Essential University Physics, 3e (Wolfson) Chapter 7 Conservation of Energy
7 . 1 Conc e p t ua lQue s t i o ns 1) Is it possible for a system to have negative potential energy? A) Yes, Yes, as long as the kinetic energy is positive. B) Yes Yes,, as long as the total energy is positive. C) Yes Yes,, since the choice of the zero of potential energy is arbitrary. ) !o, beca"se the kinetic energy of a system m"st e#"al its potential energy. $) !o, beca"se this %o"l& have no physical meaning. Ans%er' C (ar' 1
) *%immers at a %ater park have a choice of t%o frictionless %ater sli&es as sho%n in the fig"re. Altho"gh both sli&es &rop over the same height, h, sli&e 1 is straight %hile sli&e is c"rve&, &ropping #"ickly at first an& then leveling o"t. +o% &oes the spee& v1 of a s%immer reaching the en& of sli&e 1 compares %ith v, the spee& of a s%immer reaching the en& of sli&e ?
A) v1 v B) v1 - v C) v1 v ) !o simple relationship e/ists bet%een v1 an& v beca"se %e &o not kno% the c"rvat"re of sli&e . Ans%er' C (ar' 1
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4) 5%o stones, one of mass m an& the other of mass m, are thro%n &irectly "p%ar& %ith the same velocity at the same time from gro"n& level an& feel no air resistance. 6hich statement abo"t these stones is tr"e? A) 5he heavier stone %ill go t%ice as high as the lighter one beca"se it initially ha& t%ice as m"ch kinetic energy. B) Both stones %ill reach the same height beca"se they initially ha& the same amo"nt of kinetic energy. C) At their highest point, both stones %ill have the same gravitational potential energy beca"se they reach the same height. ) At its highest point, the heavier stone %ill have t%ice as m"ch gravitational potential energy as the lighter one beca"se it is t%ice as heavy. $) 5he lighter stone %ill reach its ma/im"m height sooner than the heavier one. Ans%er' (ar' 1
7) 5%o i&entical balls are thro%n &irectly "p%ar&, ball A at spee& v an& ball B at spee& v, an& they feel no air resistance. 6hich statement abo"t these balls is correct? A) Ball B %ill go t%ice as high as ball A beca"se it ha& t%ice the initial spee&. B) Ball B %ill go fo"r times as high as ball A beca"se it ha& fo"r times the initial kinetic energy. C) 5he balls %ill reach the same height beca"se they have the same mass an& the same acceleration. ) At its highest point, ball B %ill have t%ice as m"ch gravitational potential energy as ball A beca"se it starte& o"t moving t%ice as fast. $) At their highest point, the acceleration of each ball is instantaneo"sly e#"al to zero beca"se they stop for an instant. Ans%er' B (ar' 1
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8) A bo/ of mass m is presse& against 9b"t is not attache& to) an i&eal spring of force constant k an& negligible mass, compressing the spring a &istance x. After it is release&, the bo/ sli&es "p a frictionless incline as sho%n in the fig"re an& event"ally stops. If %e repeat this e/periment %ith a bo/ of mass m
A) the lighter bo/ %ill go t%ice as high "p the incline as the heavier bo/. B) :"st as it moves free of the spring, the lighter bo / %ill be moving t%ice as fast as the heavier bo/. C) both bo/es %ill have the same spee& :"st as they move free of the spring. ) both bo/es %ill reach the same ma/im"m height on the incline. $) :"st as it moves free of the spring, the heavier bo/ %ill have t%ice as m"ch kinetic energy as the lighter bo/. Ans%er' A (ar' 1
2) A bo/ of mass m is presse& against 9b"t is not attache& to) an i&eal spring of force constant k an& negligible mass, compressing the spring a &istance x. After it is release&, the bo/ sli&es "p a frictionless incline as sho%n in the fig"re an& event"ally stops. If %e repeat this e/periment b"t instea& compress the spring a &istance of x
A) the bo/ %ill go "p the incline t%ice as high as before. B) :"st as it moves free of the spring, the bo/ %ill be traveling t%ice as fast as before. C) :"st as it moves free of the spring, the bo/ %ill be traveling fo"r times as fast as before. ) :"st as it moves free of the spring, the bo/ %ill have t%ice as m"ch kinetic energy as before. $) :"st before it is release&, the bo/ has t%ice as m"ch elastic potential energy as before. Ans%er' B (ar' 1 4 Copyright 0 12 3earson $&"cation, Inc.
;) A bo/ of mass m is presse& against 9b"t is not attache& to) an i&eal spring of force constant k an& negligible mass, compressing the spring a &istance x. After it is release&, the bo/ sli&es "p a frictionless incline as sho%n in the fig"re an& event"ally stops. If %e repeat this e/periment b"t instea& "se a spring having force constant k
A) the bo/ %ill go "p the incline t%ice as high as before. B) :"st as it moves free of the spring, the kinetic energy of the bo/ %ill be t%ice as great as before. C) :"st as it moves free of the spring, the spee& of the bo/ %ill be times as great as before. ) All of the above choices are correct. $) !one of the above choices is correct. Ans%er' (ar' 1
<) 6hen an ob:ect is solely "n&er the infl"ence of conservative forces, the s"m of its kinetic an& potential energies &oes not change. A) 5r"e B) =alse Ans%er' A (ar' 1
>) A ball &rops some &istance an& gains 4 of kinetic energy. o !@5 ignore air resistance. +o% m"ch gravitational potential energy &i& the ba ll lose? A) more than 4 B) e/actly 4 C) less than 4 Ans%er' A (ar' 1
1) A ball &rops some &istance an& loses 4 of gravitational potential energy. o !@5 ignore air resistance. +o% m"ch kinetic energy &i& the ball gain? A) more than 4 B) e/actly 4 C) less than 4 Ans%er' C (ar' 1
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11) Block 1 an& block have the same mass, m, an& are release& from the top of t%o incline& planes of the same height making 4 an& 2 angles %ith the horizontal &irection, respectively. If the coefficient of friction is the same in both cases, %hich o f the blocks is going faster %hen it reaches the bottom of its respective incline? A) 6e m"st kno% the act"al masses of the blocks to ans%er. B) Both blocks have the same spee& at the bottom. C) Block 1 is faster. ) Block is faster. $) 5here is not eno"gh information to ans%er the # "estion beca"se %e &o not kno% the val"e of the coefficient of kinetic friction. Ans%er' (ar' 1
1) A girl thro%s a stone from a bri&ge. Consi&er the follo%ing %ays she might thro% the stone. 5he spee& of the stone as it leaves her han& is the same in each case, an& air resistance is negligible. Case A' 5hro%n straight "p. Case B' 5hro%n straight &o%n. Case C' 5hro%n o"t at an angle of 78 above horizontal. Case ' 5hro%n straight o"t horizontally. In %hich case %ill the spee& of the stone be greatest %hen it hits the %ater belo%? A) Case A B) Case B C) Case C ) Case $) 5he spee& %ill be the same in all cases. Ans%er' $ (ar' 1
14) 6hich, if any, of the follo%ing statements concerning the %ork &one by a conservative force is !@5 tr"e? A) It can al%ays be e/presse& as the &ifference bet%een the initial an& final val"es of a potential energy f"nction. B) It is in&epen&ent of the path of the bo&y an& &epen&s only on the starting an& en&ing points. C) 6hen the starting an& en&ing points are the same, the total %ork is zero. ) All of the above statements are tr"e. $) !one of the above statements are tr"e. Ans%er' (ar' 1
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7 . 2 Pr o bl e ms 1) An <.kg block is release& from rest, %ith v1 . ms, on a ro"gh incline, as sho%n in the fig"re. 5he block moves a &istance of 1.2m &o%n the incline, in a time interval of .< s, an& ac#"ires a velocity of v 7. ms. +o% m"ch %ork &oes gravity &o on the block &"ring this process?
A) D<1 B) D1 C) D1 ) <1 $) 1 Ans%er' A (ar' 1
) Yo" &o 1;7 of %ork %hile p"lling yo"r sister back on a s%ing, %hose chain is 8.1 m long. Yo" start %ith the s%ing hanging vertically an& p"ll it "ntil the chain makes an angle of 4. %ith the vertical %ith yo"r sister is at rest. 6hat is yo"r sisterEs mass, ass"ming negligible friction? A) .> kg B) 1>.8 kg C) 2.4 kg ) <.7 kg Ans%er' A (ar' 8D
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4) An athlete stretches a spring an e/tra 7. cm beyon& its initial length. +o% m"ch energy has he transferre& to the spring, if the spring constant is 8.> !cm? A) 74 B) 74 k C) 74 k ) 74 Ans%er' A (ar' 8D
7) A tennis ball bo"nces on the floor three times. If each time it loses .F of its energy &"e to heating, ho% high &oes it rise after the thir& bo"nce, provi&e& %e release& it from the floor? A) 11 cm B) 11 cm C) 11 mm ) 17 cm Ans%er' A (ar' 8D
8) It re#"ires 2. of %ork is nee&e& to p"sh a .kg ob:ect from point A to point B of the frictionless ramp as sho%n in the fig"re. 6hat is the length s of the ramp from A to B?
Ans%er' .21 m (ar' 1
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2) A . g bea& sli&es along a frictionless %ire, as sho%n in the fig"re. At point A, the bea& is moving to the right b"t %ith negligible spee&.
9a) 6hat is the potential energy of the bea& at point A? 9b) 6hat is the kinetic energy of the bea& at point B? 9c) 6hat is the spee& of the bea& at point B? 9&) 6hat is the spee& of the bea& at point C ? Ans%er' 9a) . G 1 9b) . G 1 9c) 7.7 ms
9&) . ms
(ar' 1
;) A roller coaster of mass <. kg is moving %ith a spee& of . ms at position A as sho%n in the fig"re. 5he vertical height above gro"n& level at position A is m. !eglect friction.
9a) 6hat is the total mechanical energy of the roller coaster at point A? 9b) 6hat is the total mechanical energy of the roller coaster at point B? 9c) 6hat is the spee& of the roller coaster at point B? 9&) 6hat is the spee& of the roller coaster at point C ? Ans%er' 9a) 1.;4 G 18 9b) 1.;4 G 18 9c) 28.; ms 9&) 47.7 ms (ar' 1
<) A mass is presse& against 9b"t is not attache& to) an i&eal horizontal spring on a frictionless horizontal s"rface. After being release& from rest, the mass ac#"ires a ma/im"m spee& v an& a ma/im"m kinetic energy K . If instea& the mass initially compresses the spring t%ice as far' A) Its ma/im"m spee& %ill be v an& its ma/im"m kinetic energy %ill be K . B) Its ma/im"m spee& %ill be v an& its ma/im"m kinetic energy %ill be K . C) Its ma/im"m spee& %ill be v an& its ma/im"m kinetic energy %ill be K . ) Its ma/im"m spee& %ill be v an& its ma/im"m kinetic energy %ill be 7 K . $) Its ma/im"m spee& %ill be 7v an& its ma/im"m kinetic energy %ill be K . Ans%er' C (ar' 1
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>) A . kg mass is moving along the xa/is. 5he potential energy c"rve as a f"nction of position is sho%n in the fig"re. 5he kinetic energy of the ob:ect at the origin is 1 . 5he system is conservative, an& there is no friction.
9a) 6hat %ill be the kinetic energy at . m along the D xa/is? 9b) 6hat %ill be the spee& of the ob:ect at 2. m along the D xa/is? Ans%er' 9a) 7 9b) . ms (ar' 1
1) An <.m massless ro& is loosely pinne& to a frictionless pivot at 0, as sho%n in the fig"re. A very small 7.kg ball is attache& to the other en& of the ro&. 5he ball is hel& at A, %here the ro& makes a 4 angle above the horizontal, an& is release&. 5he ballro& assembly then s%ings freely %ith negligible friction in a vertical circle bet%een A an& B. 5he tension in the ro& %hen the ball passes thro"gh the lo%est point at D is closest to
A) 12 !. B) !. C) 1 !. ) < !. $) 7 !. Ans%er' A (ar' 1
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11) In the fig"re, a 7.kg ball is on the en& of a 1.2m rope that is fi/e& at 0. 5he ball is hel& at point A, %ith the rope horizontal, an& is given an initial &o%n%ar& velocity. 5he ball moves thro"gh three #"arters of a circle %ith no friction an& arrives at B, %ith the rope barely "n&er tension. 5he initial velocity of the ball, at point A, is closest to
A) 7. ms B) 8.2 ms C) 2.4 ms ) 2.> ms $) ;.> ms Ans%er' (ar' 1
1) In the fig"re, a very small toy race car of mass m is release& from rest on the looptheloop track. If it is release& at a height R above the floor, ho% high is it above the floor %hen it leaves the track, neglecting friction?
A) 1.2; R B) . R C) 1.8 R ) 1.44 R $) 1.8 R Ans%er' A (ar' 1
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14) In the fig"re, a 8.kg block is moving at 8. ms along a horizontal frictionless s"rface to%ar& an i&eal massless spring that is attache& to a %all. After the block colli&es %ith the spring, the spring is compresse& a ma/im"m &istance of .2< m. 6hat is the spee& of the block %hen it has move& so that the spring is compresse& to only onehalf of the ma/im"m &istance?
Ans%er' 7.4 ms (ar' 1
17) A 2.kg person &rops from rest a &istance of 1. m to a platform of negligible mass s"pporte& by an i&eal stiff spring of negligible mass. 5he platform &rops 2. cm before the person comes to rest. 6hat is the spring constant of the spring? A) .82 G 18 !m B) 4.> G 18 !m C) 8.78 G 17 !m ) 7.1 G 18 !m $) <.<4 G 17 !m Ans%er' (ar' 1
18) A springloa&e& &art g"n is "se& to shoot a &art straight "p into the air, an& the &art reaches a ma/im"m height of 7 meters above its point of release. 5he same &art is shot "p a secon& time from the same g"n, b"t this time the spring is compresse& only half as far 9compare& to the first shot). +o% far "p &oes the &art go this time? 9!eglect friction an& ass"me the spring is i&eal an& massless.) A) 2. m B) 1 m C) 4. m ) 7< m Ans%er' A (ar' 1
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12) A block sli&es &o%n a frictionless incline& ramp. If the ramp angle is 1;. an& its length is fin& the spee& of the block as it reaches the bottom of the ramp, ass"ming it starte& sli&ing from rest at the top. A) 14.1 ms B) 1; ms C) >.2 ms ) 7. ms Ans%er' A (ar' 8D
1;) Consi&er the motion of a 1.kg particle that moves %ith potential energy given by U 9 x) 9. Hm) x D 97. Hm) x. *"ppose the particle is moving %ith a spee& of 4. ms %hen it is locate& at x 1. m. 6hat is the spee& of the ob:ect %hen it is locate& at x 8. m? A) .14 ms B) 4. ms C) 7.2< ms ) 4.2; ms Ans%er' (ar' 1
1<) A car on a roller coaster starts at zero spee& at an elevation above the gro"n& of 2 m. It coasts &o%n a slope, an& then climbs a hill. 5he top of the hill is at an elevation of 12 m. 6hat is the spee& of the car at the top of the hill? !eglect any frictional effects. A) 17 ms B) 1< ms C) 1 ms ) >. ms $) 2. ms Ans%er' A (ar' 1
1>) A pro:ectile is fire& from gro"n& level at an ang le of 7. above horizontal at a spee& of 4. ms. 6hat is the spee& of the pro:ectile %hen it has reache& a height e#"al to 8.F of its ma/im"m height? A) 2. ms B) ;.7 ms C) <.; ms ) 2.; ms $) <.1 ms Ans%er' (ar' 1
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) A very small 1g ob:ect is attache& to one en& of a massless 1cm ro& that is pivote& %itho"t friction abo"t the opposite en&. 5he ro& is hel& vertical, %ith the ob:ect at the top, an& release&, allo%ing the ro& to s%ing. 6hat is the spee& of the ob:ect at the instant that the ro& is horizontal? A) .;1 ms B) 7. ms C) 1.7 ms ) .< ms $) .< ms Ans%er' (ar' 1
1) A .kg ob:ect is moving %itho"t friction along the xa/is. 5he potential energy c"rve as a f"nction of position is sho%n in the fig"re, an& the system is conservative. If the spee& of the ob:ect at the origin is 7. ms, %hat %ill be its spee& at ;. m along the D xa/is?
A) 7. ms B) 7. ms C) 7.7 ms ) 7.2 ms $) >.< ms Ans%er' B (ar' 1
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) A small hockey p"ck sli&es %itho"t friction over the icy hill sho%n in the fig"re an& lan&s 2. m from the foot of the cliff %ith no air resistance. 6hat %as its spee& v at the bottom of the hill?
A) .< ms B) 1;.7 ms C) 17.7 ms ) 14.; ms $) 7.;1 ms Ans%er' (ar' 1
4) An ob:ect is attache& to a hanging "nstretche& i&eal an& massless spring an& slo%ly lo%ere& to its e#"ilibri"m position, a &istance of 2.7 cm belo% the starting point. If instea& of having been lo%ere& slo%ly the ob:ect %as &roppe& from rest, ho% far then %o"l& it then stretch the spring at ma/im"m elongation? A) 14 cm B) >.1 cm C) 2.7 cm ) 1< cm $) 2 cm Ans%er' A (ar' 8D
7) In the fig"re, a st"nt car &river negotiates the frictionless track sho%n in s"ch a %ay that the car is barely in contact %ith the track at the top of the loop. 5he ra&i"s of the track is >.> m an& the mass of the car is 1< kg. =in& the magnit"&e of the force of the car on the track %hen the car is at point A. Yo" can treat the car as a point mass.
Ans%er' 84, ! (ar' 1
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8) A 8.kg skier starting from rest travels m &o%n a hill that has a . slope an& a "niform s"rface. 6hen the skier reaches the bottom of the hill, her spee& is 4. ms. 9a) +o% m"ch %ork is &one by friction as the skier comes &o%n the hill? 9b) 6hat is the magnit"&e of the friction force if the skier travels &irectly &o%n the hill? Ans%er' 9a) 1.1 G 17 9b) 88.4 ! (ar' 1
2) A 8.kg ob:ect moves clock%ise aro"n& a 8. cm ra&i"s circ"lar path. At one location, the spee& of the ob:ect is 7. ms. 6hen the ob:ect ne/t ret"rns to this same location, the spee& is 4. ms. 9a) +o% m"ch %ork %as &one by nonconservative 9&issipative) forces as the ob:ect move& once aro"n& the circle? 9b) If the magnit"&e of the above nonconservative 9&issipative) forces acting on the ob:ect is constant, %hat is the val"e of this magnit"&e? Ans%er' 9a) 1;.8 9b) 8.8; ! (ar' 1
;) In the fig"re, a block of mass m is moving along the horizontal frictionless s"rface %ith a spee& of 8.; ms. If the slope is 11. an& the coefficient of kinetic friction bet%een the block an& the incline is .2, ho% far &oes the block travel "p the incline?
Ans%er' 4.; m (ar' 1
<) An ob:ect of mass 7. kg starts at rest from the top of a ro"gh incline& plane of height 1 m as sho%n in the fig"re. If the spee& of the ob:ect at the bottom of the incline& plane is 1 ms, ho% m"ch %ork &oes friction &o on this ob:ect as it sli&es &o%n the incline?
Ans%er' 1> (ar' 1
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>) An .<kg block is hel& in place against the spring by a 2;! horizontal e/ternal force 9see the fig"re). 5he e/ternal force is remove&, an& the block is pro:ecte& %ith a velocity v1 1. ms "pon separation from the spring. 5he block &escen&s a ramp an& has a velocity v 1.> ms at the bottom. 5he track is frictionless bet%een points A an& B. 5he block enters a ro"gh section at B, e/ten&ing to E . 5he coefficient of kinetic friction over this section is .4>. 5he velocity of the block is v4 1.7 ms at C . 5he block moves on to D, %here it stops. 5he spring constant of the spring is closest to
A) 4> !m. B) 2 !m. C) !m. ) 12 !m. $) 11 !m. Ans%er' A (ar' 1
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4) A 1.4;kg block is hel& in place against the spring by a ;7! horizontal e/ternal force 9see the fig"re). 5he e/ternal force is remove&, an& the block is pro:ecte& %ith a velocity v1 1. ms "pon separation from the spring. 5he block &escen&s a ramp an& has a velocity
at
the bottom. 5he track is frictionless bet%een points A an& B. 5he block enters a ro"gh section at B, e/ten&ing to E . 5he coefficient of kinetic friction over this section is .7. 5he velocity of the block is
at C . 5he block moves on to D, %here it stops. 5he initial compression of
the spring is closest to'
A) .; cm. B) 1.7 cm. C) .>2 cm. ) 8.4 cm. $) 4.2 cm. Ans%er' A (ar' 8D
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41) A 1.<2kg block is hel& in place against the spring by a <1! horizontal e/ternal force 9see the fig"re). 5he e/ternal force is remove&, an& the block is pro:ecte& %ith a velocity v1 1. ms "pon separation from the spring. 5he block &escen&s a ramp an& has a velocity
at
the bottom. 5he track is frictionless bet%een points A an& B. 5he block enters a ro"gh section at B, e/ten&ing to E . 5he coefficient of kinetic friction over this section is .<. 5he velocity of the block is v4 1.7 ms at C . 5he block moves on to D, %here it stops. 5he height h of the ramp is closest to
A) 11 B) ;.4 C) 18 ) 1; $) 1< Ans%er' A (ar' 8D
4) A .8kg bo/, sli&ing on a ro"gh horizontal s"rface, has a spee& of 1. ms %hen it makes contact %ith a spring 9see the fig"re). 5he block comes to a momentary halt %hen the compression of the spring is 8. cm. 5he %ork &one by the friction, from the instant the block makes contact %ith the spring "ntil is comes to a momentary halt, is .8 .
9a) 6hat is the spring constant of the spring? 9b) 6hat is the coefficient of kinetic friction bet%een the bo/ an& the ro"gh s"rface? Ans%er' 9a) 17 !m 9b) .71 (ar' 1
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