 a. h ln cclan f, th vco h t nd noml accran N nd gntl cpnn f clti lnt F 374 Th nt vct e nd e ol nd getil th h, cvly i h intanan as c1 te
saaos eter of rtao
377 r = 8 A 3758 V VB w x J = V 8 A8 = fi w x (w x r " 37.59
Dertion
Th liv on, , vloty, nd ln , with c t a anatng R alcltd q 3757 Eq 37.5, and q 3759 pctvy Th ng vocty w aa ai a h agid th iv p v, r 8 S Fg. 375.
Fgue 374 Nrmal anc Tagena/ Co1pts y
Equt 3 5 Tug Eq 3 .59 Rlt Mt wt Tsltg xs
w
Fg 37.5 asati Axs y
x
x
Equt 352 Tug Eq 356 Vt Qutts P Cul Mt
Equt 3.60 Tug Eq 3.62 Rlt Mt wt Rttg xs
3752
+ NO
37.53
f + me
b
37.54 3755
37.5
V = V X
j
+ x ' w x (w x A8 2 x v 8 P P
•
376 3761
376
w w w l
378
es ii
M C H A , C L
V W
M A N A L
Vrtns
v
Eq 70 E 37 E 37 gve h v p vy e wh espe ng s, epvy Se Fg 37G
=
r(2 /)
v
Fiur 376 t A:
\2
Dsc tin
Eu 373 hgh Eq 37.5 le g vey v nge m ee a evly m he spng ng vbes If he ph u s w< e n l m h e n he ar lengh ve, s ue m Eq 37
13 LNER ND ROTIONL RIBLES
A p mvg ve ph w hv
s n vy n ne ee Th n ves w be ee gny h ph n hee wn s angental elty n 1ia acleton espvey (See g. 377.) n ge h ne ves be by mul py h l vbes by he ph
Exae
F he epg pump shw, he he s 03 m he n p s 350 pm Tw en e b pmp ve he g ps pn A s 35 v,
Fiur 377 e es
Eqato 3.63 og Eq 37 66 Rlatoshps Btw La ad gula aabls
=
-rw2
37 3
a =
3
s=
P P
Solution
= rw
[toward the cnt] o el rO
l s s m
Wh m ney he ge vy p A w ses �e he epn pump s ve? (A) 0 m/ (B) m/s C 0 m/ D m/s
3 5
3
U he ehp ewee h ng g vs w = ng vey f h /s ev 2 350 n v 0 mn = 35 /
UsP E 7 v 03 ) 3GG5 as a i a vl i i s h e a h rank = rw =
(
q 1tle Mo
v()
=
The t nge l t
37-9
I N EMA ICS
ath of proj eci le
o o
nsw s
pnoi ias plcl wihoio t.he igh Thehassrce act indg nheips.)ojee ding tdhag,lanhone towhe pr1woajedctils in io oln i s ed o h w h ) acclrton spea cas otin Bshown ihg 3znt8r Thelplcanile seais he poivi 1l whe , v has i s e d gj hfliswipagrls he tavl The a voi i s ea l The ng i ai when B 5° Th L h is ea h o Th sttlignaryht ojct wod is iapex1.alo h h ( K
x
PROJECILE MOION
o
a
ov i k eL
a l a
Negeting ai pon n y aration (ie s own
vi at a e g Proji moton nd onn
of
nto o o 1 l a e o t
C a gnral fom
pjeie t mm o I ac of ra o o a oo l e t boi o i t a velocy, v0 =
h pojee o tae om t aun pont o t o im o tav om apx impact pon
Lim t projile o avel om t pah o impat h a tm e ta o l straght down om tat ig
Equato 37 .67 oug Eq. 37 72 Equatos o Projetle Moton
V= co O)
v si n B ) v cosB) VO
v = -gl y
Vratons
=
-f2 /2 + vo sin
37.67
368 369 30
3.1
{B)t + Yu
3.
y()
y(t Vy,Ot
g
�g£
scii
i d h e Theawsqat i o s d o unir aeran and c e ai n atgsps olvel grglndnitalatiallthat25c ds a h50 awid tave a au iiia ahw1gl oa wi45 l 325 586< di e i neathe dsanendtavled by th go0 t.hhe to 0 v lo ao 0acg E 3772 ltimonown d /2 v si{B) y 0 2 si sin{B) E 377 ando(solB)v The s a .inhe i o 0pac i voO2v sin 0 = 25 s 2 2598m sin5 637 6 o poi moon a v n v n of
Eam
o
A
a
m
tat
pond
a ro th pond, itting
v a
m/ o zal la olf b tv?
trvels
Am )
(8)
m
(D)
m
Solto
o e m al t m of mp m At a tim o f and tm m a t levaon of th . R o solv im tittin g a a e the evaton at a n of impt
= g = -g + v v
g
Y
+0
1
Sbt h e}1Jio t g
m.
+
v
g
o45°
Om
m
m
T answ s (0.
PPI
•
w w w p s s
l
2 3
5 G
nroio 381 m 38I wo ir a Secn< aw o oion . . . . . . . . . 38- igh . . . . . . . . . . . . 383 iio . . 383 iti o Pri . . . . . . . 38
Nomenlat·e
I p
, I' ;1:
eetion e giil celetion, Jss mu1e1 of ii ms OJe noml mnen slnt ime li weigh discmn posiin
/s2 N 9.81
/s� kg m2
kg N- N N-s N s m/ N
Sl
f t
nd/s
l ltin ceie f it0 s of n n ngle
eg
Sbcript ()
k 8
iniil n i o iinl initil dymic no m pin p t it dil tn sic tnget.! nse
1 INRODCON
2. MOMENUM . .
·
h ie mmenum momenm)
i n
b lowig aio. I a h a irio a loi or o wi i i ca a d M m a i o r i g. N· . = mv 1\oret wu o wh o al r a u a pr f o r on a ail h oi a iro o ari ar ag. l f e mem a a h lia o i hagd f o baand r a on arl T do no oib a ad lo o haging owr O h rod o a a oi i oa
x
3 NEWONS FIRS ND SECOND LWS OF MOION
Nen' f la f mn
a a a aril wil or wi oi t mv wih oan loi l a naad a r a o i i aw a ao b a o oaio o mm: I t a xal for aig o a pri i zro, inr o o ai l i e en a n oaio o o ) a ha h alraio o a pari i ry oorioal o r aig o a i inr rpori o h pr m T iron o al rao i h a a lirion o r rmi in a a o r
t
Equato 3 ad Eq. 32 Nwo's Sod Law o a Pa l
LF d(m LF = m d = m [ntn ma]
38. 1 382
Kne i 11e o oio a h r t
oio ini inud a i o h rlaioi bwn r a a r aaa oion a wn oq ad m iia r oaional oio Nwon' law rm h bai o h gog ho in b o ii.
= dp
P P
•
w w w . 2 p s s . c
38-2
E C I C L R E I EW M U L
sci
Newon eond w an b te in term of th r co i o u cl1u i1 motm Tl1 el a orc eal .o the te of cge o li mome11u. Fo a otnt ms, 38 lie. xa
A 3 kg blok moig t ee of 5 m /. e e rqurd o i1 block to o i arly
i mot
O 1 ojcc i 1 uy ctio1 L t (lt f rc o .he celetio of the obecL's cenid in letio. Te eletio "eite by e object i11Ll m ion 34 i o l momem ad eles the e el momen o oue o n objet bot ceoi o te l otol ctio, o Lhc coi dl a"i: T anglar aPraion i id by te objet eidl m 11omut o iueti /,.
M
e ion t obje oe ot etoil
i Th ctoid m iony lm of t ig d b. 385 r i ns o otio bo y rti ular m whee t eenic to om ax to objcts cetoidal i
(A) kt (B 1 3 N (C) 1 kN
(D) 19 kN
Exal
A e bled toue ct o 50 kg yie t i
Sluton
Pom ewo' econ lw, 3, foc u to to o m of kg moig at seed o 5 m s is
IF=
m dvdl
= (3 k)
=
low t o ot ou i loitdi cil l nd has a rd i o 40 m d m moment of iti o 4 kgm . Te
o ctole baring The
cyinder acceeres om a standstil with a anguar celetio of 5 d /
n(t /.l)
(8
m m O
10-� ) 10
= 175 kN
kN)
= 50 k
T answ s D
Wat i mo ly th ubaaced oe on yl i de?
Equao 383 oug Eq. 38.5: Nwtos Sod Law o a Rgd Body
IF = f(,.
M,< > I I1r
33
= l a
=
x
3 35
sii
A rigid body i comex se th caot be descbed
rtcl. Gelly, igid boy i 1 1omogeeo (ie e cee o ms does no coicide wi olu metic ete) o i cotucted o f ub comonen t I thoe cass, lyig blc c wll c otio wel tnlio Newton's eco lw of moio (cotio o momntm be lie to gd oy t the lw m e led twice: oce lin momem once ngl momentm Eio 33 ti to i momnt n t
1 q. 38.3 thrgh Eq. 385, he NCES FE Rferwc Hmt&oo (NCEES Haionk) 11SS ld rmcNs lo i�nt vetor qantiand ). Retilinar np1111ts of v·; 111y he i e, F 1\ a; a, s-pr ne t s s fo 111ultil\ In st u c o h( quatil• i digdd aio howC\', h a only t 1iud th qantt a sd.
PPI
•
. c o m
the
A 2 Nm (B) 0 N1 C 20 N·m (D 200 N·11 Solutin
Uing E. 38, te mgi11e o the momet tg on . cyi i
M., =
= kg·m )
= 20 N11
5 rad
asw s (A
'
2Te NCEE Hulbo is osi in its m an g of I rfrs lo t u!ion f th centi whi h Eq. 383 NCEE H1dJoo ls "11as 1 1l ." do' not ma11 o1Lant aCPti1 s L did rir i h NCE Ha/ Oyn111i o.
Th NEE Hdoo si t t in d<iat ing th 11t idl pa111rs. rs' t s e lo of th ntrd in hrcas Eq 383 nd f p t s h C ! nt id al 111111nt o inrti in h anga aaEq 84 h sbipt h bn 11td 11 tn a th etoda axi q 8
\V
l< E l S
Equato 386 hrough Eq 381 2: Retear Equatos or Rgd Bodes
L F . = 1 L F 1 LJ\, = I LF, 1(a LFv = (i , L Mc = ; =
Destn
'8
388 38 9 381
\ = ,)/
38 1 1 381
h eut iou ar salar fo o eons son
The wqh o a object the objec xs d o is pot gvitio1l ld
386
=
L
Descton
383
P
Exme
A nan weigh hief twie 1 1 aor Whn he evto s wigs 73 N; whn e lvo s.rts 0ving pwad he wi g 8H N Wha is os eay the an' ct ss? ( 7 g 8) 3 g (C) 78 ) 83 g Solo
The mss o he a can dtrm ed o his wigh a r
W=
I
g =
73 N 8
72. 7 g
s2
)
(73 kg
Th answ s
5 RICON 4 WEGH
Fon a e tha aways rsist mon or -
Equato 3813 Weght o a Objet
38 3
The magnid of he ictiona r dpnds on e noal rce, N a ont rn, 1 b tw body ad e conaig su
4 quatn :8< th ou h 388 are preed in the Handbok w h , "ihon los o eerality the ody 1y •1
h
NCEES
in h ; Tli tnt sunds as thuh al n simplifid to l11 111otion, which is no rue h m g11 C l thr dimensionl c;1s is not sifilly preented so here o nlity to lose !11 ct, 388 reprents the s of 11o1ts n any point so his uto is the more enel ce ot th l5 nl se (2) Equato 38! Eq 3810 d Eq 3811 r ntinally th same q 386, 87, ad E 388 nd ae edd Both of eqio r l< o he plane (3 he sbipts oidl o cnt m) d G (enter of avity) ef to th �n hg he n i noion ussay. 4 he subsipt G ad P ot dfd. (5) h ubit not deied, b pobbly epses n no 1 hoice he summaio ial normay Sinc d aX'a he sio sym, th mang of mut b i nfr 6 uation 38. spcifis he pit thugh whh h rtl x p· d not spi n axi, d Eq 388 Snce th uatons ae lmid to the plan, h rtain n on ly be paall o th z-xis s in . 88 he 11g11lar acelatin 7 he sub ha ben omit d about the cn of m, in q 88 and Eq 38 l
g oto I alays a arl t.he co t ai ng ur f t e body ovg, th to w a ynam on. I th body atoay to kow a sa /on
x
n
o
c
�(l) h NCEES Hum/b introduces 813 wih h 'tin head Conp of Wih! Units of eih a pie s nwtons Jn ft th once . of eght is eniey bs in h S system Oy he concepts of 11� and e re . Th SI systm d
"
u
PI
•
w w w p a s s
38-4
F E
M E C 1 A N A
V I E W M A N U I .
n E
sati< oit of fton i lly ent wit e bipt s wile e dynai (i., kiti) oit of ion i oe wi e bip k /lk is oe and o 75% o t v of Tee efii r ple union o r propi. Epientaly eeine v: vai ona ng niio a1 d i andol.
Equai 3.1 4 Tgh Eq. 31 7 aw Fici
F 1_, N F < 1
a boy sg a ozonta s, te norl re N, i t weig W, f te boy I e ody s a ii srfe te oal e i alulate a e pnent of weg oa o tat sfa as ilt i Fg 8. l. A i g 8 ae eied a paalel a1 ppia to te iine plae.
W v
gu 3 1 nl n Nom or
R W
mpedig motio
<
arctaµ5
e iional e as oy i spone o a iting an it nea a te itubing neas e oion of a taioay oy is ipng we e distig e ea e a iona /ts. ige 38 sows t oditio of ipeing oto a bk on a plane. Ju ee otio sas e ant of e tioa oe a oal oe quas te wegt o te l e angle a w oon is jus i1pdig a be alae o e oeie of at iio
[0 l o·ru 1 ring]
[pi 1
F = 1
3815
Pdn �p]
1
sLip oc-m �]
8 7
Vu
estio
JV = mg c < Wo ¢
F ,N
38 14
= atan /
T aw of ton sta e a e 1 i vale o e total itin e F i inepeet of e agite of aa of o a naxi oa iio e i popotiona t e nl e, . Fo lw veoiie o ig ax toal iioal is nary idepenen e veloiy oweve epeien v ta te oe nesay o iiiate lip is gate ta a nsay o anai1 e oio Em
A oy pu a d wi a ass o 35 g ozo1ay ove a srae wi a yai oeie of ton f 0. \Vat i os ay o d o oy o pll se?
(A 4 () 2 N (C) 55
D 8 N Souto
is te noal e, an / i te yai oeint of itio at t oy us p wit, Fb st e age enog to oveoe e iional . Fo Eq. 87
One oio 1s, t oet of itio op lgty, an a owe iioa e opposs Hov. i ilstd i Fig 8.2
Fb = F = = /m
(05)(35 g) 8
igu 38.2 inl r Vrsus Dsurb ore
55 g·/s
(5 )
Th as s
6 KI NETICS OF A PARTICLE -·······- ··
�
µN
distubn foc
PI • pss
-··· -···-··-····- ··--··
ewo s seo aw an e appe sepatey to ay ieion i w es ae eoved into opoens Te aw an e epe n eangula ooi ae (.e n e o ;i� ad pont es) i poa ooia o ( in angenial a oa opo nen) n aal an tanvee oponnt
N E T I S
Dscrti
Eqain 38. 1 8 Nw' Scd Law
= F /
8. 8
Vaa
If F i y h i i e g y E 38H, E 82 821
Eqa 38.22 Tg Eq. 324 Eqai f M wi Cna Ma ad Fc
F 1, scrit
Ei 88 i Nw e w i 1r ri r t i i e �ir Sii i wri r he iri r y crit ieti. g1, F ti ti i r eiy
( '
A 68 N
824
Il
scrii
C 680 N •1200
822 82
+ F 1) F,(. - to)2 :1(t) = :i·u o+
A r 70 h h 17 h c r PPe i t 0 i y B 2 N
F/ l ( - lf/2 v a
Variat is
Examl
D
38-5
t i i1e Li p t i t te qi i r gie E 822, 82 E 82
N
Solton
U Nw' w
Examl
r r/ F ma
(170 g 4
2 I th by A re 5 y iii L ppey w r i i y e re!
A) 1.5 11
680 g (680 N
C 9
D) 2
Th answ is (C.
Sofon
Eqain 3.1 9 g Eq 38.21 Eqain M wi Cnan Ma and Fc a a Fnci f m
at = Fl/
.t(I)
o()dl +
t t +
Vara
v(I)
8 9
t
82
8.2
ri i ig Nwt w, E 822
r/ 1 5 N
09 /
s
2
r y ergig t L wih ii eiy 1 i1i i 0 p i 2 hrz p1 E 8.2
( /2 l) + (4 2 - 0 ( 1
0
2
88 1 ( 1
s
(2
0
+0
T answ is (C). PPI
•
wwwi2assm
38-6
F E
EC A N I C A
EV I E W
M A N U
Eqai 3825 ad Eq 3826 Tagia ad Nmal Cmp
'F1 = 'F =
= vf
rf
38
\·:/p)
386
esrito
ov l ila h, he tangnl al p cl l vly td
P I
w w . p p p s s
Radial ad Ta Cmp
o µtl vi l l h th d a vs omo f a
l
2. 3
Ms Mm er e M Rg By i Ab Fe xs r efg rce Bng rve
39 396 396 39 398
. MASS MOMENT OF INERTIA Equain 39.1 Tg Eq Mmn nia
39.:
Ma
NmenLue
A
d
H 1;
g
I
L
[
R
v
/s2 1
acceleaion area nmbr of itataneo cnter leg c gravitaiona accelrao 981 ig angular mome mas momn of inertia lngh lengh ma ass mome quantiy adi of gyaion an adis im vlociy wigh
p
m/s
m·s kgm
g g N11
m / N ra< /2 d
angular acceration angular osiio cofcin o f ricio deny angular vloiy
kg/ 3 ra</
Subp
0
f
G
0
! / I
m N
Smbs
a·
·
inial nriugal or ceoial icioa cnr of gaviy mss norma origin o eer t aic angn.ial
9 1
+ _)dm
92
(:1? dm
9.3
./)r 1
9.4
+
Dertion
The mass oment of ·inerta mee bj ec' essce chgs r see b sefc xs Eq 39 shws h he ms mme eri i ce s e c men e ms.2 Wen he g cre sysem s e he bj e ceer m, he s, r, he fereni eemen cn be cc frm he cmens s r
+
z2
+
r hmgee by wh eiy Eq. .1 c e wie I=
2 \I
I e h m mmes f er wh espec he - xe, reecivey. Tey re cmes s e.
Th NCEES F Rfe1"Cc H NCEE Hd is inn ssnt in it s nmnclatr a It u h d M t it th mas f a bj. I 'nall int h tl nnari (.. lindr) Cr mu tk hn sin bl invlvi bh as and mmn, a quat fr th ntis us h sm syb
2(1) t w sl dn si in h NCEE ladboJ ld "� �m nt h vn h sm i l hw q 3!. l is imlit tril intl (vum intrl), mr h or
P P
w w w . p p a s s
39-2
C f I C
V I W
NU
Equai 395 ad Eq 396 Paal lel Axi Teem '
" I, + /
Equai 39 7 Ma Ra diu f Gya
39.5
39
Variati
Vaiati
39. 7
T = m rii
rti
The cnoidal as ont o in1ia be h h i f h xs cs h Lh bjs ee f gv. he paall :i horm ls h tanC ais h, i h s mmeL f e b xs E 395, d s he e he e f & he e m bje he le xs heem m b ]l h f h ci bj, h e v eq
Te a rd o gyration, r, f l bje epe s h s m h s L hh h be's ee mss l be le h hg he m e f . Equai 39 ug Eq . 3919: Ppeie f Uifm Slede Rd
y
xamle
he 5 lg sl h h f 0 e g f he s ss h he s ee f vi. he eil m me f e s m
ma a nd enrid
;
' y
£/2
39 9
0
39 1
Y<
2cm 0 ax 1
x
5c
Vh is s Lh s m f f h b he m he lef? A 0 2 ·m (I) 0.33 g · 2 () 0.5 ·m (D 091 g
398
39 1 1
ma m om n of -.ia
0 y
39 1
l
39 13
/3
39 1
m
(radiu o gyri o2
= 0
2 L/12 £/3 2 ,.
Souio
Th is 2 he xs he f vi f e l hl s g. se Eq. 395 1 = le + m d2 =
42 g· m + (2 0 )( 2 . 5 cm + 2 m )
045 g·m 3uao
P P I
39.5
i a c o
39 1 7
0 J ! = 0
39 18 39 1 9
erii
a E. 396 boh aea o he sae ae i h e NCEES Jandbook iffr oao The ioL ubri a G boh rr o h ae oe: roi er o i r of
39 1
pdut of itia
00 g
Th s s (C.
39 15
i 398 hg q 39 9 gve he ee f sldr s The ee f m cee f gvi i lce (xc Yi z), eige p h
I N I C
mas; is the cs-sectioal aa ppedicular t h logiudial axs is h ass dnsity qa to he mass divded by the volume ! s he mass mmet f ierta abut he sbcptd axis ued i calclat ig taial acceleraio and Jous ab ha axs ad i he radius o gyio a dsace m the dsiguatd axs rm he eoid where al l f the mass ca be assumed be cceaed Iv is h poduc o ne'a a mau of symmetr� with spc o a pa cotaiiug h sbscipd axes The dc of era is ze if th bjec is symme rical aot a axis perpedicular t the plae dened by he sbsciped axs
O
O I O N
I
= l = M2 I = M I 3I/2 3
1 R-? r» l',1
=
?
'JR· /2
e aswer s (
Equaon 320 Toug Eq. 334 Po of Slnd Rng
39
39 39 9 39 3 39 .3 1
J,
39.32
=
0
= R I,
l: = ly = 0
39
= 3R'! I,
15 kg)20 3= 3 = 20 kgm
395
oduc o a
A 5 kgm B 2 kgm (C ) 27 kgm (D) 3 gm
A ui form d is 2. 0 m lg ad has a ass f 5 kg
Frm Eq 39 4 the mas mmet of ierta of the d is
394
dus of gylon 2
=
Soltio
9
on f t
Exam
Vhat is m ealy h rd's mas mm f irta?
O I O N
3933 39 34
Euai 392 t lgh Eq. 393 give th pperties o led rgs The cetr of mas ( cr f graiy is ) desigated poit ad mased lad aL i om h mean adius f he ig. M s th ttal mas; A s h oecional area of th rg p is the mass desiy eual o the mass dividd b h vlm; I s mass omt of ieria abo he sscipted axis sd caculaing oationa accltio ad mme abu that axis ad is he radius o gyration a dis ace fom he designaed xis m h utoi< wher all f th mass ca be assmed t e coceaed is the radius f gyri h g abut a ax to he -ax: ad passig thrugh he cerd. f,y s h prduct of ira a masue of symmery wih espec t a plane cnainig the sbspted axes h prdc of itia is r f th obct is ymmetal abut an axis pepedicuar t the pa dfnd y h ssipd axs.
Y
Exal
Th peri of ocllain a clck baac wel is 03 s a slnder rig wth is 3 g he whel is cousrted mass ccetatd at a 06 cm radius \hat is most
ma nd enid M
ealy the wheel moment f itia
2rRpA
39
A 1 .
x
10 kg·m
391
( 16
x
1-G
(C
21
x
10G kg·m
2.6
x
0 kgm
= Z = 0
39 39 3
kgm
P P I
•
s s
39-4
M C H N I C
V I W
N L
roonl crton nd on bou h nd h rdu of rto1 duc om th dntd om h cnrod wh ll of h m n b d to b concnttd th d of ron of h cndr ot n x pll to th x nd pn through th cnrod h pod uct of nrt r of r wh rpc o pln conn h cpd x. h poduc f nrt zro f h ojct tl abou a n x prpdclr to h pl dnd h ubrpd .
r
Souo Fr
Eq
9.25, l whl mont of nr
MR
( : ) 30 g 10 g g
08
is
2
Q. m 00 c
0- gm (1
0 gm)
e swe s (
Exme
A 50 g old clndr h h gh of 3 nd rdu of 05 m Th clndr on h ;a.-s nd ontd wth ongi dnl paralel o h x Wh
Eqai 39.35 Tg Eq. 3946: Ppi Clid
?
ot nrl h · .
mass
mon of nr bout th
J
4. gm ( k· (C) 41 gm (A)
h
D
IG O
k·
x Soo
nd h m on of nr n Eq 394
m nd en.rid \ = rR�ph
95
! Q /2 0
ma moen t of
96 9 98
M(3R 4) / 2
(50 g) 3)0.5 m 2 + ( ) (3 m) 2 153 g·m (50 gm)
he awe is (
M3R + )/12 [ M /2 3R + ) / 2
99
9 4 9 4 1
Eqa 3947 Tg Eq 395 Ppi f Hllw Cid
( rad1 o gyrton)2
z (3R )/12
r;
9 4
R/2 (3R )/12
9 4
I I I
39. 44
h
-1
product o ertia
x 9 45 9 46
Derto
uton 3935 hro gh Eq. 394 h proprt of old (rht) cndr h cnr of m cnr o r lod (xc, Ye z) , dntd point M h ot m th m dnt ql to th m ddd b th ol 1 h m mon of nrt abot h ubcrpd x ud n ccun
P I
w w w p p p a
mas a d entrod M
i R) X 0 Y /2 Zr =
0
9 4 7 9 48 949 95
I N E T I C S
39 1 39 39 3
( rad1 o gylion) 2
? + = 3R = 2 ,.v = ,.2! R 2 r r: = 3 + 3R� 2
39
1
pod o netia 39 7 398
scition
Euin 34 hugh E. 3. gv h prpti f hl righ cylindr u ymry th prpr t r th m r R i h utr diu, d i th ir diu Th ctr f m ctr f gviy i lcd (x Y Z, dignd pin Mi h tl m i th m dy, qu t th m dvdd by th vum i h ms mmt f nri bu th ubciptd i, ud clcutng rt tin cct d mmn but h d i th rdiu gyri, ditc rm th dgted i m th cnrid whr ll f th m cn b umd ccttd th rd f gyrtin h hw cyind but i prll t th i nd psing thugh h ctrid ! i h prduct f urt mur f ymmtry, with rpct t p niing th ubcripd Th prduct f irt i r f th bct i ymmtric but pr pdiculr t th pln dd by th ubcripd .
I
39-5
h ut diu , d i rdiu, R, 1 = 0 m R = T 0 = 0 = U Eq. 33. M3R + 3R� + 12 g 30. m + 30 m + 4 m 12 = 07 g·m
39 6
( I/ = (
O T I O
Solu/ion
39
=
O A I O N A
\h mt nrly th cyindr m nt f irti ut i prpndculr th cylindr' lgitudl d ctd t th cyindr' nl (A) gm (B 07 gm () 7 gm gm
ma ome o nta
2
O
·
h ns is (C).
Eqa 3959 Tg Eq . 3969: Pp S
Exm
A hl yidr h f 2 g high 1 m, n
ur dmr f 1 m nd i dimtr f 0 m.
I m ;: '
8
mas and cenrod 399 396
Y = 0
Z mom en of ner
l:
396 396
=0
= = 2R
2R = = 2MR P P I
•
3963 39 6 396
w w p p a s s o m
39-6
F
N I
R V I W
N U
( ds of 9yralio1)2
fixed
. = 'R2 /5
396
=
h / R15 n
r
w dy d pd) wt h h v f y p h dy Fr r vt; th bdy d t . Th t tr td y d w p r wh h bt vy dr ar kw drw prdr t h w v w rt. at h i11sla1aneous tr Th rp prd hty dff f w v ar pr F 9. hw) r r w r p f wh th ppr ur
967 39
podut of en 3
Deco
u 3959 h 399 v prpr f pr. Th tr tr f vy) d Y, , dtd pt M h tt h dy u h dvdd by h v h f r th rpd ud rt a d t bt d rd f ya d dtd h rd wr f b d b ratd prd f r any p p u h rd zr b h b y bt prpr t h p
e 39 Gph Mhod Fdg sos
2. ANE MOTION OF A RIG ID BODY
a rd dy p such as r wh r t, d k b rpt tw d (ie., h p ) Pane b dd t p d r b d , rd F 39. Fgu 9 ompos o fP Moo
T b vy f y p P a wh r sec 9) wh rt vy v ud by y A w pd poin. C d rt w t u vy w = v / . dr p' vy w ppdur t f h w h r d h p v v ·= w =
r
Fie 393 Ious of Rog Wh
pl motio
Eqain 39. 70 Knndy Ru
tion
- )
otio
39
3 ROTATION ABOUT A FXED AXIS Deo
Inanan Cn Ran
Ay f rt p f a d bdy' p pd f f dy' saaous c is kw h r d y v p p ry Th kw s d IC) i a p a P P I
p p i p . o m
Th f tr d p fo y sch p pd y ! r/ bt eedy 's u w, hr t) an d t hp d h wh hy a no ovious, h w d-rk d br k h. Kdy' r u th y thr k (d, by
I E I C
dsind 2 u< o mnim ( ht y hv mor thn hr k d undon oon tv o on nohr wl tly t o id innous ns, C. C 1 , nd C2 nd ;o hr s n wl l ou stiht n Eu o 397 lu h uumbr o ou nr r y ubr of lin c umb o nnnous r, d 1 th mbr o ln Exme
ow mny ns1o d h l own v?
O
R O A T I O N AL
M O T I O N
397
nd b dnud fo11 vco by of th rnd l oss poduc . S 7 For id body ron bout n p throu is of rvty lod t po 0 h sr v o nlr momntum is vn by 3972 lw f cos·i of gla om1m t t no l o upo n objt nl momum nno hn nl momnnm b and ar n l ou s ppd s sm Euon 39 73 xprsss nl momtum ov w ym ontn o mulp ass.
Eqai 3974 ad Eq. 39.75: Chag i Aga Mmm Hu
( Ho,1
d(ow/ di M
9
l{11; + L M
9.75
Vaat os
A 3 B C) 5
= H dt
= I
(D) 6
d dt
Solution
This is fobr nk Th u br oni o d k btwn pos 0 nd C U E 7 umbr o insnou nt s
la
Deco
9.7
Ahoh wous ws do ot pily d wiL oton t n noou rlonsp bw pplid momn (toru d c i nul momu o otin body momn ( ou Il, ruid to h h nur momnm iv by E 974 Tl1 oto of body wll b bout n o rvty ul body s onsrd owi Th l m o E. 39 7 for ont momn o ini s hown t ond vion or oltio of prs, E 39.74 my b xpndd own n E 3975 Euto 375 drms th ur momum tim om the nu momtum t tm d t nur mpu of th 2Mo;. momn btwn t1 d
Th angulr mment.um u o pot 0 t mom o lnr momtum vor nur momum hs un of ditn x o m , ·s m do s o o vor
�The NCEES H is in on it nt i is rprnation of h r bth d in h ntroidaJ mss mmen of inti Oyn11i cion fr h m c 5ln Eq 3973 th nonsanrd ntain "yt hould inter retd as h limis o summion i.e., u11at er ll mses in h c11). h wld norma!· wrien omhing ilr.
-
c -
n(n
1)
2
-
44 -
-
6
he anse Is (
Eqati 39.7 Tg Eq 3973: Aga Mmm
Ho
=
H0
yl. H
=
r x
mv
9.7 1
= f1v
92
syst H)2
ecto
I
1
)[f
P P I
w w w s s o m
39-8
E
M E A I A
EV I EW MA UA
Equ 39. 76 Trg E q 396: Ra Abu a Abiary Fxd Ax s
O
LM' dw
3
[enm1IJ
38
dO [gtucrl]
'
c/
w=clI
d
W1 +
3.8
Vos
w=
a d
3.85
\J"d
wo
+
3.8
q
The otaon aut a arbitay axs is n m Eq 3976. Eqain 977 thh Eq 979 ay when he auar aeeaLio1 the oat boy is aiab qati 980 hrh Eq 982 ay wh the auar acceeratio o th rati by i c saut6 Eqa 98 huh Eq 985 ay whn the mm ae o he e a s csan The chae in kinec nry e he wk e accerae f w) i caat Eq 98 Eme A 5 whee ha
a ma men o ina o km The whe bee to a coan m qe ha s mos eary th aar iy f the wh 5 afte th torue i a A) 5 ra B a C 5 ras (D ras
of sbcipt n C Hdb Dynami ctio to dign agl ai om d tri Si sbsipt outl na to igt troil ( m ct) ipt l mit
P P
a
05
Fr E.
984,
=
,;0
= 2.5
the anar oiy a 5 i +
=
a
a
+
a
O.fJ
a s
5
s
e answe s ()
. p p a .
4 CENTRIPETA AND CENTRIFUGA FORCES
ewLo s end aw ats tha the s a rce r ey acerin ha a boy exeri. For a by vin aun a rve at, he ota accat an be seaae n tannia a noma m 11 By wtn' ecn aw here ae cnin cs in he tia an noma eis h c s cat wih he ra aeraL!on nwn a h epel oe The cetieta ce s a e ce h by twar the cete atin The soca efugal foe an aare ce o he by rc away from the ene ain The cnipetal a cenria rc ae qua in ante b oie in in he centrua re on a by mass wh sance fm the cene f a th ee f ma
Decrion
&The
N·m / I q <·m
384
O 12 qw1/2•
a=
he
383
w Lo +
38
( J ]
I
Ue Eq 98 n he auu acceatu f w h whe sbjc a 1 n
38
f
W C w w •2or(O - O) 0
St
3
(genrlJ
di
L
Fc = m
v� = r
5 BANKIN G OF CURES
a ehice e n a rcar ath n a a an with intaanou ai an taena oty i w xeiene an aar cetua ce h cnri a ce se by a ombnaio o oaway bank eean an way fiction. The vehicle weh W, creos he nrma re Fo sma bak an the maximum ictina rc s If
Fr are baki an, the eta fre onib o he nma ce I th away is bak that ti i uie to rit the enua ce, h suereeaio aue, , a be cacua m ta
=
\
gr
oton 2. <n EnKY . . . . . . . . . . . . . Pot1 ngy . . ne Coevi1 ncple . . 5 Lnar Impls . . . 6 mct . . . l.
4- 0 - 405 40-G 407
E
F
"
k
p
p
T
fint of rs.itulo enrg e graiationa aclao, 98 hig ping on a momn linar o1nel owr dc
oiion k i n oil ry vo or limn hizonal <ilamn
J N
I
N/ k
1-2
f
F
g
Eqai 401 Tug Eq. 406 Wk
{/ (/ =
N·
F=
I
U
OJt
nomal pring wig
1 INTRODUCTION The enrgy of m epeen e pcty he m wok Sh en n be oe n elae Thee ae y m ha e ene n tk
[varable fce
J
mig om a t at 2 tr r o lsi al o iional c intntanos nr
} -;u e
F o
/S
d </s
4 4
= F co O)8 '6 'igh) Uw
J
1
F-dr
·
Subps
1/ 2
Syls
gu loi
Nonue
1lng mhanal thmal lra and 1t: . ny ov aa nty, Hlhngh h ha in r an b th poitve o ga.v Wok ', th a o hnging th nry o ma Wrk i a gd scl quani. \ok i poi whn a c c n t h rto1 moton an mov a m om one loao1 o noh \ok is ngve whn a ac o oppo oio1 (ton f xampe lwy oppo th dc o moon and nly do gat wk} he nt wok on on n by n1or hn n f b fo y poto
=
s2
-
[ol
M(
43 4 4
ing]
45 4
l
rtin
e wok pmd by s clud a dot prot o h e or ting hrgh a plemnt vcor a hown in Eq ne h o pdct o two vo is s work i a s qntty Th inegr in q 0 i sntilly a mmato ovr ll s atg at in J th omponent o e in th etn of moo o wok In E 4 an Eq. 10.3, omnnt o c in the cn of moin FoO, wh repn h ce ang bewen th c nd the irton o o a snge
Eq 0 o Eq 06, ia r ok is gie U cotny wih h NCEES FB Rfw Hk ( Hk) Nll i is v a 2() CEE lk atemps to listinuis bt wk and stor Fo exmpe, k-nerg pinp (ll prinl f wrk and energ in th CEE Hk nt iy preseted U U 1� Ho h is o ener�· sto e 2 ciad il1 frc, sy mog a nrro fiiol< surfce, wi i on h s.�ibl appcti f Eq 43. () W noin o siated with a tin y, t BE mbk bt nd U ) T S llardbok is icsn it o t b U ch hs g: wo stod ney a n i sed n (4) NCES Hrk s incoin in used vinble ud o ndia n or disnc Bt 1nl
H
in
i.
P P I
p i m
40-2
E
E C A N I C A
omL o,
R V I E \
AN UA
rl itl rpp h fnia ds p wh / i q 4 Equon 44 pn h wok o i movig wih I' vri i: y g i t vitial Eqtio 05 ps L wrk oiL wi a la11g sio or copsio i prg wih prg o k." Eq.in 406 is h work pm y a op i.P a 1 n ) M, otig hg h g
earth's
INEIC ENERGY K
u
i of ul y ;std wth movig or oi o Equa 40 7 iea Kec Eegy 7 T=
v-
7
in kinetc ngy f a od mvig ih i1s an tou liur vloy v i 11l fm E 0
"h NG H/ook s inconsl1· i use o l h• b pt c. I q 403 F a s a o ta n o F i nu a rce rd hog :troil �(!) Th u is not a�I it wok u h nodat!'l wi obj (i.e ight) is 110\el 2) Th 1Cnng of th nga· go A s ;u1d to 1u !- he ngae s ul spo1 t a hoa fir w itai e , wk s ngate wh h o11d1gs d wr 11 th st). lowc\•r, q ·, Eq 4 q. 03 a Eq 0 d o a ngatve sg the qutio1L� seem o wle to th thdynac nventon Ay ul ma and t gtive p> m nl g c cov1. &l Eqn s i ico r prd in NG J1wk The 1/2 1ultiplier ino rtly show i l the pn N' Th n mathm as wh th p ing n k ao he gt outi f t pareths� (J) Whreas d Al i E •10J, 4< a q •06 e pp(!< ipt F, te subct $ 5 s ce. () h> te sbcis F, I' a M n 403 Eq .4 an 10.G dvd om h soce of t rg (i. fom the C ht movs), h• sbsript i q ·05 ivd from the lnd vaiabl hat aes. If a �iila ovn in ha bee ollw with Eq :, q a q 40.6 h i those qnons ol v' U an F a mpron g al upo y easng so th ative sn i nort f thi ppaton 1 a thermodyn sem stanpont (6) This equ io s hn in a sbeq l of hs in f the NEE Hanook as U2 k ? which t onl a dint rmat l us iead f hae manig of chag in ( to stord C'n ; N Hdoo aOt'S . JG wi t h uple omt" commo1 tm A prop1 pl h nt t imp o it prpa to spa f h momt o n ple Simila propery of hrrcae s it wid sped, b r 1 ng h h cn itel hu sd wl be impopr I q 40G mean scribe a ur momnt ang oato t n traio11, tr pure mo torque wonll nil ppopra. 71 T NGRBS do diffn vmiab' pent k er. 1 ts stio o U,
h
,
w
•05
\•
d
I i
I
ir an h o is 0GO, 1< th pP o h w i. !s .h wall 2 m/s \a is mos Pl Lh nr h mpr mt s i1 order pv l o Lh st f t ar?
(A
70 .]
( 40 J C 22 .] ( D) :{6 J
Ung Eq •17, t kii g of t i
llr
35 kg) 2 = / T v 2 2
w . p p 2 p m
7 J
Eqa 40: Raina Kiec Eegy 4.8 Dc Th in kc nrgy o boy movig wh ano u velocy w e y E 408
1 0 hog ik of 5 ri ra a f ngh 5 an t bo poi A ik on i o ol a orzol oor
cm
0.5 m
=
T/e swe is (
V 5 <
35 kg vlug 65 m/h is Th h a wl 3 t T oicint o fricio bt w
Slon
Dsci
l
Cvn n gu voiy o a/ ot h i a1< - r/ ao the ax t.hc kn rg o th i i mot ly
(A 0.62 J B) 1 7 C 8 J (D) 34
N R G
Solto
2
Amg th s p h k h k hs S n -\s n x rt h k, h k gy m y 2/
f,
= lc
1
y(v 1
l tv
wJ
w·z + ni} + lm = �m1: 2
=
! ( 0 kg)
(
+ +
=
4
D
(
{A 0 .J B) 00 () <00 {D ) !GOO J
+o
5 cmm (30 r 0 s
Soo
)
( 1 kg)(0. 5 m)
(
+ 0 kg) 5 cmm 0 U
he r vy s
_
2)
(17 J)
)
(
9
/
l s
ng vy s 8 S W V = 05 r
I
= 55.5 /s
v w2/ ,2 )/ + I / ( 2" + , w
409
g Eq 09 h k rgy =
1vi/ + T w/2
A nm k wth ms 0 kg mr f . m s wth spng n t z , s shwn
�m R )
n•6
2
40 10
E 09 gv h k ene g by Fr g p mn n whh the r s n rtn mpnnt, e kn rg he sm n rtn m Eq ' v t.hc k nrgy r mn �IT! ne xape
)
T = 1 1 2 / +
escpt
)
( � ( 00 k mn (GO mm (
= =
Equai 40.9 and Eq. 40. 0 Kic Engy Rigid Bdi
-Il\
40-3
h hz vy s 50 km/, te kn ngy h k m ry
e answe is ()
T
fN 0l
(m l o f (s5a
( kg) ( 89
kg)
447
(00 J)
h answ s ()
Eqain 401 Chang in Kiic Engy 40 t 1
esio
The hg kn eer fm he f n f s f vy, fro h square f he vy ne (, m (v - vT}/ (v v ) /).
P P I
s s o
404
C H A N I C A
EV I E W
A N U
3. POTENTIAL ENERGY
Eqain 4. 4 Elaic Pnal Engy
Equan 4 2 Pnial Engy in Gray Fild U
myh
4
escrti
a as _mvtat.oal pteal eh y Potnial mss CY) ee fm ea aan kwn
U, s
kJ'2 / escit
n a p wh cs1. (es k h p's elastc potal eergy i m •1
pd
a
e Pel eey lost wh he e f a mss ee h s pe ny y s ee e eney ha Equain 4 3 F in a Sing Hk aw
xml
•1 mss, hw e y a p a y a nss hen h mpee spn ees h ns ey ehs p. he spn osat, s 000 N/ a p ms 0 .
4
1 m
A s n ne se e be mpe s ha he y pem wk I a ee pn
=6m
F,
=
4
esci
he m of ee s s qa he w qe mp he spig y he s y he mss he n 013 Lh e n n whh he p f h pring contan (sffne), k n he pmen he n m Exame A sp
h s 0 N/ h s h ey a m s h en h sp e sae 30 cm fm em pn he me ms s m m h ahe he en s ffeen sn h s peme s 2 m Wh s m ey he n nsn he s p (A) 46 N/m B) 6 N/ C 0 N m D) 3 N/
Th
e h m he h s Fm Hook' w Fs k kzx2 0 m k2 � 2 m X2 6 / 1
X
Th swe s (C
P P I
•
w w p i p s s m
m e he eey s he n (A) 380 J
W
(8) 750
J
100 J D 200 J
C
Souto
s 0.4 h Ha eney s
u /2 = 000 0 m) 375 1 (38 J)
=
Souton
4 3 N
cmss ps
sw s )
Eqan 4 5 Cang in Pnia Engy 4 15
N G Y
A N
40-5
.
..
won
sco
4 ENERGY CONSERATION PRINCIP . .. .. .. E
T ang poia ngy so< i L sping w daio i spg chags m poi on o posiio s d om Eq 158
Arding o 1ry csa 7'1/e r nno b rad or dd Howvr ngy a b nd o dirt rs hrfor m of l rs of a sym s on
:
Eqae Sp Csa
Th ir ppld lod is l by ah spring in a ri o sprng linkd nd-od o) i prng ri is 1 1 l J rie
k,I
LE= ontn
v
[
=- -+ ·
k1
k2 3
pri1gs
Sprng n parll g concnri prg sh th ppid oad qvaln spring osan for spings in pal i
[
pnr� lc sprmgs
Equa 4 6 Th Eq 41 Cmbed Peial Eey I'
+ \,
0 6
± \, +/2
0 1
=
0 8
us rgy an ihr b rd or dsd xal wrk promd o a onsrvt ysm m go ino hnging systm tal nry. Tis is nown as [h work-energy pri1cip/.
aly, th pip of consvion of n s app to manc n pobm i onvsio of wor no in o pia ng) Coso of o rm of rg' io ano do no voat osaio of ngy aw � probms nolving convrson o ng are aly spcal cs Po ampl osdr a faling body a i td po b gtaional for T ovrso o poni nrgy no t ng' ca inrprtl a qat ig t wo lo t ot gavitiol rc o hag in kti nrgy
scrti
In mhia sstms h ar wo oo opo nns o wht i omally rd o s potnta rgy: gvational pon ngy ad rai nrg For a sysm oniing a lr ast spg a s lotl om vaon a gviaoa fld, Eq gvs th oa of wo omponns Eqao 07 giv h ponial ngy of wg i n a gravaioal l Eqao 40 givs an rg in a in ar, at prg
c a o HS a �
i
he NCEES 1tlbook n h ntation to de psiti , Eq 44, n lo hang ength (i. a h g in tin in Eq 5 n EJ •05 and q 18, th NCES /buk ins f . ns in q •l0.15, but h meaning i th � 1 n Equn 0 6 is n liit to mechanica m Pnil rngy re xiss 'l agnei d numt n t 1 11 l ytm s 1 1 Athu P8 u in th Unit Sn f h NS anfnerg in th bk ntify pt ntial nry n U s in'd I Dyw111irs t n th N!'S HfJok intu 1 w Oud th onrvtn f ·rg quti fo tnil h nw via d em to ud lsw i th NCES I h< prvisy ( th NCS rdk mk. (:) pent' hy 1 og hr 3 Th suhpl nd c 1d1, h gvtoa clato inlil f Hin i ca ut ot ta n elat ta y 1 uto I01 whl h uatio in t Dynami ion o th N b :d h. i impliiy th itance om om ri val r which = 0 i gd (2) o± u t1l wi h a thnlnamic i terpettion gy, w apnt usd Eq •t. The in he ener lly ri r it whih an e psit or C� i\C Bv n ± th imlitin i tht s lwy a pii is <\C w e rfeqa1l ty, rc·ges f wheth th cn tun
)
(
n0
g b \9
!
y
Eqai .9 aw Csera f Eey Cseaie Sysem 0 9
sit
)
or ner1v yem whr r o ds sipaion o gan t toa ng of is ql to h m of t kinti d potna grvaiol lati) rgs xam
A projtl wi a as of L O kg d dirty pwrd
om gund lvl wt iiial oi of 0 /s. Ngling h s of ir sistnc wa wil b h pd of th pojtl whn i ipas h grod? A) 7 m/s B 980 m/s C) 1 m/s D) 00 m/s
L •
Equ1 t 08 ha prsly >' p ba t i lwy tive. - ·l) utin 4018 hu not be trprt \ w t k
.
� PPI
• s
406
C
A N I C / L
E V I E W
A N U
ltn
Fgwe mpue
Ue h lw 1sv er
F
I
v mgh- v + 1.'1
mp
=
Equai 4021 ad Eq 40.22 Impul· Mmtm Picip f a Paic
/ = F
dv Fdl
00 / I r retce s lctd, th m velcy wll th m h t vlc
Va a o
answ s .
Dscrio
=
(2
40 Dscro
e fc1) ccte y the wr y th ervv rc mv w t d s , W 2 I he c ervive rcs cr he energy th sys I1_2 µve I th csvv r deree h eey h sys1 W ev Nonconsrvative forces
5. INEAR IMUSE
s vt ty e th he vr mmetm s f mpe e he sm the r l mmem: N· Fu h r ht mpuls s rprd y th d h rcm rv m1
!
h ppld oce s cst mpl ely
ca1le.
he ch 11 mmtm s q h me Th kw s th momntum ·e Fr < em wh s e ms I= • www.p
4
= L(m
xam
60 00 mv <t m/h s l .ry l h vly he w s r cp 0 m/ (i .he il d. f mt) he c s mpld 05 wh s mst y h vr imsve rc th ? A) N (B 990 (C D 93 N A
l
Slt
he l velty f he GO rlc
� h v= 60 1111 h = 077 m/ 1
1 00
Go
lllll
Use
he
impsemt prl
F = m
P I
4.
Th mlstm ie r 11st e 1d m mts ht h memmem pr lw rcly m ew's s1l w
Equa 4020 aw Ca f Egy Ncati Sym
mp
1,
v v = 1qh mh) + v - v) = o + \22 - 2I
1 - 02 77 (6000 kg)0.27 ) 11(VJ V2 F0 s - 0 s - 2 -93 N 930 ) [oppo g1 dieo]
answ is (D
G Y
F, d
40
(A 010 /
8 O.GO / (C 0. /
L m;v; ad
(D 1 k/
th liar 11on1 a i 1 and te rspecively r a syste (ie olletio of arti The le o rcs o ti . 1 o is
6
40-7
1 /h iantaoly oled to a atio11ay 0 0 kg railar Wat is os nary the sed of e opl ar?
esctio
f
W O K
xa A GO g raiar ovig at
Eqai 423 ImleMmem Pici f a Syem f Paicle
) L
A N D
F
lutin
Us .l onsrvaton of oentu prciple
F; d f
1 + m v2 = (GO 00 kg
IMPACTS
ccordg o Nwo's sond law oe is oseved les a oy s aed uon y an xa r h as 1viy or ito 1 I a1 i11a o collision coac is very ief, ad he effe of eal r g1ia1t rf, ot s cosvd even .ogh nerg• ay e los tog eat geeration and deorg the odie
1
m )
4 g ( O g+ = OGO /h
4 g
e ase s ()
Equai 4025: Ceficie f Riin
Coidr two patics, i11tally ovig wi. vloi.is v 1 ad v2 o a ollision path a ow i Fg 4 coervaio of ot qato an e used o nd .e veloits afer a v� and v
405 Vaes
ure 4 c Ca mac
nlasc pcly ilastc pati pefely elasti e ia is sad to e a at mpa f etc ee : lo e iact is sai d o a or j a i te wo atices stik togethr ad ove o w ith th a ial vloity Th ipac s sad Lo e a i ma. oy f ietc eer s oneved
<
e=O
= 1
sritio
Th fi sil, s the tio o eaiv veoty diferee alog a t al staigt l Wn ot iac vlociis are not directe along e a sraght lne, e coefcint of iio sold e alca< sartly r ac velocit cooen
Eq 5, t sscrp ndias a the veloty Eqain 4024 Cna f Mmem 404 escitio
to e sed i aculaig t cofcie o stttio ol e vlociy copot oal o la e of ac
W an objc rods o a satonary oect i iely ive lane te taioary ojcts a ad al vlocitis a zro ha as h bu y a e allad o oly the oject' veloits
e a mm m eqaio i usd o nd vlociy of wo ari cls aft colliso 1 ad v are iiial vlo.s o Ie ariles ad ad v are t vloii alr ac
'
P= P P I
1:1
s s . o
408
E
M E C A I C A E V I E W A U A
l L ii1t f ri1n an se grz t lsn s esti i;ti Fr efey 1i n (ie as.ic csn) wn Lw is stk tgtr t Piin f rtn i Fr a rcy sic l1 nt rtitin i F m ls1s fit estn w bwn n ii1g (prtily) ielastic clisin Eml
y ll oing a ae f m/s lis w 5 kg ll y 1ig i1 me dr.in a f / t m nery te inl ciy f o lls i y st gr tr liing? (A) m/ 8) /s C m/ D 9 /s 1
Soutio
Fn t fien f retittin enitin, E 5
= Frm t nsrn f mnm E 1 2 2 ( + 2) 2 = 11 ++ 2 2 kg)4 � (5 kg) � kg + kg 8 m/s (9 m/s) =
he answer is
P P I
w . s s
Eqai 4026 ad Eq 4027 ey A Impa
' _ 2(2) + ·+(
( )
V ( I
)( 1 )
406 407
esrit io
fiint f reitin is nwn Eq 406 y b ul t ls er it
1 3 7 8 9. 0
Ts Viats 1- 4 de Cmes 4 Stc Dclc 4 "ree Vrt Amlde Os . . . . -4 Vetic verss Hritl Osclti - Nr Feqec Trsl ree Vit 1- 41- Udmed Fred Vrts Vti si d Ctl . 1- slt m Ave Be 41 Vits Shs -
u
ms
rau r ener li iplacemnt.
8
Symos tin angular pn pr naural frqu
Subscrip 0 nial mna ng naural palar p a rsal
E
I
D
1
•
Nomenur aleain a mp A fin u lamng spamn ul a.y rqun F re graianal alra.in 9.8 g har lu G plar mas mn na plar ara moment f neta spring sa nh
J k �;
m /
N· m Pa
1 YPES OF IBRAIONS
\/ s sciltr i lJut n qlr it he is he reslt distur rce ht is ld cc nd hen emved the mtn is w s fr ba rc ims s lid d t sstem the mt s w s fd \h th he teries ta d rcd ris re he scteres de d dmed vrts ther s n dam7g n rcti) stem wil exeree r vras de iel This is w s fr d s m See F 411 Fir 41.1
/
aturl nam r)
Pa
kg·
lkgN/m 1
J /
Vb vbrati
Hz N m/s
T
ap
nap
ap
The ermce evir s sim ssts c e dened sle vrile Sh sstems erd t s sg dg f frdm (SDOF) ss Fr exme he st mss i rm sri s dend the e vle Sst req w r mre ves dee the sts l rs rc w s dgr f frdm (MDO sm See ) 2 DEA COMPONENTS
a rad/
\h sd t dcrie cmes v ss em he dectves c d ·da ell im ar nd he sece icti d dm he ehvr ar cml c dscd iner eq Fr exmle he er equt F kx dscris lr sr hwever the qdric q t F v descris nnli dsh. Siil F a d F v re er el vscs rcs sctvl =
=
1
=
=
Althgh he vent s y n ns nversl the vrle ny sd the stn vale n sllatr syst evn whn he n s n te ert y drtn
P P I
s s
412
FE
412
ECAIAL V I A U A
Sil d Mlpl D Fdm Sym
414 Smple Spg-Ma Stem
t+
x
poiio o ai ilib
dd qibi posio
-
( S D O sym
(b
MDOF
sy
415 e bt
. . ., . . . .. . .
x
3. STATIC DELECTION
A imr ce e clcltig h bv vibtg sysem is e dee·o, 6• Ti is e eeci f mechic systm u t vt ti re le Te sbig ce i ci eel clcltig t stic lcti it is xteml imprtt itigi bewee ms wigh g ue 4 3 ilustrts tw css sttic dflect.ion
(a)
Sr
siio
Exam
_ wghL3
_
st
A mp wi m f 30 kg i re y sg with sprg cst 120 Nm mtr is sre lw y veric mveme W i ms rly t tc ti te srig
48 £/ (b)
=
{A 01 m (B) 09 { C) 02 m {D 031
4. REE IBRATION
Te simple mss ie srig ilse i ig 4 is xpl systm t exeiece ee vbrt he sysem i iiilly rs he mss is gg sig t eqlbrium psii is the sic eflei Afer te ms isl els it wl ilte w Sice ee is frci e he virti is undampd), scillts wil ctie revr (See Fg 45)
Soluton
Il
clt th stc elt
k gk
-
mg=
Equ 4 1 Sym R 41
2The em c/eformati1
synyLlY w dje01.
P P I ass
w
Te sti eeti 6,1i s e eect te grvitl rce e s the mss systm is te rvitl ccelet (98 rn/s2), k i th ems sprg ct
gu 413 Emple Stt Deet
o
=
The answer is ().
Il
(30 kg s 250 023
Il
(02 m
V I
;- (k/ Vrn
m
0
4.3
44
m �: =
k8s1 :c Ot)
W a sime sras s i sur a wwar i. e ma i ue wwar fm i ai eei a reae) a L iia i i i rm mas w e a u rri re a ieria r (mg). Ea 12 u E 44 ar iar iffria uai f m. Equai 4 1 5 Tug Eq. 4 .8 Ga Si Siml Sg-Ma Sym
s) + si) 2 2 2g C
f
\
I 0 NS
41�3
A a f 25 k is ai m a ri wi a i a f 44 /m. f e ma i u w a re, wa is ms ar eri f iai? (A) 5 ( 1.2 ) 5 D 21 Solo
si n
x(t)
A I
m
Equai 4 2 Tug Eq 4 4 F iba Eqai f Mi f Sim Sg-Ma Sym
+ k.r
BA
rm E. . e ei is 2 = 1. = 2 4 .25 k
ns s
Equai 4 9 Scific Sl Sml Sig-Ma Sym 4 46 47 18
\ 8
Vai ati os
419 esritn
ia ii iiia i a i) a e u ermi e sas f a a 2 i E 415 Euai 4.9 he e ia ae re xml
mas i u frm a r wi aus e sri isa 2 . T mass s w G m a a Wa i ms ar i f e mass afr 12 s (A) G m ( 2 ) 4 D 8 A
2 = f1 = w sritin
a C2 a as f iai a e ia iam a f mass is w as e nau ral freu ncy of vbtin a1gula fruncy I a uis f aias er i e am a e lnar frquncy wi as ui f e e pid f oscillaio, i ia f e ia eu Te e aura ru f rai a aua i f ia ar i E. 1. a E 18 s ua 17 a e u wi a are f m iui e i ams safs a aes
Souto
Il
Frm E. 17 aa fru f ssem 98 m m m tvn = 2 m = 22 /
I
• w i a s s .
414
r:
C A N IC A
V I W
M A U /
Thposio veh1iayohmmaFrsoimJqad/ !and hhpi iiianl h ma c(i ) v/)( 2 s ) c 22 ( 0 + 22 ° � ; 22 m 0. 6 m) Thoppegat iive fgnhdiucarsath(aqhiblriaumpoin stmh whr h sys wa ra
7 ···· NATURAL ···· ··· FREQUENCY
T' a th 8a quipriuumcpp rqun tqual h kiuhc plarem d ast icFonrh aspthgpuyomaxm hown mm di Fg 6 h nry vat quaion V 2 2 hef hvl picityifnmucioi div y g h divaiv w/ v) d:v cos hutpgvtohus quat h hw thnerva a v" quaSubo i o nrgy r i v s h WLiosh aiuiraaudampd vi b a i o t h e i n a l n a f h l p an vl o i y ) 1 o ampTides ma: oh scmaaularliohwpiowidlobIhociafspflainogd.iasThishown i i a l y i pl a ed b o tha,amheHwv eqy h xcosilaiou ado ahposi wilal i Thyz in a man hwgi Fig4h spcagmaaass wslaedpd sdspyseme om, hheiinnmaahaldiwioscpl arcmnt elatuio oha pprof co1iatin G
:() =
oso f 9
1
U=k
k:c�unx
005!!
=
( l)
l asw s ()
=
=
:wx
)
(/
=
= i:;1ax·
5 AMPLITUDE OF OSCILATION
l clr u o ib o. 2
(<
A
k
= 1
TORSIONAL FREE BRATION
7
ooal pdum
iu 41 .7
Trsional Peulum
L
6 ERTCAL ERSUS HORIZONTA OSCILATION
Ahwginaigic tin6iaa h e w o as o os c l a quiit)vaAhoghi., wiil mayhavemh sam qucy a ampl eheiweanghxofra hvibytaioalis omprcewyhca vriceal moythaoh,hetpp qieqlibbiimm ph esparieqig vvalccalwhoci hril ahioousysaabl oislaai ab h uss p u 41 6
Vertcal nd Hozon l Osclations
Equain 4 0 and Eq. 4. 1 Difn ia Equan f Min f Simpe Tina Spin
. p 2 p a s s
41.1
Oc) + {/n() D Thgardsihmasa adquammn o m iin,Eqa ofh0, dihafs Thi q.su i hly anadf gosnloequath lioutishow h spigmas ym Ot)
P I
" + k,/ 0
=
4111
V 0N S
Equain 41 2 Trnal Sprng Cnan
= CJ/L
Usig
Eq 4
qun
h dmd aul i
W1
Vitin
I Dsii
41-5
=
0 /s
06 N· d 0 k·
e swe s (D)
T n n n r si n
um is und fr he a mu si G, mm of ina J, nd he h a f ngh L
Equain 4 1 1 5 Undamped Naual rid
T
w, =
3
L
Dsitn
Ea. n 3 vs
fqn
t h ndamd t i d sung d usd
l sing sing he i m E 4 h undm iu n qun n b rwn s Eq 44 h mmn nrt f he verical suppor, wih u i 4 I i h o ms et of n e t th il ig dik wh uis of gm Th n h sm
Examl
A n
5 05 a l
kg ur dk pdulu ih adus m tchd t s e d i nt T si n spng ott i N / rd igdg h ms h od wh mot nry h und med nu qun t.hc rsi n nlum?
5 0
a
/
2r .
Equan 4 1 3 and Eq 41 1 4 Undamped Ccua Naural quency
{ T = C:.J
2n
5
k,
Dsiti
Simil o h undm ped f brn r sm s Eq 8 h udmd na d inl sm b m
Eq 5
9 UNDAMPED ORCED BRATONS
Whn dtrbg rce, F t, n he s:m m is sid b A I ugh fin fnctio ly osr o be e iod c need n be s h cs imul s nd andom ntions 3 Hweve , nii dsu (i wh mas dsld nd lsd to il l s n n m ng S Fig 48) Cd p wi h cng qn nd mimum lu o F0
Th d ifr quin min i
J\ 0 ad/ /s (C) 4 ad/s (B)
Fig 1.8 Fod Vbos
D) 2 d/s Sout
Th r
()
h dsk yide)
(5 k)(05 m 063 gm
I = MR /
h nsdl ce is impo n, n o ir rsrms an b ued md n�' ng fnn ion i 1 o
P I
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