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158 C HAPTER 5 Apply ing Newton 's Laws
SUMMARY The goal of Chapter 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 solver if you adhere to it as you do the homework problems. The Dynamics Worksheets 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.
Check lhat the object is either at rest or moving with constant velocity (0 ~ 0).
Identify forces and show lhem on a free-body diagram.
SOLVE Use Newton 's second law in component form:
L,Fx= l1Iax = 0
L,F}. = ilia), = 0
"Read" the components from the freebody diagram.
ASSESS Is your result reasonable?
IMPORTANT CONCEPTS
Dynamics Problems Object accelerating.
PREPARE Make simplifying assumptions. Make a visual overview:
Sketch a pictorial representation.
.Identify known quantities and what the problem is uy ing to find.
Identify all forces and show them on a free· body diagram.
SOLVE Use Newton's second law in component form:
"Read" the components of the vectors from the free· body diagram. If needed, use kinematics to find posit ions and velocities.
ASSESS Is your result reasonable?
Objects in Contact Two or more objects interacting.
PREPARE Make a visual overview:
Sketch a pictorial representation.
Iden tify aU forces acting on each object.
Identify action/reacLion pairs of forces acting on objects in the system.
Draw a separate free·body diagram for each object.
SOLVE Write Newton 's second law for each object. Use Newton's third law to equate the magnitudes of actionlreaction pairs. Determine how the accelerations of the objects are related to each other.
ASSESS Is your result reasonable?
Specific information about three important forces: Newton's laws are vector expressions. You must write them out by components: Weight iii = (mg, downwilld)
Friction 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)
Drag jj ~ dpA v2, direction opposite the motion) for motion in air
APPLICATIONS
Apparent weight is the magnitude 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)
Apparent weight equals your true weight \II = mg only when ay = O.
A falling object reaches terminal speed
v1erm ~)4mg
pA
Terminal speed is reached when the drag force exactly balances the weight force: a = O.
jj
(Fnd)" = L,F.. = max
(Fnd) ), = L,Fy = may
For equilibrium problems, a.( = 0 and lIy = 0.
Strings and pulleys
A string or rope pulJs what it 's connected to with a force equal to its tension.
F"'f'< "" ... 11 = !.en;,ion :.
The tension in a rope is equal to Fh• noI.: .. " ,!", = ten~ion the force pulling on the rope. ..,,.: .... ___ _
The tension 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.
tMP)TM For homework assigned on MasteringPhysics, go to !:!!/ www.masteringphysics.com
Problem difficulty is labeled as I (straightforward) to 11111 (challenging),
QUESTIONS
Conceptual Questions
I. An objcGt is subject lO two fo rces that do not point 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 bjects described here in stat ic equili brium, dynamic equilibrium, or not in equili brium at all? a. A girder 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 plane 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 slide as the truck stops.
3. What forces are acting on you ri ght now? What net fo rce is acting on you right now?
4. Dec ide whether each of the fo llowing is true or fa lse. Give a reason! a. The mass of an object depe nds on its location. b. The we ight o f an object depends on its location. c. Mass and weight desc ri be the same thing in di fferen t units.
5. An astronaut takes hi s bathroo m scale to the moon and then stands on it. Is the reading of the scale hi s true weight? Explain.
6. A light block of mass m and a heavy block of mass M are attached to the ends of a rope. A stu-dent holds the heavier block and lets the li ghter M
block hang below it, as shown in Figure QS.6. Then she lets go. Air resistance can be neg lected. a. What is the tension in the rope while the
blocks are falling, before e ither hi ts the grOllnd?
b. Would yo ur answer be d iffe rent if she had been ho lding the lighter block initially?
7. Four baUs are thrown straight up . Figure QS.7 is FIGURE 05.6
a "snapshot" showing 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, Fnc11 ' Fnc12 '
Fnel3 , Fnel4 , act ing on the ba ll s. Some may be equa l. Give yo ur
answer in the form A > B = C> D, and state your reason ing.
~m/' ~ ~ 3 m/s 3 m/s
3oo g '2 3oo g
3
4oo g
4
5 m/s
200 g
FIGURE 05 .7
8. Suppose you attempl to pour oul 100 g of salt , us ing a pan balance for measureme nts, whi le in an elevator that is acce lerating upward. Will the quanti ty of salt be too muc h, too li tt le, o r the correct amoun t? Ex plain .
Questions 159
Prob lems labeled til can be done on a Workbook Dynamics
Worksheet; INT integra te significant material from earlier chapters;
BIO are of biological or medical interest.
9. a. Can the normal force on an o bject 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 ight up . Taking the 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 Figure QS. II . Rank in order, from largest to smallest, the three drag forces 0 1,
O2, and 0 3, Some may be equal. Give 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 skydive r is falling at her termi nal speed. Ri ght after she opens he r parachute, which 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; some fall quite quickl y, others quite slowly. Why might thi s be true?
14. An a irp lane moves through the a ir at a constant speed. The j et engine's thrust applies a force in the di rect ion of motio n. Reducing thrust will cause the plane to fl y at a slower-but st ill constan t-speed. Ex plain why thi s is so .
IS. Is it poss ible for an object to trave l in air faster than its terminal speed? If not, why not? If so, ex plain how thi s might happen .
For Questions 16 through 19, determine the tension in the rope at the poin t indicated with a dot.
16.
AU objec ts are at rest. The strings and pulleys are massless, and the pull eys are Frict io nless.
17. 18. __ ~-.I
FIGURE 05.16
FIGURE 05.17
FIGURE 05.18
160 CHAPTER 5 Apply ing New ton 's Laws
19.
FIGURE 05.19
20. The noor is fr ic ti onless. In which direction is the kineti c fri ct ion force on block I in Figure Q5.20? On block 2? Expla in.
FIGURE 05.20
Multiple-Choice Questions
2 1. II The wood block in Figure Q5.2 1 is at rest on a wood ramp. In which di rect ion is the stati c friction force on block I?
22.
23.
A. Up the slope. B. Down the slope. FIGURE 05.21 C. The frict ion force is zero. D. There 's not enough information to tel l. II A 2.0 kg ball is suspended 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
J Y FIGURE 05.22
I Whi le stand ing in a low tunnel , you rai se your arms 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 accelerat-ing downward at 1.20 m/s2•
a. What is the magni tude of the normal force of the e levator noor on the dog? A. 34 N B. 43 N C. 49 N D. 55 N E. 74 N
b. What is the magn itude of the force of the dog on the elevator noor? A. 4.2N D. 43 N
B. 49 N E. 74N
C. 55 N
25. I A 3.0 kg puck slides due east on a horizon tal fr ictionless surface at a constant speed of 4.5 m/s. Then a force of magn itude 6.0 N, directed due north , is applied for 1.5 s. Afterward, a. What is the northward component 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 rocket in space, initi ally at res t, fi res its main engines at a constant th rust. As it burns fue l, the mass of the rocket dec reases. Which of the graphs in Figure Q5.26 best represents the velocity of the rocket as a fu nct ion of time?
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 downward at 1.7 m/s2
. What is the approximate read ing on the scale?
28.
29.
30.
A. 0 N B. 400 N C. 500 N D. 600 N I The two bJocks in Figure Q5.28 are at rest on frictionless surfaces. What must be the mass of the ri ght block in order that the two blocks remain stationary? A. 4.9 kg B. 6.1 kg C. 7.9 kg D. 9.8 kg E. 12 kg
-----~~~---------~Q FIGURE 05 .28
I A footba ll player at pract ice pushes a 60 kg block ing sled across the fi eld at a constant speed. The coeffic ient of ki net ic fri ction between the grass and the sled is 0.30. How much force must he apply to the sled? A. 18N B.60N C. 180N D.600N I Two football players are pushi ng a 60 kg blocki ng sled across the field at a constant speed of 2.0 m/s. The coeffic ient of ki netic friction between the grass and the sled is 0.30. Once they stop pushi ng, how far will the sled slide before coming 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 vehicles could cl imb a slope of 45°. For thi s to be poss ible, what l1lust be the mi nimum coeffi c ien t of stalic fr ict ion between the ve hicle's tires and the road? A. 0.5 B. 0.7 C. 0.9 D. 1.0
32. II A truck is trave li ng at 30 m/s on a slippery road. The dr iver slams 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 far will the tfllck skid before stopping? A. 230 In B. 300 m C. 450 m D. 680 m
PROBLEMS
Section 5. t Equilibrium
I. I The three ropes in F igure PS. l a re tied to a smaU, 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 shown. What are 71 and 1;, the magni tudes of the tension forces in the first two ropes?
O.GOm Rope 2
O.80m
Rope I
FIGURE P5 .1 FIGURE PS.2
2. III The three ropes in Figure PS.2 are tied to a small , very li ght rI ring. Two of these ropes arc anchored to walls at right angles
with the tensions shown in the fi gure. W hat are the magnill1de and d irect io n of the tens ion f3 in the thi rd rope?
3. 1111 A 20 kg louds peaker is suspended 2 .0 III below the ce ili ng by & two cablcs that are each 30° from vert ical. What is the te nsion
in the cables? 4. II A 1000 kg steel beam is sup
PI ported by the two ropes shown in Figure P5.4. Each rope can Rope I
support a max imum sustained tens ion of 5600 N. Do the ropes break?
5. J A cable is used to raise a g 25 kg urn from an underwater FIGURE PS.4
archeological site. There is a 25 N drag force from the water as the urn is ra ised at a constan t speed . What is the tension in the cable?
6. JIll When you bend your knee, BID the quadriceps muscle is
stretched. Thi s increases the tens ion in the quadri ceps tendon att ached to your kneecap (pate ll a), which, in turn , increases the tens ion in the patella tendon that attaches your kneecap to your lower leg bone (ti bia). Simultaneously, the end of your upper leg bone (femu r) pushes out ward on the pateUa. Figure P5.6 shows how
FIGURE PS.6
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 each tendon is 60 N?
7. II The two angled ropes used to rJ support the crate in Fi gure P5.7
can withstand a max imum tension of 1500 N before they break. What is the largest mass the ropes can support? FIGURE PS.7
Problems 161
Section 5.2 Dynamics and Newton 's Second Law
8. II A force with x-component Fx acts on a 500 g object as it moves along the x-ax is. The objecl' s acceleration graph (ax ve rsus t) is shown in Figure P5.8. Draw a graph of F, versus t .
1.0 2
0.5
o.0t--~~-~-r- I (s) 0 +---I--~-~-'l- I (s) 4 234
- 0.5 - I
FIGURE PS .8 FIGURE PS.9
9. II A force with x-component 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 Fx versus I is shown in Figure P5. 1O. Draw an acce lerati on graph (at versus t) for thi s object.
f~ (N)
1.5
1.0
0.5
0.0 t--+~-~---1cr- 1 (s)
- 0.5
FIGURE PS.l0
)
2.0N ---4 __ ~~~~L-x
4.0N 3.0 N
FIGURE PS.11
II . II The forces in Figure P5. 11 are act ing on a 2.0 kg object. Find the values of ax and a". the x- and y-components of the object 's acce leration.
12. I The forces in Figure P5.1 2 are acti ng on a 2.0 kg object. Find the values of (/, and {/y . the x- and y-components of the object's accelerati on.
13. I A horizon tal rope is tied to a 50 kg box on frict ionless ice. What
2.0N
1.0N
y
3.0 N
4.0N
2.0 N
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?
x
14. 1111 A crate pushed along the fl oor with ve loci ty Vi slides a distance d afte r the pushing 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? Explain.
b. If the initial ve locity of the crate is doubled to 2Vi but the mass is not changed, what distance does the crate slide before stopping? Explain.
15. II In a head-on colli sion, a car stops in 0. 10 s from a speed of 11 14 m/s . The dri ver has a mass of 70 kg, and is. fortunately.
tightly strapped into hi s seal. What force is applied to the driver by hi s seat be lt duri ng that fraction of a second?
162 CHAPTER 5 Apply ing New ton 's Laws
Section 5.3 Mass and Weight
16. I An astronaut's we ig ht on earth is 800 N. What is hi s weight on Mars, where g = 3.76 m/s2?
l 7. I A woman has a mass o f 55.0 kg. a. What is her we ight o n earth? b. What are her mass and her we ight o n the moon, where
g = 1.62 m/s2? 18. III A box with 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 moving upward, a. What is the passenger 's true we ight? b. What is the passenger's apparent we ight?
19. II a. How much force does an 80 kg astronaut exe rt o n hi s chair while s itting 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 skyscraper 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 passenger 's apparen t weight a. Before the e levator starts moving? b. Whi le the elevator is speeding up? c. After the elevator reaches its cru ising speed?
2 1. II Zach, whose mass is 80 kg, is in an elevato r descendi ng at 10 m/s. The e levator takes 3.0 s to brake to a stop at the fi rst floor. a. What is Zach 's apparent we ight before the elevator starts
brakin g? b. What is Zach's apparent we ight whi le the elevator is brakin g?
22. III Figure P5.22 shows the \' (m/s)
INT ve locity graph of a 75 kg pas- Y:8V senger in an elevator. What is ~ the passenger's apparent weight at I = 1.0 s7 At 5 .0 s? A t 9.0 s? +-~-~~-~--'~ 1 (s)
o 2 4 6 8 to
FIGURE PS.22
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 lide 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 hild . 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 II fl oor to the ir getaway car. The safe s lides with a constant speed
if C lyde pushes fro m behi nd with 385 N o f force while Bonnie pu lls forward on a rope with 350 N o f force. What is the safe's coeffic ie nt of kinet ic fric tion on the bank floor?
26. III A 4000 kg truc k is parked on a 15° s lope. How big is the fri cg tion 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 ra wel concrete where f-Lk = 0.60. How long are the skid marks? 28. I A stubborn 120 kg mule s its down and refuses to move. To ra drag the mule to the barn , the exasperated farmer ties a rope
aro und the mule and puBs with hi s max imum force of 800 N.
The 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 mul e?
29. II I A 10 kg crate is placed o n a horizontal conveyor bel l. The II materials are such that f-Ls = 0.50 and f-Lk = 0.30.
a. Draw a free-body di agram showing all the fo rces o n the crate if the conveyer be lt runs at constant speed.
b. Draw a free-body d iagram showing 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
30. II P;:LI't c, what is the accelerati on o f the c rate?
What is the minimum down-II ward fo rce o n the box in Figure
P5.30 that will keep it from sl ipping? The coefficients of static and kinetic fri c ti on be tween the box and the floor are 0.35 and 0.25, respective ly.
Section 5.6 Drag
FIGURE PS.30
125 N
3'1. II What is the drag force on a 1.6-m-wide. IA-m-hig h car travel ing at a. 10 mls C"22 mph)? b. 30 m/s C"65 mph)?
32. 1111 A 22-cm-diameter bowling ball has a terminal speed of 77 m/s. What is the ball's mass?
33. 11 111 A 75 kg skyd iver can be- modeled as a rectangular "box" with d imensions 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
34. II I A 1000 kg car pushes a 2000 kg truck that has a dead battery. II When the dr iver steps o n the accele rator, the drive wheels 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. What is the magn itude of the force of the truck o n the car?
35. 11 111 Blocks with masses of 1.0 kg, 2.0 kg, and 3.0 kg are lined up in II a row on a fri c ti onless table . All three are pushed 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?
Section 5.8 Ropes and Pulleys
36. III What is the tension in the rope of Figure g P5.36? 37. II A 2.0-m-Iong, 500 g rope pu ll s a 10 kg
g block of ice across a horizontal, friction less surface. The block accelerates at 2.0 m/s2
.
How much force pu ll s forward o n (a) the block of ice, (b) the rope?
38. 111 Figure P5.38 shows two 1.00 kg blocks FIGURE PS.36
~ connected by a rope. A second rope hangs beneath the lower b lock. Both ropes have a mass of 250 g. The ent ire asse mbly is acce lerated upward at 3.00 m/s2 by force F. a. What is F? b. What is the tens io n at the top end of rope I? c. What is the tension at the bottom end of rope l ?
100 kg
d. What is the tens io n at the top end o f rope 2? FIGURE PS.38
39. II Each of 100 identical blocks sill ing on a frictionless surface II is connected to the next block by a massless string. The first
block 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 tension in the string connecting block 50 10
block 51? 40. " Two blocks on a frictio nless table, A and B, are connected by II a mass less Siring. When block A is pulled with a certain force,
dragging block B, the tension in the string is 24 N. When block B is pulJed by the same force, dragg ing block A, the tension is 18 N. What is the ratio III AIIII B or the blocks' masses?
General Problems
4 1. III A 500 kg piano is being lowered into position by a crane II while two people steady it wi th ropes pulling 10 the sides. Bob's
rope pull s 10 the left, 15° below horizontal, with 500 N of tension. Ellen 's rope pull s toward the righ t, 25° below horizontal. a. What tension must EUen mainta in in he r rope to keep the
piano descending verticaUy at constant speed? b. What is the tension in the vertical main cable supporting the
piano? 42. " Dana has a sports medal suspended by a long ribbon from her II rearview mirror. As she acce lerates onto the highway, she notices
that the medal is hanging at an angle of 10° from the verti cal. a. Does the medal lean toward or away from the windshie ld?
Explain . b. What is her acceleration?
43. 11 Figure P5.43 shows the ve locity graph of a 2.0 kg object as it INT moves along the x-ax is. What is the net rorce act ing on thi s
object atl = I s?At4 s? At 7 s?
';;r~ oJ~ /(s) o 2 4 6 8
:'~(N) 2 ' o I I (s)
2 I 4 I 6 -2 --
FtGURE PS.43 FtGURE PS.44
44. II Figure P5.44 shows the net rorce acting on a 2.0 kg object as INT it moves along the x-ax is. The object is at rest at the ori 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 the tension in the
tI rope if a. The box is at rest? b. The box has Vy = 5.0 m/s and is speeding up at 5.0 mls2?
46. II A 50 kg box hangs from a rope. What is the tens ion in the
II 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 can withstand a force of about 6.0 kN before II fracturing, wh ile your cheekbone can only 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 orthe rorce that SLOpS the baseball? c. What rorce does the baseball apply to your head? Explain. d. Are you in danger of a fracture ir the ball hits you in the
forehead? In the cheek? 48. II I Scat be lt s and air bags save lives by reducing the forces II exerted on the driver and passengers in an automobile coll ision. BID Cars are des igned with a "crumple zone" in the front of the car.
Problems 163
In the event of an impact. the passenger compartment dece lerates over a distance of about I m as the front of the car crumples. An occupant restra ined by seat belts and air bags decelerates with the car. By contrast. an unrestra ined occupant keeps moving forward wi th no loss of speed (Newton 's firs t law !) until hitting the dashboard or windshield, as we saw in Fi gure 4.2. These are un yield ing surfaces, and the unfortunate occupan t then decelerates over a di stance of only about 5 mm. a. A 60 kg person is in a head-on colli 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 ultimately 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? 49. 11 111 Bob, who has a mass of75 kg, can throw a 500 g rock with a II speed of 30 m/s. The distance through which hi s hand moves as he INT accelerates the rock forward from rest until he relealies it is 1.0 m.
a. What constan t fo rce must Bob exert on the rock 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?
50. 111 11 An 80 kg spacewalk ing astronaut pushes orr a 640 kg sa te 1-Il lite. exerting a 100 N force for the 0.50 s it takes him to !NT straighten his arms. How far apart are the astro naut and the
satellite after 1.0 min? 5 1. II What thrust does a 200 g model rocket need in order to have II a vertical accelerati on of 10.0 111/s2
a. On earth? b. On the moon, where g = 1.62 m/f,2?
52. III A 20.000 kg rocket has a rocket motor that generates II 3.0 X 105 N of thrust.
a. What is the rocket'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 or fue l has it burned? 53. 111 You've always wondered abolltlhe acceleration or the eleva
tors in the lOl-story-tali Empire State Building. One day, whi le visiting New York , you take you r bathroom scales in to the e levator and stand on them. The sca les read 150 Ib as the door closes. The reading vari es between 120 lb and 170 lb as the elevator travels 10 1 floors. a. What is the magnitude 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? 54. 1111 A 23 kg chi Id goes down a 30 N II straight slide incl ined 38° above
hori zontal. The child is ac ted on by hi s we ight, the normal rorce from the slide, kine ti c frict ion, and a hori zontal ro pe exert ing a
30 N force as shown in Figure FtGURE PS .S4 P5.54. How large is the normal rorce of the sl ide on the child?
55. II Josh starts his sled at the lOp of a 3.0-m-high hill that has a II constant slope of 25°. After reach ing the bOlLom, he slides INT across a horizontal patch of snow. The hill is fri ctionless, but
the coeffic 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?
56. II A wood block, after be ing given a start ing push, slides down II a wood ramp at a constan t speed. What is the angle or the ramp
above horizontal ?
164 CHAPTER 5 Applying Newton 's Laws
57. BID INT
II Researchers often use force plates to meas ure the forces that people exert aga inst the floor during movement. A force plate works like a bathroom scale, but it keeps a record of how the reading changes with time. Figure PS.57 shows the data from a force plate as a woman jumps straight up and then lands. a. What was the vertical component of her acceleration during
push-off? b. What was the 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 plate? e. How high did she j ump?
F(N)
1500
1000
500 t------' o ~------~-----------'---------
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 his accelerat ion is paralle l to the ground. Give yo ur answer as a magnitude and an angle with respect to the hori zonta l.
b. This 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 bone on the foot is equal but opposite to the force of the track on the foot. If the lower leg bone is 60° from horizontal, what are the components of the leg's force on the foot in the directions parallel and perpendicular to the leg? (Force components perpendicular to the leg can cause di slocation of the ankJejoint.)
59. 11111 Sam, whose mass is 75 kg, takes off across level snow on hi s rJ jet-powered ski s. The ski s have a thrust of 200 N and a coeffi
cient of kinetic fri ction on snow of 0.1 O. Unfortunately, the ski s run out of fuel after only lO s. a. What is Sam's top speed? b. How far has Sam traveled when he fin ally coasts to a
stop? 60. 1111 A person with compromi sed pinch II strength in hi s fingers can onl y exert a norBIO mal force of 6.0 N to either side of a pinch-
he ld object, such as the book shown in Figure PS.60. What is the heaviest book he can hold onto vertically before it sl ips out of hi s fi ngers? The coeffic ient of stat ic friction of the surface between the fingers and the book cover is 0.80. FIGURE P5 .60
61.
ill
62.
ill
III A 1.0 kg wood block is pressed against a vertical wood wall by a 12 N force as shown in Figure P5.61. If the block is initiall y at rest, will it move upward , move downward , or stay at rest? 1111 A 50,000 kg locomoti ve, with steel FIGURE P5 .61
wheels, is traveling at 10 mls on stee l rail s when its engine and brakes both fail. How far will the loco-motive roll before it comes to a stop?
63. II An Airbus A320 jetl iner has a takeoff mass of 75,000 kg. It II reaches 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 rolling fri ction.
64. 11 111 A 2.0 kg wood block is launched up a wooden ramp that is II 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 back down to its starting point?
65. III Two blocks are at rest II on a frictionl ess incl ine,
as shown in Figure P5.65. What are the tensions in the two strings?
66. II Two identi cal blocks II are stacked one on top
of the other. 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 blocks is 0.3S. What is the maximum hori zontal force that can be applied to the lower block without the upper block slipping?
67. III A wood block is sliding up a wood ramp. If the ramp is very II steep , the block wiLl reverse direction at its hi ghest point and
slide back down. If the ramp is shallow, the block will stop when it reaches its highest point. What is the smaLles t ramp angle, measured from the hori zontal, for which the block will slide back down?
6S. 11111 The fastest recorded skydi ve was by an Air Force offi ce r who j umped from a hel ium balloon at an eleva tion of 103,000 fl , three times hi gher than airliners fl y. Because the density of air is so low at these altitudes, he reached a speed o f 614 mph at an elevatio n o f 90,000 fl , then gradually s lowed as the air became marc dense. Assume that he fell in the spread-eag le pos ition o f Example 5.15 and that hi s lowa ltitude terminal speed is 12S mph . Use thi s information to determine the density of air at 90,000 f1.
69. 11 11 A 2.7 g Ping-Pong balJ has a diameter of 4.0 cm. a. The ball is shot straight up at twice its terminal speed. What
is its initial acce leration? b. The ball is shot straight down at twice its terminal speed.
What is its initial acceleration? 70. 1111 Two blocks are connected by a string as in Figure P5.70. II What is the upper bloc k"s acceleration if the coefficien t of
kinetic frict ion between the block and the table is 0.20?
5.0 kg
1.0 kg
FIGURE PS .70 FIGURE PS .71
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 rJ speed. A second wood block, with mass 3.0 kg, is allached to
the first by a hori zontal string. What is the magn itude of the force pulling on the first block?
73. 1111 A magician pulls a tablecloth out II 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.
74. III The 100 kg block in Figure PS.74 II takes 6.0 s to reach the floor after being
tOO kg
tNT released from rest. What is the mass of the block on the left? FIGURE PS.74
Problems 75 and 76 show free-body diagrams. For each,
a. Write a realistic dynamics problem for which thi s is the correct free-body diagram. Your problem should 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! 75. I g
y
9.8 N
__ ~4~.9~N~t-____ t>- x
20 N
9.8 N
FIGURE PS.7S
76. III g y
!2,000 N
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, YO Li are to
a. Write a realistic problem for which these are the correct equations.
b. Draw the free-body diagram and the pictorial representation for your problem.
c. Finish the solution of the problem.
77. II - 0.8011 = (1500 kg)"., 11 - ( 1500 kg)(9.8 m/s2) = 0
Stop to Think 5.1: A. The lander is descending and slowing. The acceleration vector points upward, and so Filet points upward. This 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 supporting 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 friction 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.
Problems 165
78. T - 0 .211 - (20 kg)(9.8 O1/s') sin 20' = (20 kg)(2.0 m/s2) II - (20 kg)(9.8 m/s') cos20' = 0
79. (100 N)cos30' - I. = (20 kg)a, II + (100 N)sin 30' - (20 kg)(9.8 m/s2) = 0 I . = 0.2011
Passage Problems
Sliding on the Ice
In the winter sport of curling, players give a 20 kg stone a push across a sheet of ice. The Slone moves approx imately 40 m before com ing to rest. The final position of the stone, in principle, only depends on the initial speed at which it is launched and the force of friction between the ice and the stone, but team members can use brooms to sweep the ice in front of the stone to adjust its speed and trajectory a bit ; they must do thi s without touching the stone. Judicious sweeping can lengthen 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. Ignoring 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
81. 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 ing the coeffic ien t of friction between the stone and
the ice. e. Changing friction from kinetic to static. D. Changing fTiction from static to kinetic.
82. I Suppose the stone is launched with a speed of 3 mls and travels 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 longer di stance before com
ing to rest. B. The stone would now lravel a shorter di stance before com
ing 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. This 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. FQonH = FHonQ and FPonQ = FQonP because these are action/reaction pairs. Box: Q is slowing down and therefore must have a net force to the left. So from Newton's second law we also know that FHollQ > FponQ '
Stop to Think 5.6: Equal to. Each block is hanging in equilibrium, with no net force, so the upward tension force is mg.
