Chapter 4 Newton s Laws

Transcription

1 Chapter 4 Newton s Laws Conceptual Problems 3 [SSM] You are ridin in a limousine that has opaque windows that do not allow ou to see outside. he car is on a flat horizontal plane, so the car can accelerate b speedin up, slowin down, or turnin. Equipped with just a small heav object on the end of a strin, how can ou use it to determine if the limousine is chanin either speed or direction? Can ou determine the limousine s velocit? Determine the Concept In the limo ou hold one end of the strin and suspend the object from the other end. If the strin remains vertical, the reference frame of the limo is an inertial reference frame. 9 You are ridin in an elevator. Describe two situations in which our apparent weiht is reater than our true weiht. Determine the Concept Your apparent weiht is the readin of a scale. If the acceleration of the elevator (and ou) is directed upward, the normal force eerted b the scale on ou is reater than our weiht. You could be movin down but slowin or movin up and speedin up. In both cases our acceleration is upward. 3 Suppose a block of mass m rests on a block of mass m and the combination rests on a table as shown in iure ell the name of the force and its cateor (contact versus action-at-a-distance) for each of the followin forces; (a) force eerted b m on m, (b) force eerted b m on m, (c) force eerted b m on the table, (d) force eerted b the table on m, (e) force eerted b Earth on m. Which, if an, of these forces constitute a Newton s third law pair of forces? Determine the Concept (a) he force eerted b m on m. (b) he force eerted b m on m. (c) he force eerted b m on the table. (d) he force eerted b the table on m. (e) he force eerted b Earth on m. Normal force, contact Normal force, contact Normal force, contact Normal force, contact Gravitational force, action-at-a-distance 69

2 70 Chapter 4 he Newton s third law force pairs are the two normal forces between the two blocks and the normal force between the table and the bottom block. he ravitational force has a third law force pair that acts on Earth and, so, is not in the question set. A.5-k object hans at rest from a strin attached to the ceilin. (a) Draw a free bod diaram of the object, indicate the reaction force to each force drawn, and tell what object the reaction force acts on. (b) Draw a free bod diaram of the strin, indicate the reaction force to each force drawn, and tell what object each reaction force acts on. Do not nelect the mass of the strin. Determine the Concept he force diarams will need to include forces eerted b the ceilin, on the strin, on the object, and forces eerted b Earth. (a) r b strin on object.5 k Object r b Earth on object r r orce b strin on object b Earth on object hird-law Pair r b r b object on strin object on Earth (b) r b ceilin on strin Strin r b Earth on strin r b object on strin r r r orce b ceilin on strin b Earth on strin b object on strin hird-law Pair r b strin on ceilin r b r b strin on Earth strin on object Estimation and Approimation 7 Estimate the force eerted on the oalie s love b the puck when he catches a hard slap shot for a save. Picture the Problem Suppose the oalie s love slows the puck from 60 m/s to zero as it recoils a distance of 0 cm. urther, assume that the puck s mass is 00. Because the force the puck eerts on the oalie s love and the force the

3 Newton s Laws 7 oalie s love eerts on the puck are action-and-reaction forces, the are equal in manitude. Hence, if we use a constant-acceleration equation to find the puck s acceleration and Newton s second law to find the force the love eerts on the puck, we ll have the manitude of the force eerted on the oalie s love. Appl Newton s second law to the puck as it is slowed b the oalie s love to epress the manitude of the force the love eerts on the puck: Use a constant-acceleration equation to relate the initial and final speeds of the puck to its acceleration and stoppin distance: m a () love on puck v v0 + puck puck puck ( Δ ) puck a Solvin for a puck ields: a puck v v 0 ( Δ ) puck Substitute for a puck in equation () to obtain: loveon puck m puck ( v v ) ( Δ ) puck 0 evaluate love on puck : love on puck ( ) ( 0.00 k) 0 ( 60 m/s) ( 0.0 m) 3.6 kn Remarks: he force on the puck is about 800 times its weiht. Newton s irst and Second Laws: Mass, Inertia, and orce 35 A bullet of mass k movin at 500 m/s impacts a tree stump and penetrates 6.00 cm into the wood before comin to rest. (a) Assumin that the acceleration of the bullet is constant, find the force (includin direction) eerted b the wood on the bullet. (b) If the same force acted on the bullet and it had the same speed but half the mass, how far would it penetrate into the wood? Picture the Problem Choose a coordinate sstem in which the + direction is in the direction of the motion of the bullet and use Newton s second law and a constant- acceleration equation to epress the relationship between stoppin and the mass of the bullet and its displacement as it is brouht to rest in the block of wood. (a) Appl Newton s second law to the bullet to obtain: stoppin () ma

4 7 Chapter 4 Use a constant-acceleration equation to relate the bullet s initial and final speeds, acceleration, and stoppin distance: vf vi + aδ or, because v f 0, 0 v a Δ a i + vi Δ Substitute for a in equation () to obtain: evaluate stoppin : stoppin vi m () Δ ( ) ( 500 m/s.80 0 k ) 3 stoppin ( 6.00 cm) 3.8 kn where the minus sin indicates that stoppin opposes the motion of the bullet. (b) Solvin equation () for Δ ields: or m m and Δ Δ : Evaluate this epression for m' m to obtain: vi Δ m (3) stoppin v Δ' m' i stoppin vi Δ' m (4) 4 stoppin Dividin equation (4) b equation (3) ields: v m Δ' 4 Δ m or ' Δ Δ i stoppin vi stoppin evaluate Δ : ( 6.00 cm) 3.00 cm Δ' Mass and Weiht 45 [SSM] o train astronauts to work on the moon, where the acceleration due to ravit is onl about /6 of that on Earth, NASA submeres them in a tank of water. If an astronaut, who is carrin a backpack, air conditionin unit, oen suppl, and other equipment, has a total mass of

5 Newton s Laws k, determine the followin quantities. (a) her weiht, includin her backpack, etc., on Earth, (b) her weiht on the moon, (c) the required upward buoanc force of the water durin her trainin for the moon s environment on Earth. Picture the Problem We can use the relationship between weiht (ravitational force) and mass, toether with the iven information about the acceleration due to ravit on the moon, to find the astronaut s weiht on Earth and on the moon. (a) Her weiht on Earth is the product of her mass and the ravitational field at the surface of Earth: evaluate w: (b) Her weiht on the moon is the product of her mass and the ravitational field at the surface of the moon: Substitute for her weiht on Earth and evaluate her weiht on the moon: w m Earth w Earth moon ( 50 k)( 9.8m/s ).45 kn w m m w moon 6 6 w.453 kn earth (.453 kn) 409 N moon 6 (c) he required upward buoanc force of the water equals the difference between her weiht on Earth and on the moon: w buoanc w w Earth.45 kn 0.4 kn.04kn moon ree-bod Diarams: Static Equilibrium 49 In iure 4-38a, a k block is suspended at the midpoint of a.5-m-lon strin. he ends of the strin are attached to the ceilin at points separated b.00 m. (a) What anle does the strin make with the ceilin? (b) What is the tension in the strin? (c) he k block is removed and two 0.50-k blocks are attached to the strin such that the lenths of the three strin sements are equal (iure 4-38b). What is the tension in each sement of the strin? Picture the Problem he free-bod diarams for parts (a), (b), and (c) are shown below. In both cases, the block is in equilibrium under the influence of the forces and we can use Newton s second law of motion and eometr and trionometr to obtain relationships between θ and the tensions.

6 74 Chapter 4 (a) and (b) (c) (a) Referrin to the free-bod diaram for part (a), use trionometr to determine θ : (b) Notin that, appl ma to the k block and solve for the tension : θ cos 0.50m 0.65m 36.9 sinθ m 0 because a 0 and m sinθ 37 evaluate : ( 0.500k)( 9.8m/s ) 4.N sin36.9 (c) he lenth of each sement is: ind the distance d:.5m 3 d m.00m m m Epress θ in terms of d and solve for its value: θ cos cos 0.47 m d 0.97 m m

7 Newton s Laws 75 Appl ma 0.50-k block: to the evaluate 3 : Appl ma to the 0.50-k block and solve for the tension : evaluate : m 3 sinθ m 0 3 sinθ 3 ( 0.50k)( 9.8m/s ) sin N 3 cosθ 0 since a 0. and 3 cosθ N ( N) cos N B smmetr: N 5 A 0-k object on a frictionless table is subjected to two horizontal forces, r and r, with manitudes 0 N and 30 N, as shown in iure ind the third force r 3 that must be applied so that the object is in static equilibrium. Picture the Problem he acceleration of an object is directl proportional to the net force actin on it. Choose a coordinate sstem in which the + direction is the same as that of r and the + direction is to the riht. Add the two forces to determine the net force and then use Newton s second law to find the acceleration of the object. If r brins the sstem into equilibrium, it must be true that r 3 + r + r 0. Epress r 3 in terms of 3 r and Epress r and r in unit vector notation: and r r r r r : 3 () r iˆ (0 N) {( 30 N)sin 30 } iˆ + {(30 N)cos30 } ˆj ( 5 N)ˆ i (6 N) ˆj + Substitute for r and r in equation () and simplif to obtain: r ( 0 N) iˆ [( 5 N)ˆ i + (6 N) ˆj ] ( 5.0 N) iˆ + ( 6 N)j ˆ 3

8 76 Chapter 4 ree-bod Diarams: Inclined Planes and the Normal orce 6 A 65-k student weihs himself b standin on a scale mounted on a skateboard that is rollin down an incline, as shown in iure Assume there is no friction so that the force eerted b the incline on the skateboard is normal to the incline. What is the readin on the scale if θ 30º? Picture the Problem he scale readin (the bo s apparent weiht) is the force the scale eerts on the bo. Draw a free-bod diaram for the bo, choosin a coordinate sstem in which the positive -ais is parallel to and down the inclined plane and the positive -ais is in the direction of the normal force the incline eerts on the bo. Appl Newton s second law of motion in the direction. 30 r r n Appl ma to the bo to find n. Remember that there is no acceleration in the direction: cos30 n 0 or, because m, m cos30 0 n Solvin for n ields: n m cos30 evaluate n : ree-bod Diarams: Elevators n ( 65k)( 9.8m/s ) 0.55kN cos30 63 (a) Draw the free bod diaram (with accurate relative force manitudes) for an object that is hun b a rope from the ceilin of an elevator that is ascendin but slowin. (b) Repeat Part (a) but for the situation in which the elevator is descendin and speedin up. (c) Can ou tell the difference between the two diarams? Comment. Eplain wh the diaram does not tell anthin about the object s velocit.

9 Newton s Laws 77 Picture the Problem (a) he free bod diaram for an object that is hun b a rope from the ceilin of an ascendin elevator that is slowin down is shown to the riht. Note that because >, the net force actin on the object is downward; as it must be if the object is slowin down as it is movin upward. (b) he free bod diaram for an object that is hun b a rope from the ceilin of an elevator that is descendin and speedin up is shown to the riht. Note that because >, the net force actin on the object is downward; as it must be if the object is speedin up as it descends. r r r r (c) No. In both cases the acceleration is downward. You can onl tell the direction of the acceleration, not the direction of the velocit. ree-bod Diarams: Several Objects and Newton s hird Law 7 A block of mass m is bein lifted verticall b a rope of mass M and lenth L. he rope is bein pulled upward b a force applied at its top end, and the rope and block are acceleratin upward with an acceleration of manitude a. he distribution of mass in the rope is uniform. Show that the tension in the rope at a distance (where < L) above the block is (a + )[m + (/L)M]. Picture the Problem Because the distribution of mass in the rope is uniform, we can epress the mass m of a lenth of the rope in terms of the total mass of the rope M and its lenth L. We can then epress the total mass that the rope must support at a distance above the block and use Newton s second law to find the tension as a function of. L m r rope r Set up a proportion epressin the mass m of a lenth of the rope as a function of M and L and solve for m : m' M L m' M L

10 78 Chapter 4 Epress the total mass that the rope must support at a distance above the block: m + m' m + M L Appl ma to the block and a lenth of the rope: m + m' m + m' ( ) ( ) a Substitutin for m + m ields: Solve for and simplif to obtain: m + M L m + M ( a + ) m + L M L a 73 A 40.0-k object supported b a vertical rope is initiall at rest. he object is then accelerated upward from rest so that it attains a speed of 3.50 m/s in s. (a) Draw the object s free bod diaram with the relative lenths of the vectors showin the relative manitudes of the forces. (b) Use the free bod diaram and Newton s laws to determine the tension in the rope. Picture the Problem A net force is required to accelerate the object. In this problem the net force is the difference between r r r and ( m). (a) he free-bod diaram of the object is shown to the riht. A coordinate sstem has been chosen in which the upward direction is positive. he manitude of r is approimatel.5 times the lenth of r. r m (b) Appl to obtain: ma to the object m ma Solvin for ields: ma + m m( a + ) r Usin its definition, substitute for a to obtain: Δv m + Δt

11 evaluate : ( 40.0 k) 59 N Newton s Laws m/s m/s s 79 A 60-k housepainter stands on a 5-k aluminum platform. he platform is attached to a rope that passes throuh an overhead pulle, which allows the painter to raise herself and the platform (iure 4-57). (a) o accelerate herself and the platform at a rate of 0.80 m/s, with what force must she pull down on the rope? (b) When her speed reaches.0 m/s, she pulls in such a wa that she and the platform o up at a constant speed. What force is she eertin on the rope now? (Inore the mass of the rope.) Picture the Problem Choose a coordinate sstem in which the upward direction is the positive direction. Note that r is the force eerted b the painter on the rope and that r is the resultin tension in the rope. Hence the net upward force on the painter-plus-platform is. r (a) Lettin m tot mframe + mpainter, appl ma to the frame-pluspainter: m m Solvin for ields: m ( a ) evaluate : tot tot + tota ( 75k)( 0.80 m/s + 9.8m/s ) 398N Because : 398 N 0.40kN (b) Appl 0 to obtain: mtot 0 mtot evaluate : General Problems ( 75k)( 9.8m/s ) 0.37kN 87 he mast of a sailboat is supported at bow and stern b stainless steel wires, the foresta and backsta, anchored 0 m apart (iure 4-6). he.0-mlon mast weihs 800 N and stands verticall on the deck of the boat. he mast is positioned 3.60 m behind where the foresta is attached. he tension in the

12 80 Chapter 4 foresta is 500 N. ind the tension in the backsta and the force that the mast eerts on the deck. Picture the Problem he free-bod diaram shows the forces actin at the top of the mast. Choose the coordinate sstem shown and use Newton s second and third laws of motion to analze the forces actin on the deck of the sailboat. Appl ma to the top of the sinθ B sinθb 0 mast: ind the anles that the foresta and backsta make with the vertical: Solvin the -direction equation for B ields: θ tan and B θ tan B 3.60m 6.7.0m 6.40m 8..0m sinθ sinθ B sin6.7 sin 8. evaluate B : B ( 500 N) 305N Appl 0 to the mast: cosθ cosθ 0 mast B B Solve for mast to obtain: cosθ + mast B cosθ B evaluate mast : ( 500 N) cos6.7 + ( 305 N) cos N mast he force that the mast eerts on the deck is the sum of its weiht and the downward forces eerted on it b the foresta and backsta: mast on the deck 748N N.55kN 93 he masses attached to each side of an ideal Atwood s machine consist of a stack of five washers, each of mass m, as shown in iure he tension in the strin is 0. When one of the washers is removed from the left side, the remainin washers accelerate and the tension decreases b N. (a) ind

13 Newton s Laws 8 m. (b) ind the new tension and the acceleration of each mass when a second washer is removed from the left side. Picture the Problem Because the sstem is initiall in equilibrium, it follows that 0 5m. When one washer is moved from the left side to the riht side, the remainin washers on the left side will accelerate upward (and those on the riht side downward) in response to the net force that results. he free-bod diarams show the forces under this unbalanced condition. Applin Newton s second law to each collection of washers will allow us to determine both the acceleration of the sstem and the mass of a sinle washer. r r ' 4m r 5m r (a) Appl washers: ma to the risin 4 m 4m a () ( ) Notin that, appl ma to the descendin masses: Eliminate between these equations to obtain: ( m) a 5 m 5 () a 9 Use this acceleration in equation () or equation () to obtain: 40 m 9 Epressin the difference Δ between 0 and ields: Δ 40 5m m m Δ evaluate m: m 9 5 ( N) 9.8 m/s 55.0 (b) Proceed as in (a) to obtain: Add these equations to eliminate and solve for a to obtain: m 3ma m 5ma 3 and 5 a 4

14 8 Chapter 4 evaluate a : a 4 ( 9.8m/s ).45 m/s Eliminate a in either of the motion equations and solve for to obtain: evaluate : m ( k)( 9.8m/s ).03N