Download College Physics: A Strategic Approach

158
C HAPTER
5
App ly ing Newton 's Laws
SUMMARY
The goal of Cha pter 5 has been to learn how to solve problems about motion in a straight line.
GENERAL STRATEGY
All examples in this chapter follow a three· pillt strategy. You ' ll become a beller problem so lver if you adhere to it as you do the
homework problems. The Dynamics Worksh eets in the Student Workbook wiLl help you structure your work in thi s way.
Equilibrium Problems
Object at rest or moving at constant
velocity.
PREPARE
Make simplifying assumptions.
Dynamics Problems
Objects in Contact
Object accelerating.
Two or more objects interacting.
Make simplifying assumptions.
Make a visual overview:
PREPARE
PREPARE
(0 ~
0).
Use Newton 's second law in
component form:
Iden tify aU forces acting on each
object.
.Identify known quantities and what
the problem is uy ing to find.
Identify forces and show lhem on a
free-body diagram.
SOLVE
Sketch a pictorial representation.
Sketch a pictorial representation.
Check lhat the object is either at rest
or moving with constant velocity
Identify action/reacLion pairs of forces
acting on objects in the system.
Identify all forces and show them on
a fre e· body diagram.
Draw a separate free·body diagram
for each object.
SOLVE Use Newton's seco nd law in
component form:
SOLVE Write Newton 's second law
for each object. Use Newton's third
law to equate th e mag nitudes of
actionlreaction pairs. Determine how
the acceleration s of the objects are
related to each other.
L,Fx= l1Iax = 0
L,F}. = ilia), = 0
"Read" the components from the freebody diagram.
ASSESS
"Read" the components of the vectors
from the free· body diagram. If needed,
use kinematics to find posit ions and
velocities.
Is your result reasonable?
ASSESS
Make a visual overview:
ASSESS
Is your result reasonable?
Is your result reasonable?
IMPORTANT CONCEPTS
Newton's laws are vector ex pressio ns. You
must write them out by components:
Specific information about three important forces:
Weight iii = (mg, downwilld)
Friction
Drag
ls = (0 to J-LslI , direction as necessary to prevent motion)
Jk = (J-L kll , direction opposite the motion)
Jr = (P,r"' direction opposite the motion)
(Fnd)" =
L,F.. = max
(Fnd ) ), =
L,Fy = may
For eq uilibrium problems, a.( = 0 and
jj ~ dpA v2 , direction opposite the motion) for motion in air
lIy
= 0.
APPLICATIONS
Apparent weight is the
magn itude of the contact force
supporting an object. It is what
a scale would read, and it is
your sensation of weight:
Wapp
= m(g
+ ay )
Appare nt weight equals your
true weight \II = mg only when
ay = O.
A falling object reaches
terminal speed
v
1erm
~)4mg
pA
Terminal speed is reached
when the drag force
exactly balances the
weight force: a = O.
jj
Strings and pulleys
A string or rope pulJ s what it 's
connected to with a force eq ual
to its ten sion.
F"'f'< "" ...11 =
The tension in a rope is eq ual to
the force pullin g on the rope.
F h• noI.: .. " ,!", =
!.en;,ion
:.
ten~ion
..,,.:...
. _ _ __
The ten sion in a massless rope is
the same at all points in the rope.
Tension does not change when a rope passes over
a mass less, frictionless pulley.
Que stions
tMP)TM
!:!!/
For homework assigned on MasteringPhysics, go to
www.masteringphysics.com
Problem difficulty is labeled as I (straightforward) t o 11111 (challenging),
Prob lems labeled
Worksheet;
INT
til
159
can be done on a Workbook Dynamics
integra te significant material from earlier chapters;
BIO are of biological or medical interest.
QUESTIONS
Conceptual Questions
I. An objcGt is subject lO two fo rces that do not po int in oppos ite
direct ions. Is it possible to choose the ir magni tudes so that the
object is in eq uil.i brium ? Explain.
2. Arc the o bj ects described here in stat ic equili brium, dynami c
eq uilibrium , or not in equili brium at all ?
a. A g irder is lifted at constant speed by a crane.
b. A g irder is lowered by a crane. It is slowing down.
c. You' re strai ning to ho ld a 200 Ib barbeH over your head.
d . Ajet pl ane has reached its crui s ing speed and altitude.
e. A roc k is fall ing into the Grand Canyon.
f. A box in the back of a truck doesn't sli de as the truck stops.
3. W hat forces are acting on you ri ght now? What net fo rce is acting o n you right now?
4. Dec ide whether each of the fo llo wing is true or fa lse. G ive a
reason!
a. The mass of an object depe nds on its locatio n.
b. The we ight o f an object depends on its locatio n.
c. Mass and weight desc ri be the same thin g in di fferen t units.
5. An astronaut takes hi s bathroo m scale to the moon an d then
stands on it. Is the reading of the scale hi s true weight ? Explain.
6. A light bloc k of mass m and a heavy bloc k o f
mass M are attached to the ends of a rope. A student holds the heav ier bloc k and lets the li ghter M
bloc k hang below it, as shown in Figure QS.6.
Then she lets go. Air resistance can be neg lected .
a. W hat is th e tension in the rope whil e the
bl oc ks are fallin g, before e ither hi ts the
grOllnd?
b. Wou ld yo ur answe r be d iffe rent if she had
bee n ho lding the lighter bloc k initially?
FIGURE 05 .6
7. Four ba Us are thrown straight up . Fi gure QS.7 is
a "snapshot" show ing their ve loci ties . They have the same size
but d ifferent mass. Air res istance is negligible. Ran k in order, from
largest to smallest, the magni tudes of the net forces, F nc11 ' F nc12 '
F nel3 , F nel4 , act ing on the ba ll s. So me may be equa l. G ive yo ur
answer in the form A > B = C> D, and state your reason ing.
5 m/s
200 g
~
3oo g '2
9. a. Can the normal force o n an o bj ect be directed horizon tally?
If not, why not? If so, prov ide an example.
b. Can the normal force on an objec t be d irected downward? If
not, why not? If so, prov ide an example.
10. A ball is thrown stra ig ht up . Takin g th e drag fo rce of a ir into
account, does it take longe r for the ball to travel to the top of its
mot ion or for it to fall bac k dow n again?
II . Th ree objects move through the air as shown in Figu re Q S. II .
Rank in order, fro m largest to smallest, the three drag forces 0 1,
O2 , and 0 3 , Some may be equal. G ive your answer in the form
A> B = C and state your reason ing .
~~~/'
20cm x 30cm
20cm x 20cm
30cm x 30cm
FIGURE 05 .11
12. A skyd ive r is fallin g at her termi nal speed. Ri ght after she opens
he r parachute, whi ch has a very large area, what is the direct ion
of the net force on her?
13. Raind rops can fa ll at d ifferent speeds; so me fall quite qui ckl y,
others quite slow ly. Why mi ght thi s be true?
14. A n a irp lane moves thro ugh the a ir at a co nstant speed. The
j et engine's thrust app lies a force in the di rect ion o f moti o n.
Reducing thrust w ill cause the plane to fl y at a slower-but st ill
constan t-speed . Ex pl ain why thi s is so .
IS. Is it poss ible for an obj ect to trave l in air faster than its termin al
speed? If not, why not? If so, ex pl ain how thi s mi ght happen .
For Questions 16 through 19, determine the tension in the rope at the
poin t in dicated w ith a dot.
16.
A U objec ts are at rest.
Th e strin gs an d pulleys are massless, an d the pull eys are
Frict io nless.
18. _ _~-.I
17.
m/' ~ ~
3 m/s
3oo g
3 m/s
4oo g
3
4
FIGURE 05 .7
8. S uppose you attempl to pour oul 100 g of salt , us ing a pan balance for measureme nts, whi le in an elevator th at is acce leratin g
upward. W ill the quanti ty of salt be too muc h, too li tt le, o r th e
correct amoun t? Ex pl ain .
FIGURE 05 .16
FIGURE 05 .17
FIGURE 05 .18
160
CHAPTER 5
Appl y ing New ton 's Laws
19.
25. I A 3.0 kg puck sli des due east on a hori zon tal fr ictionless surface at a constant speed of 4.5 m/s. Th en a force of magn it ude
6.0 N, directed due north , is applied for 1.5 s. Afterward,
a. What is the northward componen t of the puck's veloc ity?
A. 0.50 m/s
B. 2.0 m/s
C. 3.0 mls
D. 4.0 mls
E. 4.5 mls
b. What is the speed of the puck?
A . 4.9 m/s
B. 5.4 m/s
C. 6.2 mls
D. 7.5 mls
E. II mls
26. A roc ket in space, initi all y at res t, fi res its ma in eng ines at
a constant th rust. As it burn s fue l, the mass of the rocket
dec reases. Whi ch of the graphs in Figure Q5.26 best represents
the veloc ity of the rocket as a fu nct ion of time?
FIGURE 05 .19
20. The noor is fr ic ti onless.
In whi ch directi on is the
kineti c fri ct ion force on
block I in Figure Q5.20?
On block 2? Expla in.
FIG URE 05 .20
Multiple-Choice Questions
2 1. II The wood block in Fi gure Q5.2 1
is at rest on a wood ramp. In which
di rect ion is the stati c friction force
on bloc k I?
A. Up the slope.
B. Down the slope.
FIGURE 05 .21
C. The frict ion force is zero.
D. There's not e nough informati on to tel l.
22. II A 2.0 kg ball is suspen ded by two light strings as shown in
Figure Q5.22. What is the tension T in the angled string?
A.9 .5N
B. 15N
C. 20N
D.26N
E. 30N
FIGURE 05.22
JY
23. I Whi le stand ing in a low tunnel , you rai se your arm s and push
aga inst the ce iling with a force of 100 N. Your mass is 70 kg.
a. What force does the ceiling exert on you?
A. ION
B. lOON
C. 690N
D. 790N
E.980N
b. What force does the noor exert on you?
A. ION
B. lOON
C. 690N
D. 790N
E. 980 N
24.
A 5.0 kg dog sits on the noor of an e levator that is accelerating downward at 1.20 m/s 2•
a. What is the magni tude of th e norm al force of th e e levator
noor on the dog?
A. 34 N
B. 43 N
C. 49 N
D. 55 N
E. 74 N
b. Wh at is the magn itude of the force of the dog on the elevator
noor?
A. 4.2N
B. 49 N
C. 55 N
D. 43 N
E. 74N
A.
B.
c.
D.
FIGURE 05 .26
27. I Eri c has a mass of 60 kg. He is standing on a scale in an elevator that is accelerating dow nward at 1.7 m/s 2 . Wh at is the
approx imate read ing on the scale?
A. 0 N
B. 400 N
C. 500 N
D. 600 N
28. I The two bJocks in Fi gure Q5.28 are at rest on frictionless surfaces. What must be the mass of the ri ght bloc k in order that the
two blocks remain stati onary?
A. 4.9 kg
B. 6. 1 kg
C. 7.9 kg
E. 12 kg
D. 9.8 kg
-----~~~---------~Q
FIGURE 05 .28
29. I A footba ll player at pract ice pushes a 60 kg block ing sled
across the fi eld at a co nstant speed. The coeffic ient of ki net ic
fri cti on betwee n the grass and the sled is 0.30. How much fo rce
mu st he appl y to the sled?
A. 18N
B.60N
C. 180N
D.600N
30. I Two football players are pushi ng a 60 kg blocki ng sled across
the fiel d at a constant speed of 2.0 m/s. The coeffic ie nt of ki netic
fri cti on between the grass and the sled is 0.30. Once they stop
pushi ng, how far will the sled sli de before com in g to rest?
A. 0.20 III
B. 0.68 m
C. 1.0 m
D. 6.6 m
3 1. II Land Rover ads lIsed to claim that their vehi cles could cl imb
a slope of 45 °. For thi s to be poss ible, what l1lu st be the mi nimum coe ffi c ien t of stali c fr ict ion between the ve hi cle's tires
and the road?
B. 0.7
C. 0.9
D. 1.0
A. 0.5
32. II A truck is trave li ng at 30 m/s on a slippery road. The dr iver
slam s on the brakes and the truck starts to skid. If the coeffi cient
of ki net ic fr ict ion between the ti res and the road is 0.20, how fa r
will the tfllck skid be fore stopping?
A. 230 In
B. 300 m
C. 450 m
D. 680 m
Problem s
161
PROBLEMS
Section 5. t Equilibrium
Section 5.2 Dynamics and Newton 's Second Law
I. I The three ropes in F igure PS. l a re tied to a sma U, very li ght
II ri ng. Two of the ropes are anchored to wall s at right angles, and
the thi rd rope puUs as s hown. What are
71 an d 1;, the m agni -
8. II A force with x-componen t Fx acts on a 500 g object as it
moves along the x-ax is. The objecl' s accelerati on graph (ax ve rsus t) is shown in Figure P5.8. Draw a graph of F, versus t.
tudes of the te nsion forces in the first two ropes?
1.0
2
O.GOm
Rope 2
0.5
O.80m
Rope I
FIGURE P5 .1
FIGURE PS .2
2. III T he three ropes in Figure PS.2 are tied to a small , very li ght
rI rin g. Two of th ese ropes arc anchored to walls at right angles
w ith the tension s show n i n the fi gure. W hat are the mag nill1de
and d irect io n of the tens ion in the thi rd rope?
3. 1111 A 20 kg louds peaker is suspe nded 2 .0 III below the ce ili ng by
two cablcs th at are eac h 30° from vert ical. Wh at is the te nsion
f3
&
4.
PI
5.
g
6.
BID
7.
rJ
in the cables?
II A 1000 kg steel beam is supported by the two ropes show n
in Figure P5.4. Eac h rope can Rope I
support a max imum sustained
tens ion of 5600 N. Do the ropes
break?
J A cabl e is used to raise a
25 kg urn from a n underwater FIGURE PS .4
archeolog ical site. T he re is a 25 N drag force fro m th e water
as the urn is ra ised at a constan t speed . What is the tension in
the cab le?
JIll When you bend yo ur knee,
the quadriceps muscle is
stretc hed. Thi s increases the
tens ion in th e quadri ceps te ndon att ac hed to your kneecap
(pate ll a), whi ch, in turn ,
increases the tens ion in the
patella tendon that attac hes
yo ur kneecap to your lower leg
bone (ti bia). Sim ult aneously,
the end of your uppe r leg bone
(fe mu r) pu shes out ward on the
FIGURE PS .6
pate Ua. Figure P5.6 shows how
these parts of a knee joint are arranged. What size force does the
fem ur exert on the kneecap if the tendons are ori ented as in the
fi gure and the tension in eac h tendon is 60 N?
II T he two angled ropes used to
support the crate in Fi gure P5.7
can with stand a max imum ten sion of 1500 N before they
break. W hat is the largest mass
the ropes can support?
FIGURE PS .7
o.0 t--~~-~-r- I (s)
4
0 +---I--~-~-'l- I (s)
234
- 0.5
- I
FIGURE PS .8
FIGURE PS .9
9. II A force with x-componen t F, acts on a 2.0 kg object as it
moves along the x-ax is. A graph of Fx versus t is shown in Figure
P5.9. Draw an acceleration graph Ca.\, versus t) for thi s object.
10. I A force with x-component Fx acts on a 500 g object as it
moves along the x-ax is. A graph of F x versus I is shown in Figure
P5. 1O. Draw an acce lerati on graph (at vers us t) for thi s object.
f~ (N)
1.5
)
1.0
0.5
- --4__~~~~L- x
4.0N
2.0N
3.0 N
0.0 t--+~-~---1cr- 1 (s)
- 0.5
FIGURE PS . l0
FIGURE PS .11
II . II The forces in Fi gure P5. 11 are act ing on a 2.0 kg object. Fi nd
the values of ax and a". the x- and y-compone nt s of the obj ect 's
acce leration.
y
12. I The forces in Figure P5.1 2 are
acti ng on a 2.0 kg object. Find
3.0 N
the values of (/, and {/y . the x- and
2.0N
x
y-components of the object's accel4.0N
1.0N
erati on.
2.0 N
13. I A horizon tal rope is tied to a
50 kg box on frict ionless ice. Wh at
is the tension in the rope if
FIGURE P5.12
a. The box is at rest?
b. The box moves at a steady 5.0 m/s?
c. The box has Vx = 5.0 mls and a., = 5.0 m/s 2?
14 . 1111 A crate pu shed along the fl oor with ve loci ty Vi slides a distance d afte r the pushi ng force is removed.
a. If the mass of the crate is doubled but the ini tial veloc ity is
not changed, what distance does the crate slide before stoppi ng? Ex plain.
b. If the initi al ve loc ity of the crate is doubled to 2Vi but the
mass is not changed, what distance does the crate slide
before stoppin g? Ex plain.
15. II In a head-on co lli sion, a car stops in 0. 10 s from a speed of
m/s . The dri ver has a mass of 70 kg, an d is. fortunately.
tightl y strapped into hi s seal. What force is appli ed to the driver
by hi s seat be lt duri ng that fraction of a second?
11 14
162
CHAPTER 5
Appl y ing New ton 's Laws
Section 5.3 Mass and Wei ght
16. I A n astronaut's we ig ht on eart h is 800 N. What is hi s weight
on Mars, where g = 3.76 m/s 2 ?
l 7. I A woman has a mass o f 55.0 kg.
a. W hat is her we ight o n earth ?
b. Wh at are her mass and her we ight o n the moo n, w here
g = 1.62 m/s 2?
18. III A box w ith a 75 kg passenger ins ide is launched straight up
in to the air by a g iant ru bber band. After the box has left the
rubber band but is st ill mov ing upward,
a. W hat is the passenger 's true we ight?
b. W hat is the passenger's apparent we ight?
19. II a. How m uch force does an 80 kg astronaut exe rt o n hi s
chair while s ittin g at rest o n the lau nc h pad?
b. How much force does the astro naut exert on hi s cha ir
while accelerati ng straight lip allO mfs2?
20.
It takes the elevator in a s kyscraper 4 .0 s to reach it s crui sing
speed of J 0 m/s. A 60 kg passe nger gets aboard o n the gro und
floor. What is the passe nger 's a pparen t weight
a. Before the e levator starts mov ing?
b. Whi le the elevator is speeding up ?
c. After the elevator reac hes its cru ising speed?
2 1. II Zac h, whose mass is 80 kg, is in an elevato r descen di ng at
10 m/s. T he e levator takes 3.0 s to brake to a stop at the fi rst floor.
a. Wh at is Zach 's apparent we ight before the elevator starts
brakin g?
b . What is Zach's appare nt we ight whi le the elevator is brakin g?
22. III Figure P5.22 s hows the \' (m/s)
INT
ve locity graph of a 75 kg passenger in an elevator. What is
the passenger's apparent weight
at I = 1.0 s7 At 5 .0 s? A t 9.0 s?
Y:8V
~
+-~-~~-~--'~ 1 (s)
o
2
4
6
8
Section 5.4 Normal Forces
23. II a. A 0.60 kg bull frog is s itt ing at rest on a level log. How
large is the normal force of the log o n the bull frog?
b. A second 0 .60 kg bullfrog is o n a log til ted 30° above
horizon tal. How large is the normal force o f the log on
thi s bull frog?
24. III A 23 kg c hi ld goes down a straight s li de incl ined 38° above
horizontal. The child is acted on by hi s weight, the normal force
from the s lide, and kinetic fri ction.
a. Draw a free-body di agram of the c hil d .
b . How large is the normal force o f the sl ide on the chi ld?
Section 5.5 Friction
25. III Bo nnie and C lyde are sl id ing a 300 kg bank safe across the
fl oor to the ir getaway car. The safe s li des with a constant speed
if C lyde pushes fro m behi nd with 385 N o f force while Bon nie
pu lls forward on a rope w ith 350 N o f force. W hat is the sa fe's
coeffic ie nt of kinet ic friction on the bank fl oor?
26. III A 4000 kg truc k is parked on a 15 ° s lope . How big is the fri ction force o n the truc k?
27. III A 1000 kg car traveling at a speed of 40 m/s skids to a hah on
wel concrete where f-Lk = 0.60. How lo ng are the skid marks?
28. I A stubborn 120 kg mule s its down and refuses to move. To
drag the mul e to the barn , the exasperated farmer ties a rope
aro und the mule and pu Bs with hi s max imum force of 800 N.
g
ra
ra
II
II
Section 5.6 Drag
3'1. II W hat is the drag force on a 1.6-m-w ide. I A-m-hi g h car travel ing at
a. 10 mls C"22 mph)?
b. 30 m/s C"65 mph)?
32. 1111 A 22-cm -di ameter bow lin g ball has a termin al speed of
77 m/s. What is the ball's mass?
33. 11111 A 75 kg skyd iver can be- modeled as a rectangular "box" with
d imens ions 20 cm X 40 em X 1.8 m. What is hi s termina l speed
if he falls feet first?
Section 5.7 Interacting Objects
to
FIGURE PS .22
II
T he coeffi cie nts o f fricti on between the mule and the ground are
f-Ls = 0.80 and f-Lk = 0.50. Is the fanner ab le to move the m ul e?
29. III A 10 kg crate is pl aced o n a horizontal co nveyor bel l. T he
materials are suc h that f-Ls = 0.50 and f-Lk = 0 .30.
a. Draw a free-body di agram show ing all the fo rces o n the
crate if the conveyer be lt runs at constant speed.
b. Draw a free-body d iagram show ing all the forces o n the
crate if the conveyer be lt is speed ing up.
c. W hat is the max imum acceleration the belt can have without
the crate sli pping?
d. If acceleratio n of the belt exceeds the value determ ined in
P;:LI't c, what is the accelerati on o f the c rate?
30. II W hat is the minimu m downward fo rce o n the box in Figure
P5.30 that w ill keep it fro m sl ip125 N
ping? The coeffi cients of static and
kinetic fri cti on be tween the box
and the floor are 0.35 and 0.25,
FIGURE PS .30
respective ly.
34. III A 1000 kg car pu shes a 2000 kg truck that has a dead battery.
II When the dr iver ste ps o n the accele rator, the drive wheel s of the
car push backward aga inst the ground with a fo rce o f 4500 N.
a. What is the magn itude of the force of the car on the truck?
b. W hat is the magn itude of the force of the truck o n the car?
35. 11111 Blocks with masses of 1.0 kg, 2.0 kg, and 3.0 kg are lined up in
a row on a fri cti onl ess table . All three are pu shed forward by a
12 N fo rce applied to the 1.0 kg block. How much Force does the
2.0 kg block exert on Ca) the 3.0 kg block and (b) the 1.0 kg block?
II
Section 5.8 Ropes and Pulleys
36. III Wh at is the tension in the rope of Figure
g P5.36?
37. II A 2.0- m- Io ng, 500 g rope pu ll s a 10 kg
g block of ice across a horizontal, friction less
100 kg
surface. T he bl ock accelerates at 2.0 m/s 2 .
How much force pu ll s forward o n (a) the
block of ice, (b) the rope?
FIGURE PS .36
38. 111 Fi gure P5.38 shows two 1.00 kg blocks
~ con nected by a rope. A seco nd rope hangs
beneath the lowe r b lock. Both ropes have a mass
of 250 g. The ent ire asse mb ly is acce lerated
upward at 3.00 m/s 2 by force F.
a. Wh at is F?
b. W hat is the tens io n at the top end of rope I ?
c. What is the tension at the bottom end of rope l ?
d . W hat is the tens io n at the top end o f rope 2? FIGURE PS .38
Problems
39. II Eac h of 100 identical blocks sill ing on a frictionless surface
II is connected to the next block by a massless string. The first
bloc k is pu lled with a force of 100 N.
a. What is the tension in the string connecting block 100 to
block 99?
b. What is the te nsion in the string connecting block 50 10
block 51?
40. " Two blocks on a fri ctio nless table, A and B, are connected by
a mass less Siring. When bl oc k A is pulled with a certain force,
dragging block B, the tension in the string is 24 N. When bloc k
B is pulJ ed by the same force, dragg in g bloc k A, the tension is
18 N. What is the ratio III AIIII B o r the blocks' masses?
II
General Problems
49.
4 1. III A 500 kg piano is being lowered into position by a crane
while two people steady it wi th ropes pulling 10 the sides. Bob's
rope pull s 10 the left, 15° below horizontal, wit h 500 N of te nsion. Ellen 's rope pull s toward the righ t, 25° below horizontal.
a. What tension must EUen mai nta in in he r rope to kee p the
piano descending verticaUy at constant speed?
b. What is the tension in the vertical main cab le supporting the
piano?
42. " Dana has a sports medal suspended by a long ribbon from her
rearview mirror. As she acce lerates onto the hi ghway, she notices
that the medal is hanging at an angle of 10° from the verti cal.
a. Does th e medal lean toward o r away from the wind shi e ld?
Explain .
b. What is her acceleratio n?
43. 11 Figure P5.43 shows the ve locity graph of a 2.0 kg object as it
INT move s along the x-ax is. What is the net rorce act in g on thi s
object atl = I s?A t4 s? At 7 s?
II
II
FtGURE PS.43
INT
50.
Il
!NT
5 1.
II
52.
II
:'~
(N)
';;r~
oJ~
o 2 4 6 8
II
2
/(s)
-2
53.
'
o
I (s)
I
2
I
4
I
--
6
FtGURE PS.44
44. II Figu re P5.44 shows the net rorce actin g on a 2.0 kg object as
INT it moves along the x-ax is. Th e object is at rest at the o ri gin at
1= 0 s. What are its acceleration and veloc ity at I = 6.0 s?
45. II A 50 kg box hangs from a rope . What is th e te nsio n in th e
rope if
a. The box is at rest?
b. The box has Vy = 5.0 m/s and is speeding up at 5.0 mls 2?
46. II A 50 kg box hangs from a rope. What is the tens ion in th e
rope ir
a. The box moves up at a steady 5.0 m/s?
b. The box has v-' = 5.0 mls and is slowing down at 5.0 m/s 2?
47. I Your rorehead ca n with stand a force of about 6.0 kN before
fracturing, wh ile your cheekbone can on ly withstand aboUi 1.3 kN.
BKl a. If a 140 g baseball strikes your head at 30 mls and SlOpS in
0.00 15 s, what is the magnitude of the ball's accelerat ion?
b. What is the magn itude o rthe rorce that SLOpS the baseball?
c. What rorce does the baseball apply to your head? Exp lain.
d. Are you in danger of a fracture ir the ball hits you in the
forehead? In the cheek?
48. III Scat be lt s and air bags save lives by reducing the forces
exerted o n the driver and passengers in an automobil e coll ision.
BID Cars are des igned with a "crumpl e zone" in the fro nt of the car.
tI
54.
II
II
II
II
55.
II
INT
56.
II
163
In the event of an impact. the passe nger com partment dece lerates over a distance of about I m as the front of the car crumples. An occu pant restra ined by seat belts and air bags
decelerates with the car. By contrast. an unrestra ined occupa nt
keeps moving forward wi th no loss of speed (Newton 's firs t
law !) until hitting the dashboard or windshi eld, as we saw in
Fi gure 4.2. Th ese are un yield ing surfaces, and the un fortun ate
occupan t the n decelerates over a di stance of only about 5 mm .
a. A 60 kg person is in a head-on co lli sion. The car 's speed at
impact is 15 m/s . Est imate the net force on the person if he
or she is wearing a seat belt and if the air bag deploys.
b. Esti mate the net force that ultim ately stops the person if he
or she is not restrained by a seat be lt or air bag.
c. How do these two rorces compare to the person 's weight?
11111 Bob, who has a mass of75 kg, ca n throw a 500 g rock with a
speed of 30 m/s. The distance through which hi s hand moves as he
accelerates the rock forward from rest until he relealies it is 1.0 m.
a. What constan t fo rce must Bob exert o n the roc k to throw it
with thi s speed?
b. If Bob is standing on frictionless ice, what is hi s recoil speed
after releas ing the rock?
11111 An 80 kg spacewa lk ing astro naut pushes o rr a 640 kg sa te 1lite. exerting a 100 N fo rce for the 0.50 s it takes him to
straighten hi s arms. How far apart are the astro naut and the
satellite after 1.0 min ?
II Wh at thrust does a 200 g model rocket need in order to have
a vertical accelerati o n of 10.0 111/s2
a. On earth ?
b. On the moon, where g = 1.62 m/f,2?
III A 20.000 kg roc ket has a rocket mo tor that generates
3.0 X 10 5 N of thrust.
a. What is the roc ket'S i_nitial upward acce leration?
b. At an alt itude o r 5.0 "-m the rocket's acce leration has
increased to 6.0 m/s 2• What mass o r fue l has it burned?
111 You've always wondered abolltlhe acceleratio n or the elevators in the lOl-story-tali Empire State Bu ilding. One day, whi le
visiting New York , you take you r bat hroom sca les in to the
e levator and stand on them. The sca les read 150 Ib as the door
closes. The read ing vari es between 120 lb an d 170 lb as the
elevator travels 10 1 floors.
a. What is the mag nitude of the accelerat ion as the elevator
starts upward?
b. What is the magnitude of the acce le rat ion as the e levator
brakes to a stop?
1111 A 23 kg chi Id goes down a
30 N
straight slide in cl ined 38° above
hori zo ntal. The child is ac ted o n
by hi s we ight, the normal rorce
from the slide, kine ti c frict io n,
and a ho ri zontal ro pe exert ing a
30 N fo rce as shown in Figure FtGURE PS .S4
P5.54. How large is the no rm al
rorce of the sl ide on the child ?
II Josh starts his sled at the lOp of a 3.0-m-high hill that has a
constant slope of 25 °. After reach in g the bOlLom, he slides
across a horizontal patch of snow. The hill is fri cti onless, but
the coe ffi c ien t or kinetic fric ti on between his sled and the snow
is 0.05. How far from the base of the hill does he end up?
II A wood block, after be ing given a start in g pu sh, slides dow n
a wood ramp at a constan t speed. What is the angle o r the ramp
above horizontal ?
164
CHAPTER 5
Applying Newton 's Laws
57. II Re searc hers often use force plates to me as ure the forces
BID that people exert aga inst th e floor during moveme nt. A force
INT plate works like a bathroom sca le, but it keeps a record of how
the reading c hanges with time. Fig ure PS.57 shows the data
from a force plate as a woman j umps straight up and then
lands.
a. What was the vertical component of her acceleration during
push-off?
b. Wh at was th e vertical component of her acce leration while
in the air?
c. What was the vertical componen t of her acceleration during
the landing?
d. What was her speed as her feet left the force pl ate?
e. How hi gh did she j ump?
61. III A 1.0 kg wood block is pressed again st a
cal wood wall by a 12 N force as shown
ill invertiFigure
P5.61. If the block is initi all y at
62.
ill
63.
II
64.
II
65.
F(N)
II
1500
1000
500
t------'
o ~------~-----------'---------
66.
II
FIGURE PS .57
58. 11111 A 77 kg sprinter is running the 100 m dash. At one instant,
BIO early in the race , hi s acceleration is 4.7 m/s 2 .
a. What lotal force does the track surface exert on the sprinter?
Assume hi s acce lerat ion is parall e l to th e ground. Give
yo ur an swer as a magnitude and an angle with respect to
the hori zonta l.
b. Thi s force is applied to one foot (the other foot is in the air),
which for a fraction of a second is stationary with respect to
the track surface. Because the foot is stationary, the net force
on it must be ze ro . Thus the force of the lower leg bo ne on
the foot is equal but o pposite to the force of the track on the
foot. If the lower leg bone is 60° from horizontal, what are
the co mpone nts of the leg's force on the foot in the directi ons parall el and perpend ic ular to the leg? (Force components perpe ndicular to the leg can cause di slocati o n of the
ankJejoint.)
59. 11111 Sam, whose mass is 75 kg, takes off across leve l snow on hi s
jet-powered ski s. The ski s have a thrust of 200 N and a coe ffi cie nt of kinetic fri ction on snow of 0.1 O. Unfortunately, the ski s
run o ut of fu el after o nly lO s.
a. What is Sam's top speed?
b. How far has Sa m travel ed wh e n he fin ally coasts to a
stop ?
60. 1111 A perso n with compromi sed pinch
stre ngth in hi s fin gers ca n o nl y exert a norBIO mal force of 6.0 N to either side of a pinchhe ld object, such as the book shown in
Fi gure PS.60. What is the heav iest book he
can hold onto verticall y before it sl ips out of
hi s fi ngers? The coeffic ient of stat ic friction
of the surface between th e fingers and the
FIGURE P5 .60
book cover is 0.80.
rJ
II
67.
II
6S.
69.
70.
II
rest, will it move upward , mo ve downward ,
or stay at rest?
1111 A 50,000 kg loco moti ve, with steel FIGURE P5 .61
whee ls, is traveling at 10 mls on stee l rail s
when its e ngine and brakes both fail. How far will th e locomoti ve roll before it comes to a stop?
II An Airbus A320 jetl iner has a takeoff mass of 75,000 kg . It
reac hes its takeoff speed of 82 m/s ( ISO mph) in 35 s. What is
the thrust of the engines? You can neglect air res istance but not
rollin g fri ction.
11111 A 2.0 kg wood block is laun ched up a wooden ramp th at is
inclined at a 35° ang le. The block 's initial speed is 10 m/s .
a. What verti ca l height does the block reach above it s start ing
point ?
b. What speed does it have when it slides bac k down to its
starting point ?
III Two blocks are at rest
on a frictionl ess incl ine,
as shown in Fi gure P5.65.
What are the te nsion s in
the two strings?
II Two ide nti cal blocks
are stac ked one on to p
of th e oth er. The bottom FIGURE P5 .65
block is free to slide on a frictionl ess surface. The coeffic ient
of static fri ct ion between the bloc ks is 0.3S. What is the maximum hori zo ntal force that can be appli ed to th e lower block
without the upper block slipping ?
III A wood block is slidin g up a wood ramp . If the ramp is very
steep , th e block wiLl reverse direction at its hi ghest point and
slide back down. If the ramp is shall ow, the block will stop
when it reaches its hi ghest point. What is the smaLl es t ramp
angle, measured from the hori zontal, for which th e block will
slide back down ?
11111 The fastest recorded skydi ve was by an Air Fo rce offi ce r
who j umped from a hel ium balloon at an eleva ti on of
103,000 fl , three tim es hi gher than airliners fl y. Because the
den sity of air is so low at these altitudes, he reached a speed
o f 614 mph at an elevati o n o f 90,000 fl , th en grad uall y
s lowed as the air beca me marc dense. As sum e that he fell in
th e spread-eag le pos ition o f Example 5.15 and that hi s lowa ltitude terminal speed is 12S mph . Use thi s inform ation to
determine the density of air at 90,000 f1.
1111 A 2.7 g Ping- Pon g balJ has a diameter of 4.0 cm.
a. The ball is shot strai ght up at twice its terminal speed. What
is its initi al acce leration ?
b. The ball is shot straight down at twice its terminal speed.
What is its initial acceleration?
1111 Two blocks are connected by a strin g as in Fi gure P5.70.
What is the upper bloc k"s acceleration if the coeffici en t of
kinetic fri ct ion between the block and the tabl e is 0.20?
5.0 kg
1.0 kg
FIGURE PS .70
FIGURE PS .71
Problems
71. 1111 The 10 kg block in Figure P5.71 slides down a frictionless
~ ramp. What is its accelerati on?
72. II A 2.0 kg wood block is pulled along a wood Ooor at a steady
speed. A seco nd wood block, with mass 3.0 kg, is allac hed to
the first by a hori zo ntal string. What is the magn itude of the
force pulling on the first block?
73. 1111 A magician pulls a tablecloth out
from under some di shes. How far do
tNT the dishes move during the 0.25 s it
takes to pull out the tablecloth? The
coefficient of kinetic friction between
the cloth and the dishes is J.Lk = 0.12.
tOO kg
74. III The 100 kg block in Figure PS.74
takes 6.0 s to reach the floor after being
tNT released from rest. What is the mass of
FIGURE PS .74
the block on the left ?
rJ
II
II
Problems 75 and 76 show free-body diagrams. For eac h,
a.
Write a realistic dynamics problem for which thi s is the correct
free-body diagram. Your problem sho uld ask a question that can
be answered with a value of position or ve locity (such as "How
far?" or " How fast?"), and should give suffi cient information to
allow a solution.
b. Solve your problem!
76. III
75. I
y
y
g
g
9.8 N
__~4~.9~N~t-____t>- x
20 N
!2,000 N
9.8 N
FIGURE PS .7S
FIGURE PS .76
In Problems 77 throug h 79 you are given the dynamics equat ions
that are L1 sed to solve a problem. For each of these, YOLi are to
a.
Write a realistic problem for which these are the correct equation s.
b. Draw the free-body diagram and the pictorial represe ntation for
your problem.
c. Finish the solution of the problem.
77. II - 0.8011 = ( 1500 kg)".,
11 - ( 1500 kg)(9.8 m/s 2) = 0
Stop to Think 5.1: A. The lander is descending and slowing. The
accelerati o n vector points upward, and so Filet points upward. Thi s
can be true only if the thrust has a larger magnitude than the weight.
Stop to Think 5.2: D. When you are in the air, there is flO contact
force supportin g you, so your apparent weight is ze ro: You are
we ight less.
Stop to Think 5.3:/" > Ie = 10 = IE > IA' Situations C. D, and E
are all kinetic friction , which does not depend on either velocity or
acceleration . Kinet ic fri ction is less than the maximum static fri ct ion
that is exerted in B. JA = 0 because no friction is needed to keep the
object at rest.
78.
165
T - 0.211 - (20 kg)(9.8 O1/s') sin 20' = (20 kg)(2.0 m/s2)
(20 kg)(9.8 m/s') cos20' = 0
II -
79.
( 100 N)cos30' - I. = (20 kg)a,
+ ( 100 N)sin 30' - (20 kg)(9.8 m/s 2) = 0
I. = 0.2011
II
Passage Problems
Sliding on the Ice
In the winter sport of curlin g, players give a 20 kg stone a push
across a sheet of ice. The Slone mo ves approx imately 40 m before
com in g to rest. The final position of the stone, in principle, only
depends on the initi al speed at whi ch it is launched and the force of
friction between the ice and the stone, but team members can use
brooms to swee p the ice in front of the stone to adju st its speed and
trajectory a bit ; th ey must do thi s without touching the stone. Judicious sweeping can len gthen the travel of the stone by 3 m.
80. I A curler pushes a stone to a speed of 3.0 mls over a time of
2.0 s. Ignorin g the force of friction, how much force must the
curler apply to the stone to bring it Lip to speed?
A.3.0N
B.ISN
C. 30N
D.ISON
8 1. I The sweepers in a curl ing competit ion adjust the trajectory
of the slone by
A. Decreasing the coefficient of friction between the stone
and the ice.
B. [ncreas in g the coeffic ien t of friction between the stone and
the ice.
e. Changing friction from kinetic to static.
D. Chan gin g fTiction from static to kinetic.
82. I Suppose the stone is launched with a speed of 3 mls and
travel s 40 111 before comi ng to rest. What is the approximare
magnitude of the friction force on the stone?
A. ON
B.2N
C. 20N
D.200N
83. I Suppose the stone' s mass is increased to 40 kg, but it is
launched at the same 3 mis. Which one of the following is
true?
A. The stone would now travel a lon ger di stance before comin g to rest.
B. The stone would now lravel a shorter di stance before coming to rest.
e. The coefficient of friction would now be greater.
D. The force of friction would now be greate r.
Stop to Think 5.4: D. The ball is shot down at 30 mis, so
VUy = -30 m/s. Thi s exceeds the terminal speed, so the upward drag
force is greater than the downward weight force. Thus the ball slows
down even though it is "falling." It will slow until v)' = - 15 mIs, the
terminal veloci ty, then maintain that veloc ity.
Stop to Think 5.5: B. F QonH = F HonQ and F Pon Q = F QonP because
these are action/reaction pairs. Box: Q is slowing down and therefore
mu st ha ve a net force to the left. So from Newton's second law we
also know that F Holl Q > F p onQ '
Stop to Think 5.6: Equal to. Each block is hanging in equ ilibrium,
with no net force, so the upward tension force is mg.