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Chapter 5 I Force and Motion-l

zontal force F acts on one endof the rope. (u) Show that therope must sag, even if only by animperceptible amount. Then, os-suming that the sag is negligible, Ffiffi. 5-6* ProblemT2.find (b) the acceleration of ropeand block, (c) the force on the block from the rope, and (d)the tension in the rope at its midpoint.73 A worker drags a crate across a factory floor by pullingon a rope tied to the crate.The worker exerts a force of magni-tude F - 450 N on the rope, which is inclined at an upwardangle 0 : 3Bo to the horizontal, and the floor exerts a horizon-tal force of magnitude / - 1-25 N that opposes the motion.Calculate the magnitude of the acceleration of the crate if(a) its mass is 310 kg and (b) its weight is 310 N. ssm

74 Three forces act on a particle that moves with unchang-ing velocity i - (2 m/s)i - (7 m/s)i. Two of the forcesare F, : (2 N)i + (: N)j + (-lN)k and Fr: (-5 N)i +(A X)j + e2 N)k. What is the third force?75 A 52 kg circus performer is to slide down a rope that willbreak if the tension exceeds 425 N. (u) What happens if theperformer hangs stationary on the rope? (b) At what magni-tude of acceleration does the performer just avoid breakingthe rope?76 An 80 kg man drops to a concrete patio from a window0.50 m above the patio. He neglects to bend his knees on land-itrg,taking 2.0 cm to stop. (u) What is his average accelerationfrom when his feet first touch the patio to when he stops? (b)What is the magnitude of the average stopping force exertedon him by the patio?7 In Fig. 5-63,4.0 kg block A and 6.0 kg block B are con-n ted by a string of negligible mass. Force Fa : (12 N)i actson block A; force Fu : (24 N)i acts on block B. What is thetension in the string?

Ff;ffi. S-S3 ProblemTT.

speed; if the engine provides only 2200 N, the craft acceleratesdownward at 0.39 mlsz. (a) What is the weight of the landingcraft in the vicinity of Callisto's surface? (b) What is the massof the craft? (c) What is the magnitude of the free-fall acceler-ation near the surface of Callisto?81 An object is hung from a spring balance attached to theceiling of an elevator cab. The balance reads 65 N when thecab is standing still. What is the reading when the cab is mov-ing upward (a) with a constant speed of 7.6 mls and (b) with aspeed of 7 .6 m/s while decelerating at a rate of 2.4 mls2?

S2 In Fig. 5-65, a force F ofa FedEx box of mass wlz: L.0kg. The force is directed up aplane tilted by 0 : 37". Thebox is connected by a cord toa UPS box of mass tn1 : 3.0 kgon the floor. The floor, plane,and pulley are frictionless, andthe masses of the pulley andcord are negligible. What is the

magnitude 12 N is applied to

78 In the overhead view ofFig. 5-64, five forces pull on abox of mass m :4.0 kg. Theforce magnitudes are Fr : lIN, Fz: 17 N, Fz : 3.0 N, Fq:14 N, and Fs:5.0 N, and angle0a is 30". Find the box's acceler-ation (u) in unit-vector nota-tion and as (b) a magnitude and(c) an angle relative to the posi-tive direction of the x axis.

Vq A certain force gives an object of mass w\ arrr accelera-tion of I2.0 mlsz and an object of mass wt2 ?rr acceleration of3.30 m/s2. What acceleration would the force give to an objectof mass (u) *, - tft1and (b) *, * ml ssm

80 Imagine a landing craft approaching the surface ofCallisto, one of Jupiter's moons. If the engine provides anupward force (thrust) of 3260 \ the craft descends at constant

F$m. S-SS Problem82.

tension in the cord?83 A certain particle has a weight of 22 N at a point whereg - 9.8 m/s2. What are its (a) weight and (b) mass at a pointwhere g - 4.9 m/s2? What are its (c) weight and (d) mass if itis moved to a point in space where g - 0?

84 Compute the weight of a 75 kg space ranger (u) onEarth, (b) on Mars, where g - 3.7 mls2,and (c) in interplane-tary space, where g - 0. (d) What is the ranger's mass at eachlocation?85 A 1400 kg jet engine is fastened to the fuselage of apassenger jet by just three bolts (this is the usual practice).Assume that each bolt supports one-third of the load.(a) Calculate the force on each bolt as the plane waits in linefor clearance to take off. (b) During flight, the plane encoun-ters turbulence, which suddenly imparts an upward verticalacceleration of 2.6 mlsz to the plane. Calculate the force oneach bolt now.

S6 An B0 kg person is parachuting and experiencing adownward acceleration of 2.5 mls2. The mass of the parachuteis 5.0 kg. (u) What is the upward force on the open parachutefrom the air? (b) What is the downward force on the para-chute from the person?

87 Suppose that in Fig. 5-I3,the masses of the blocks are2.0kg and 4.0 kg. (a) Which mass should the hanging block haveif the magnitude of the acceleration is to be as large as possi-ble? What then are (b) the magnitude of the acceleration and(c) the tension in the cord?

88 You pull a short refrigerator with a constant force Facross a greased (frictionless) floor, either with F hori zontal(case 1) or with F tilted upward at an angle 0 (case 2).(a) What is the ratio of the refrigerator's speed in case 2 to itsspeed in case 1 if you pull for a certain time /? (b) What is thisratio if you pull for a certain distanc e d?

89 A spaceship lifts off vertically from the Moon, where g -1.6 m/sz.Ifthe ship has an upward acceleration of 1.0 m/s2 as itlifts off, what is the magnitude of the force exerted by the shipon its pilot, who weighs 735 N on Earth?90 Compute the initial upward acceleration of a rocket ofmass 1.3 x 104 kg if the initial upward force produced by its

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FEffi. 5-S4 Problem 78.

Chapter 6 I Force and Motion-llat 35" to the horizontal. As the child slides down the slope, hehas an acceleration that has a magnitude of 0.50 m/sZ and thatis directed up the slope. What is the coefficient of kineticfriction between the child and the slope?

ffi4 In Fig. 6-58, a stuntmandrives a car (without negativelift) over the top of a hill, thecross section of which can beapproximated by a circle of ra-dius R - 250 m. What is thegreatest speed at which he candrive without the car leaving the

F$ffi" S-Sq Problem 87.

ES A car weighingI}.T kN and traveling at I3.4 m/s withoutnegative lift attempts to round an unbanked curve with aradius of 61.0 m. (a) What magnitude of the frictional force onthe tires is required to keep the car on its circular path? (b) Ifthe coefflcient of static friction between the tires and the roadis 0.350, is the attempt at taking the curve successful? ssM

E6 A 100 N force, directed at an angle 0 above a horizontalfloor, is applied to a25.0 kg chair sitting on the floor. If 0: 0o,what are (a) the hortzontal component F1, of the applied forceand (b) the magnitude Fry of the normal force of the floor onthe chair? If 0:30.0o, what are (.) Fh and (d) F*? If 0 -60.0', what are (e) F1, and (0 F"? Now assume that the coeffi-cient of static friction between chair and floor is 0.420. Doesthe chair slide or remain at rest if g is (g) 0', (h) 30.0", and(i) 60.0"?

87 A student, crazed by finalexams, uses a force P of magni-tude 80 N and angle 0 - J0" topush a 5.0 kg block across theceiling of his room (Fig. 6-59). Ifthe coefficient of kinetic frictionbetween the block and the ceil-ing is 0.40, what is the magnitude of the block's acceleration?

ffi& A certain string can withstand a maximum tension of40 N without breaking. A child ties a 0.37 kg stone to oneend and, holding the other end, whirls the stone in a vertical cir-cle of radius 0.91 m, slowly increasing the speed until the stringbreaks. (u) Where is the stone on its path when the stringbreaks? (b) What is the speed of the stone as the string breaks?

89 You must push a crate across a floor to a docking bay.The crate weighs 165 N. The coefflcient of static frictionbetween crate and floor is 0.510, and the coefficient of kineticfriction is 0.32. Your force on the crate is directed hortzontally.(a) What magnitude of your push puts the crate on the vergeof sliding? (b) With what magnitude must you then push tokeep the crate moving at a constant velocity? (c) If, instead,you then push with the same magnitude as the answer to (a),what is the magnitude of the crate's acceleration?9$ A child weighin g 140 N sits at rest at the top of a play-ground slide that makes an angle of 25" with the horizontal.The child keeps from sliding by holding onto the sides of theslide. After letting go of the sides, the child has a constantacceleration of 0.86 m/s2 (down the slide, of course). (u) Whatis the coefficient of kinetic friction between the child and theslide? (b) What maximum and minimum values for the coeffi-cient of static friction between the child and the slide areconsistent with the information given here?

F$ffi. S-Sffi Problem 84.

road at the top of the hill?

Pt A filing cabinet weighing 556 N rests on the floor. Thecoefficient of static friction between it and the floor is 0.68,and the coefflcient of kinetic friction is 0.56. In four differentattempts to move it, it is pushed with horizontal forces of mag-nitudes (a) 222 N, (b) 334 N, (c) 445 N, and (d) 556 N. For eachattempt, calculate the magnitude of the frictional force on itfrom the floor. (The cabinet is initially at rest.) (e) In which ofthe attempts does the cabinet move? ssM

Wg A sling-thrower puts a stone (0.250 kg) in the sling'spouch (0.010 kg) and then begins to make the stone andpouch move in a vertical circle of radius 0.650 m. The cordbetween the pouch and the person's hand has negligible massand will break when the tension in the cord is 33.0 N or more.Suppose the sling-thrower could gradually increase the speedof the stone. (a) Will the breaking occur at the lowest point ofthe circle or at the highest point? (b) At what speed of thestone will that breaking occur?PS A four-person bobsled (total mass - 630 kg) comesdown a straightaway at the start of a bobsled run. The straight-away is 80.0 rn long and is inclined at a constant angle of I0.2"with the horrzontaL Assume that the combined effects of fric-tion and air drag produce on the bobsled a constant force of62.0 N that acts parallel to the incline and up the incline.Answer the following questions to three significant digits.(u) If the speed of the bobsled at the start of the run is6.20 m/s, how long does the bobsled take to come down thestraightaway? (b) Suppose the crew is able to reduce theeffects of friction and air drag to 42.0 N. For the same initialvelocity, how long does the bobsled now take to come downthe straightaway?S4 In Fig. 6-60,force F i. ap-plied to a crate of mass m on afloor where the coefficient ofstatic friction between crateand floor is p,r. Angle 0 is ini-tially 0o but is gradually in-creased so that the force vector

Ffffi. 6-6S Problem94.

rotates clockwise in the figure. During the rotation, the magni-tude F of the force is continuously adjusted so that the crate isalways on the verge of sliding. For ;tr,:0.70, (a) plot the ratioFlmg versus g and (b) determine the angle gi,,r at which the ra-tio approaches an infinite value. (c) Does lubricating the floorincrease or decreaSe 0i,t, or is the value unchanged? (d) Whatis 0i.,r for trr,, : 0.60?

ry$ In the early afternoon, a car is parked on a street thatruns down a steep hill, atan angle of 35.0'relative to the hori-zontal. Just then the coefficient of static friction between thetires and the street surface is 0.725. Later, after nightfall, asleet storm hits the area, and the coefficient decreases dueto both the ice and a chemical change in the road surfacebecause of the temperature decrease" By what percentagemust the coefficient decrease if the car is to be in danger ofsliding down the street?S6 In Fig. 6-6I, block 1 of massmass tn2: 1.0 kg are con-nected by a string of negligiblemass. Block 2 is pushed by

tnt : 2.0 kg and block 2 of

force F of magnitude 20 N andangle 0 : 35". The coefficientof kinetic friction between FXffi. 6-&t Problem 96.

each block and the horrzontal surface is 0.20. What is the ten-sion in the string?S7 In Fig. 6-62 a fastidiousworker pushes directly alongthe handle of a mop with aforce F. ffr. handle is at an an-gle 0 with the vertical, and Fsand Fr are the coefflcients ofstatic and kinetic friction be-tween the head of the mop andthe floor. Ignore the mass of thehandle and assume that all themop's mass m rs in its head. (a)If the mop head moves along the floor with a constant velocity,then what is F? (b) Show that if g is less than a certain value gs,

then F lrtitt directed along the handle) is unable to move themop head. Find 0s.

9S A circular curve of highway is designed for traffic movingat 60 km/h. Assume the trafflc consists of cars without nega-tive lift. (u) If the radius of the curve is 150 m, what is thecorrect angle of banking of the road? (b) If the curve were notbanked, what would be the minimum coefficient of frictionbetween tires and road that would keep traffic from skiddingout of the turn when traveling at60 km/h?#9 A block slides with constant velocity down an inclinedplane that has slope angle 0. The block is then projected up thesame plane with an initial speed yo. (a) How far up the planewill it move before coming to rest? (b) After the block comesto rest, will it slide down the plane again? Give an argument toback your answer. ssM

10S In Fig. 6-63, a block weigh-ing 22 N is held at rest against avertical wall by a horizontal forceF of magnitude 60 N. The coeffi-cient of static friction between thewall and the block is 0.55, and thecoefflcient of kinetic friction be-tween them is 0.38. In six experi-ments, a second force F ir appliedto the block and directed parallelto the wall with these magnitudes and directions: (a) 34 N, up,(b) 12 N, up, (c) a8 N, up, @) 62 N, up, (e) 10 N, down, and (f)18 N, down. In each experiment, what is the magnitude of thefrictional force on the block? In which does the block move(g) .rp the wall and (h) down the wall? (i) In which is the fric-tional force directed down the wall?t 0t When a small 2.0 g coin is placed at a radius of 5.0 cm on ahorizontal turntable that makes three full revolutions in 3.14 s, thecoin does not slip. What are (a) the coin's speed, the (b) magni-tude and (c) direction (radially inward or outward) of the coin'sacceleration, and the (d) magnitude and (e) direction (inward oroutward) of the frictional force on the coin? The coin is on theverge of slipping if it is placed at a radius of 10 cm. (f) What is thecoefficient of static friction between coin and turntable?t $A A child places a picnic basket on the outer rim of amerry-go-round that has a radius of 4.6 m and revolves onceevery 30 s. (u) What is the speed of a point on that rim?(b) What is the lowest value of the coefficient of static frictionbetween basket and merry-go-round that allows the basket tostay on the ride?

Ffiffi. &-&& Problem9T.

Ffiffi. e-e$ Problem 100.

Problems

A 1.5 kg box is initially at rest on a horrzontal surfacen at t:0 ahortzontal force F - (1.8/)iN (with /in sec-

onds) is applied to the box. The acceleration of the box as a func-tion of time /^is given by d:0 for 0<t<2.8s andd - (I.Zt - 2.4)i m/s2 for t > 2.8 s. (a) What is the coefficient ofstatic friction between the box and the surface? (b) lVhat is thecoefficient of kinetic friction between the box and the surface?

$ S4 A trunk with a weight of 220 N rests on the floor. Thecoefficient of static friction between the trunk and the floor is0.4I, and the coefficient of kinetic friction is 0.32. (a) What isthe magnitude of the minimum horizontal force with which aperson must push on the trunk to start it moving? (b) Oncethe trunk is moving, what magnitude of horizontal force mustthe person apply to keep it moving with constant velocity?(.) If the person continued to push with the force used tostart the motion, what would be the magnitude of the trunk'sacceleration?$S5 A warehouse worker exerts a constant hortzontal forceof magnitude 85 N on a 40 kg box that is initially at rest on thehorizontal floor of the warehouse. When the box has moved adistance of I.4 m,its speed is 1.0 m/s. What is the coefficient ofkinetic friction between the box and the floor? ssM

S S6 Imagine that the standard kilogram is placed on Earth'sequator, where it moves in a circle of radius 6.40 X 106 m(Earth's radius) at a constant speed of 465 mis due to Earth'srotation. (a) What is the magnitude of the centripetal force onthe standard kilogram during the rotation? Imagine that thestandard kilogram hangs from a spring balance at that loca-tion and assume that it would weigh exactly 9.80 N if Earthdid not rotate. (b) What is the reading on the spring balance;that is, what is the magnitude of the force on the springbalance from the standard kilogram?t ff? As a 40 N block slides down a plane that is inclined at25" to the hortzontal its acceleration is 0.80 mlsz, directed upthe plane. What is the coefficient of kinetic friction betweenthe block and the plane?{ 0S Luggage is transported from one location to another inan airport by a conveyor belt. At a certain location, the beltmoves down an incline that makes an angle of.2.5" with thehorizontal. Assume that with such a slight angle there is noslipping of the luggage. Determine the magnitude of thefrictional force by the belt on a box weighin g 69 N when thebox is on the inclined portion of the belt and the belt speed is(u) 0 and constant,(b) 0.65 m/s and constant, (c) 0.65 m/s and in-creasing at a rate of 0.20 mlsz., (d) 0.65 m/s and decreasing at arate of 0.20 mls2,and (e) 0.65 m/s and increasing at arate of 0.57m/s2. (f) For which of these five situations is the frictional forcedirected down the incline?Sffi$ In Fig. 6-64, a 5.0 kg blockis sent sliding up a plane inclinedat 0 :37" while a horizontalforce F of magnitude 50 N actson it. The coefficient of kineticfriction between block and planeis 0.30. What are the (a) magni-tude and (b) direction (np ordown the plane) of the block'sacceleration? The block's initial speed is 4.0 m/s. (c) How far upthe plane does the block go? (d) Vfhen it reaches its highestpoint, does it remain at rest or slide back down the plane?

Ffiffi. S-64 Problem 109.

each block and the horrzontal surface is 0.20. What is the ten-sion in the string?S7 In Fig. 6-62 a fastidiousworker pushes directly alongthe handle of a mop with aforce F. ffr. handle is at an an-gle 0 with the vertical, and Fsand Fr are the coefflcients ofstatic and kinetic friction be-tween the head of the mop andthe floor. Ignore the mass of thehandle and assume that all themop's mass m rs in its head. (a)If the mop head moves along the floor with a constant velocity,then what is F? (b) Show that if g is less than a certain value gs,

then F lrtitt directed along the handle) is unable to move themop head. Find 0s.

9S A circular curve of highway is designed for traffic movingat 60 km/h. Assume the trafflc consists of cars without nega-tive lift. (u) If the radius of the curve is 150 m, what is thecorrect angle of banking of the road? (b) If the curve were notbanked, what would be the minimum coefficient of frictionbetween tires and road that would keep traffic from skiddingout of the turn when traveling at60 km/h?#9 A block slides with constant velocity down an inclinedplane that has slope angle 0. The block is then projected up thesame plane with an initial speed yo. (a) How far up the planewill it move before coming to rest? (b) After the block comesto rest, will it slide down the plane again? Give an argument toback your answer. ssM

10S In Fig. 6-63, a block weigh-ing 22 N is held at rest against avertical wall by a horizontal forceF of magnitude 60 N. The coeffi-cient of static friction between thewall and the block is 0.55, and thecoefflcient of kinetic friction be-tween them is 0.38. In six experi-ments, a second force F ir appliedto the block and directed parallelto the wall with these magnitudes and directions: (a) 34 N, up,(b) 12 N, up, (c) a8 N, up, @) 62 N, up, (e) 10 N, down, and (f)18 N, down. In each experiment, what is the magnitude of thefrictional force on the block? In which does the block move(g) .rp the wall and (h) down the wall? (i) In which is the fric-tional force directed down the wall?t 0t When a small 2.0 g coin is placed at a radius of 5.0 cm on ahorizontal turntable that makes three full revolutions in 3.14 s, thecoin does not slip. What are (a) the coin's speed, the (b) magni-tude and (c) direction (radially inward or outward) of the coin'sacceleration, and the (d) magnitude and (e) direction (inward oroutward) of the frictional force on the coin? The coin is on theverge of slipping if it is placed at a radius of 10 cm. (f) What is thecoefficient of static friction between coin and turntable?t $A A child places a picnic basket on the outer rim of amerry-go-round that has a radius of 4.6 m and revolves onceevery 30 s. (u) What is the speed of a point on that rim?(b) What is the lowest value of the coefficient of static frictionbetween basket and merry-go-round that allows the basket tostay on the ride?

Ffiffi. &-&& Problem9T.

Ffiffi. e-e$ Problem 100.

Problems

A 1.5 kg box is initially at rest on a horrzontal surfacen at t:0 ahortzontal force F - (1.8/)iN (with /in sec-

onds) is applied to the box. The acceleration of the box as a func-tion of time /^is given by d:0 for 0<t<2.8s andd - (I.Zt - 2.4)i m/s2 for t > 2.8 s. (a) What is the coefficient ofstatic friction between the box and the surface? (b) lVhat is thecoefficient of kinetic friction between the box and the surface?

$ S4 A trunk with a weight of 220 N rests on the floor. Thecoefficient of static friction between the trunk and the floor is0.4I, and the coefficient of kinetic friction is 0.32. (a) What isthe magnitude of the minimum horizontal force with which aperson must push on the trunk to start it moving? (b) Oncethe trunk is moving, what magnitude of horizontal force mustthe person apply to keep it moving with constant velocity?(.) If the person continued to push with the force used tostart the motion, what would be the magnitude of the trunk'sacceleration?$S5 A warehouse worker exerts a constant hortzontal forceof magnitude 85 N on a 40 kg box that is initially at rest on thehorizontal floor of the warehouse. When the box has moved adistance of I.4 m,its speed is 1.0 m/s. What is the coefficient ofkinetic friction between the box and the floor? ssM

S S6 Imagine that the standard kilogram is placed on Earth'sequator, where it moves in a circle of radius 6.40 X 106 m(Earth's radius) at a constant speed of 465 mis due to Earth'srotation. (a) What is the magnitude of the centripetal force onthe standard kilogram during the rotation? Imagine that thestandard kilogram hangs from a spring balance at that loca-tion and assume that it would weigh exactly 9.80 N if Earthdid not rotate. (b) What is the reading on the spring balance;that is, what is the magnitude of the force on the springbalance from the standard kilogram?t ff? As a 40 N block slides down a plane that is inclined at25" to the hortzontal its acceleration is 0.80 mlsz, directed upthe plane. What is the coefficient of kinetic friction betweenthe block and the plane?{ 0S Luggage is transported from one location to another inan airport by a conveyor belt. At a certain location, the beltmoves down an incline that makes an angle of.2.5" with thehorizontal. Assume that with such a slight angle there is noslipping of the luggage. Determine the magnitude of thefrictional force by the belt on a box weighin g 69 N when thebox is on the inclined portion of the belt and the belt speed is(u) 0 and constant,(b) 0.65 m/s and constant, (c) 0.65 m/s and in-creasing at a rate of 0.20 mlsz., (d) 0.65 m/s and decreasing at arate of 0.20 mls2,and (e) 0.65 m/s and increasing at arate of 0.57m/s2. (f) For which of these five situations is the frictional forcedirected down the incline?Sffi$ In Fig. 6-64, a 5.0 kg blockis sent sliding up a plane inclinedat 0 :37" while a horizontalforce F of magnitude 50 N actson it. The coefficient of kineticfriction between block and planeis 0.30. What are the (a) magni-tude and (b) direction (np ordown the plane) of the block'sacceleration? The block's initial speed is 4.0 m/s. (c) How far upthe plane does the block go? (d) Vfhen it reaches its highestpoint, does it remain at rest or slide back down the plane?

Ffiffi. S-64 Problem 109.