some studentserroneouslyincludeforcesthat arenot actingon the object.For eachforce
you includeyou shouldbe ableto point to somethingin the environmentthat is exertilg the
force. This simpleprocedureshouldpreventyou from erroneouslyincluding a normaliorce,
for example,whenthe objectyou areconsideringis not in contactwith a surface.
4. Dmw a coordinatesystemon the diagram.In principle,the placementandorientationof the
coordinatesystemdo not matterasfar asobtainingthe cofiect answeris concemedbut
somechoicesreducethe work involved.Ifyou canguessthe direction ofthe acceleratign,
placeoneofthe axesalongthat direction.The accelerationofaa object sliding on a surface
at rest,suchasa tabletop or inclined plane,for example,is parallelto the surface.once the
coordinatesystemis drawn,label the angleeachforcemakeswith a coordinateaxis. This
will be helpful in writing down the componentsof the forcesIater.
The diagram,with all forcesshownbut without the coordinatesystem,is caileda free-body
diagram.we addthe coordinatesystemto help us carryout the next stepin the solutionof
the problem.
5. WriteNeuton'ssecondlaw in component
form: Fnet,=mar, F'6y: rflayand,if necessary,
Fngt2:ma2,The ieft sidesof theseequationsshouldcontainthe appropriatecomponentsof
the forcesyou drew on your diagram.You shouldbe ableto write the equationsby
inspectionof yow diagram.Usealgebraicslmbols to wdte them,not numbers;most
problemsgive or ask for force magnitudessoyou shouldusuallywrite eachforce
componentastheproductofa magnitudeandthe sineor cosineofan appropriateangie
6. Ifmore than oneobjectis importrant,aswhentwo objectsareconnectedby a string,you can
sometimestreatthemasa singleobject.To do this you mustknow that their accelerations
arethe same.on the otherhand,ifyou areaskedfor the forceof oneobject on another,you
must carryout the stepsgiven aboveseparateiyfor eachobject.Thereis then an additional
conditionyou must consider.Usually the conditionis that the magnitudesoftheir
accelerationsarethe same.You mustthen invokeNewton'sthird iaw: the force of the two
objectson eachotherareequalin magnitudeandoppositein direction.Usethe same
algebraicsymbolto representttre magnitudesofthese forcesanddraw their arrowsin
oppositedirectionson the free-bodydiagrams.
7. Identi$ the known quantitiesand solve for the unknowns.
Questionand ExampleProblemsfrom Chapter5
Question1
The figwe below showsoverheadviews of four situationsin
which forcesact on a block that lies on a frictioniessfloor. If
the force magnitudesme chosenproperly,in which situationsis
it possiblethat the block is (a) stationaryand(b) movirg with a
constantvelocity?
(a)
( u) zu-tu--b.-v
I aq
ir
Question2
The body that is suspended
by a rope in the figure below hasa weight of75 N. Is T equalto,
greaterthan,or lessthan 75 N whenthe body is moving downwardat decreasingspeed?
-s)
a {)+4,"b
)^+r
""A
Lo v-v</Yv;,ad,/'aAla
,l tI
iI
la
J/
t
?5N
Question3
In the figure below,two forces 4 ^a F, areappliedto a lunchboxasit slidesat constant
velocityovera frictionlessfloor.We areto decrease
theangle0 of { withoutchangingthe
magnitudeof .{ . 1a;To keepthe lunch box sliding at constantvelocity, shouldwe increase,
decrease,or maintainthe magnitudeof { f it; Wfrat happensto the normal force aswe
decreasethe angle0 of { ?
rE1
(^) llf^*l
(r)
Question4
The figure showsfour choicesfor the direction ofa force of magnitudeF to be appliedto a block
on an inclined plaae. The directionsareeitherhorizonlal or vertical. (For choicesa and b,
directionsareeitherhorizontalor vertical. (For choicesa andb, the force is not enoughto lift
the block offt}e plane.)Ralk the choicesaccordingto the magrritudeof the normalforce on the
block from the plane,greatestfirst.
d, c, a, t
ProblemI
The helicopterin tle drawingis moving horizontallyto the right at a constantvelocity. The
weight of the helicopteris 53,800N. The lift forceL generatedby the rotatingblademakesan
angleof 21.0owith respectto the vertical. (a) What is the magnitudeof the lift force?
(b) Determinethe magnitudeof the air resistanceR that opposesthe motion.
21.0"
/'-z)l'o"
L
Lax'Il,o"
tF-=f)c\*=o
l=r-mor-o
L /t.ut
t1.6o- (,
L = /^at"
=
o
4 = lr';n, al"o
=O
-- (5.?t"lDtN)
a.r' 3 l,o"
53.,8ooP
+..--_?
ooc ).1.o'
A"oJ^161
Problem2
Accelerometer.A small massm hangsfrom a thin string andcan swinglike a pendulum.you
attachit abovethe window of your car asshownin the figure below. when theiar is at rest,the
shing hangsvertically. what angledoesthe stringmake(a) whenthe car accelerates
at a
constantrateof 1.20rnls', and (b) whenthe car movesat a constantvelocity. v : 90 km&?
t_
I L6.O
r\
'kT
\
\l
6\
Ts'.o I
<---1
\
II
tI
)'
L
l-J I y
> t.r'\ ^
r \-'
r ' | r { - A Y - ^t
Trotg-Sg
=o
T= *u/*,
a) e =h^-'(Yi =
s-?
r-
=
A l-* fo4x
-]-,ti\O
ing
L
mg
Y
i
xJ
,o$/?,01x
k#'o'!j+t
= M4.^
(nTr*o) o," a = rl ex ---i bn O =
7*
y'r \
(5'
, )
.
\_-,,
q= \) nl5 Ab
|--._--_-_r
lr'.
I
IV=U T
Problem3
In the figuretelow, two forces, 4 ^a F,, act a 50.0 kg cratethat sits on a frictionlessfloor.
The magritudeof Fr is 255N andit is appliedat a 40oangle.The magnitudeof F2is 55N.
(a) What is the normalforce exertedon the crate?(b) What is the crate'sacceieration?
3 Ea-^tto'
F
rl
f ,c-ac9o"
(q)
<]
)
F
Fv = Foay :o
/.
\
s-l -
) l-y = o4v
(b,
lTt + F, A.rmqo" _ co at = O
e^. [ "aato'- f=
f,cott1o"- Yr. n^,, ?
'l\ = m? l - , A t r n4 o "
/l\^
m
C\x -( )tsN) oo-tt1o'_-.'519
=.,( Eo,"o
Fs)19.to./s.) , (Esrt) ax".,
q o-
o.oKq
Problem4
frictionlesspulley (alsoof negligiblemass).The arrangementis know asAtwood's machine.
Oneblock hasa massmr = 1.30kg; the otherhasmassm2= 2.80 kg. What are(a) the magnitude
ofthe block's accelerationand (b) the tensionin the cord?
fn a Tfqr , 9!'o.h J )e'i4 aq.-ct!-uaXQ
"+u"A Xr!i.9\ ), );-!!
azxp.pu',ah ,Jsur^ura^"AA
-> r,wrr"
* @.
ffr.y^An^"^Ed,!o$at"^a i,u
4"
D, /Ja\'<
4or e^'"9Lhen<^
,1
^x\aA4 1"""
I
i
J
4
tq
/drryw'r9[t "a + ,"
t
F.,' t t = lYtdv= o
--t'q,%'fqrq
- - - - ) T-
tfl, ?
T
/t,
m,X+$,4
(')
.'._::
I
/T./\4,ar'I:
I
?
lr
L
tq
A+N,"g d,ourr)
IFy=..)ay=o
oo+:
m^q
(r) ,v.i"1a; --+ ma3-(n,Xtrr,o) = rnla ----)
_-p:+
(a) a = ( A"Iorr - 1,1oyn)(t.to
"f s,)
( l.:ors*]"volg)
3.51-fs\
q = (rq+-rrr,)g
( n, r+ t^tr*;
( b ) ptr^WY o =3,51^/s',rnb(r)
or
(t),.,*X,*F=n;il
Z= tr.qN
Problem5
In the figure below, let the massofthe block be 8,5kg andthe angle0 be 30o.Find (a) the
tensionin th6 cord and (b) the norrnalforce actingon the block. (c) Ifthe cord is cut, find the
resultingaccelerationof the block.
;$x*
an+
&w^:
\-2
*
m34rto0
(o)
X F* - N4,r --o
7- m3ai",g= P
f - nqg^/^.re'= (SJKgx9"to*/5)On,
to-
f= 4aN
(b)
!F
t
t
(")
=frrar;o
- tngco.o--o
------1
= -Z caO = (Z"SYg)(e.so,/sr)rc^a-tJa"
T\ =?aru
m.-".t ,t5, tF-o r^t f = b
XF., = N)ay ---->-ro3n.l^"D = roq/
g= -cI3,,/r) srn3o.
QY= -a ,r".r
Problem 6
In the fgure below, a block of weight w1 = 100.0N on a frictionlessinclined planeof angle l5o is
connectedby a cord over a massless,frictionlesspulley to a secondblock of weight wz : 30.0N.
(a) What arethe magnitudeanddirectionof the aecelerationof eachblock? (b) What is the
tensionin tle cord?
ur t = loo.o N -----rfD, = 10.? l(e
Jq
ajArvyn4 w^
o-r*-oh"ofua
"la -u,""^) ^-Lr,
-r-
1d\T
*a"rhX".
v
/t
\/z
I
mi,f
/ \ $$QcU
Y/(
(l)
a*,c*"rha
f - *,,ga,vn0 = ilrg
= ,n,q
- ,^.,,t,0.,r,rg
- fcr,
q
f) = tfl-,q
'
oo
I c ).}/rrO
( r'r, +vn'')
I -lq
'-' ( b)
( to'et<g+ 3.oaKe)
=w
= o"3lr/r.r.t ]Jo^.W
(t)
-ff.,2r.r
{.-11r;
f=
7-= l\
(:)
--.----
il, jr,,rO
(.rg-m,q)
u+ pv
XF* = roa-
\.r'
\)*
Pd(')-)
u)x= 3o.og ----) Da= ?.o6l(g
m'9 - l\'9
= (:".oN)-(2."u$)(o.zrr/s,)
=EU.
rQ + rr't,?-a-fnO = (ro.rfj)(o.3 | */s)
.
15 = W
f (roon),o.i-,
Problen 7
The figure below showsa box of massm2= 1.0kg on a frictionlessplaneinclined at angle0 = 30'.
It is connectddby a cord ofnegligible massto abox m1: 3.0 kg on a horizontalfrictionlesssurface,
The pulley is frictionlessandmassless.(a) If the magnitudeof the horizontalforce F is 2.3 N, what
is the tensionin the connectingcord?(b) What is the largestvaluethe magnitudeof F may have
without the cord becomingslack?
0.f\
I
-.?:4a<91q, ".
l=l
,dab;r,rd rv$,l .a+
,
19,g
XF*=mq*
nlea"\
m^.
\
\ n-
o\t'vr\i a4t2 a2 wwa:
rf\
v
4
\ ery','er"
u'*A q
X n.,.=rr)e*
(_
(T+ a)
f o a g r C , w r O - T =M a q (a)
mxga.[n$
N\sgcddo
r^aA
Q=
f 0 r $ , A , t v r r $ *F = f q , Q + m s q
l+3.,:
rn.ga,l"nO + F = ( r . o l < g ) C 1 . 5 4 " / s l ) i / ^3oo " * a . : N
l"rl,+fq.
A = l"trn'/.*
-----J
t r
(b)
$"
@4,J,-bac"o'r! )p"^rl\
Ed dh C')
-a.3N
T=D,q-F = (r.otfu)(t.sq/s,)
f = 3"tN
,a"e*l l-= oN
-$a,o'n(a) 1
l.olrg+ l.aK9
m*3,a,u0-o= sa4
--+
q = gz n g, = (t.8n/s,) axn?o"
A
= At e 9' vn' U^>1 . \
v\
-----
b^rtt) ,
-- I\rQ
| + l-
->
F.
*lq = (:.ole)(4.qn/sJ)