CHAPTER 4 Dynamics: Newton s Laws of Motion /newtlaws/newtltoc.html

Transcription

1 CHAPTER 4 Dynamics: Newtn s Laws f Mtin /newtlaws/newtltc.html Frce Newtn s First Law f Mtin Mass Newtn s Secnd Law f Mtin Newtn s Third Law f Mtin Weight the Frce f Gravity; and the Nrmal Frce Slving Prblems with Newtn s Laws: Free-Bdy Diagrams Applicatins Invlving Frictin, Inclines Prblem Slving A General Apprach Classical Mechanics Describes the relatinship between the mtin f bjects in ur everyday wrld and the frces acting n them Cnditins when Classical Mechanics des nt apply very tiny bjects (< atmic sizes) bjects mving near the speed f light Frces Usually think f a frce as a push r pull Vectr quantity May be a cntact frce r a field frce Cntact frces result frm physical cntact between tw bjects Field frces act between discnnected bjects Als called actin at a distance Cntact and Field Frces 1

2 Fundamental Frces Types Strng nuclear frce Electrmagnetic frce Weak nuclear frce Gravity Characteristics All field frces Listed in rder f decreasing strength Only gravity and electrmagnetic in mechanics Sir Isaac Newtn Frmulated basic cncepts and laws f mechanics Universal Gravitatin Calculus Light and ptics Newtn s First Law f Mtin Newtn s first law is ften called the law f inertia. Every bject cntinues in its state f rest, r f unifrm velcity in a straight line, as lng as n net frce acts n it. External and Internal Frces External frce Any frce that results frm the interactin between the bject and its envirnment Internal frces Frces that riginate within the bject itself They cannt change the bject s velcity

3 Inertia Is the tendency f an bject t cntinue in its riginal mtin. Mass Mass is the measure f inertia f an bject. In the SI system, mass is measured in kilgrams. Mass is nt weight: Mass is a prperty f an bject. Weight is the frce exerted n that bject by gravity. If yu g t the mn, whse gravitatinal acceleratin is abut 1/6 g, yu will weigh much less. Yur mass, hwever, will be the same. Newtn s Secnd Law f Mtin Newtn s secnd law is the relatin between acceleratin and frce. Acceleratin is prprtinal t frce and inversely prprtinal t mass. F ma SI unit f frce is a Newtn (N) 1N 1 kg m s Units f Frce US Custmary unit f frce is a pund (lb) 1 N = 0.5 lb Frce is a vectr, s F ma is true alng each crdinate axis. The unit f frce in the SI system is the newtn (N). Nte that the pund is a unit f frce, nt f mass, and can therefre be equated t newtns but nt t kilgrams. 3

4 Example 1: What average net frce is required t bring a 1500-kg car t rest frm a speed f 100 km/h within a distance f 55 m? v a( x x ) ( v v ) 0 (8 m/ s) a 7.1 m / s ( x x ) (55 m) 4 F ma (1500 kg)( 7.1 m / s ) 1.1x10 N Example : An airbat with mass 350 kg, including passengers, has an engine that prduces a net hrizntal frce f 770 N, after accunting fr frces f resistance. (a) Find the acceleratin f the airbat. F 770N F ma a.0 m / s m 350kg (b) Starting frm rest, hw lng des it take the airbat t reach a speed f 1.0m/s? v v at 1.0 / 0 (.0 / ) 5.45 m s m s t t s (c) After reaching this speed, the pilt turns ff the engine and drifts t a stp ver a distance f 50.0 m. Find the resistance frce, assuming it s cnstant. v v ax 0 (1 m / s) a(50.0 m) a 1.44 m / s Fresis tan ce ma (350 kg)( 1.44 m / s ) 504N 4

5 Example 3: What happens when a persn pulls upward n a bx with a frce f 100N? The bx has a weight f mg = 98N. F F mg 100N 98N.0N y P a y Fy.0N 0.0 m / s m 10.0kg 1kg = 9.80N Example 4: Calculate the sum f the tw frces exerted n the bat by wrkers A and B. The cmpnents f The cmpnents f F A F F cs 45.0 (40.0 N)(0.707) 8.3N AX A F F sin 45.0 (40.0 N)(0.707) 8.3N AY A F B F F cs 37.0 (30.0 N)(0.799) 4.0N BX B F F sin 37.0 (30.0 N)(0.60) 18.1N BY B The cmpnents f the resultant frce are F F F 8.3N 4.0N 5.3N RX AX BX F F F 8.3N 18.1N 10.N RY AY BY The magnitude f the resultant frce: F F F (5.3 N ) (10. N) 53.3N R RX RY The angle that the net frce makes with the x axis: 10. tan FRY N F 5.3N tan 1 (0.195)=11.0 RX 5

6 Example 5: A 10.0-kg bx is pulled at a 30 angle with a frce f FP 40.0N. (a) Calculate the acceleratin f the bx: F (40.0 N)(cs 30.0 ) (40.0 N)(0.866) 34.6N PX F (40.0 N)(sin 30.0 ) (40.0 N)(0.500) 0.0N PY FPX (34.6 N) F ma a 3.46 m / s m (10.0 kg) (b) Calculate the magnitude f the upward frce: F mg F ma N PY Y F 98.0N 0N 0 F 78.0N N N Newtn s Third Law f Mtin Any time a frce is exerted n an bject, that frce is caused by anther bject. Newtn s third law: Whenever ne bject exerts a frce n a secnd bject, the secnd exerts an equal frce in the ppsite directin n the first. A key t the crrect applicatin f the third law is that the frces are exerted n different bjects. Make sure yu dn t use them as if they were acting n the same bject. Rcket prpulsin can als be explained using Newtn s third law: ht gases frm cmbustin spew ut f the tail f the rcket at high speeds. The reactin frce is what prpels the rcket. Nte that the rcket des nt need anything t push against. 6

7 The weight f an bject n the Earth s surface is the gravitatinal frce exerted n it by the Earth. Apparent weight: Yur perceptin f yur weight is based n the cntact frces between yur bdy and yur surrundings. Weight If yur surrundings are accelerating, yur apparent weight may be mre r less than yur actual weight. Weight the Frce f Gravity; and the Nrmal Frce An bject at rest must have n net frce n it. If it is sitting n a table, the frce f gravity is still there; what ther frce is there? The frce exerted perpendicular t a surface is called the nrmal frce. It is exactly as large as needed t balance the frce frm the bject (if the required frce gets t big, smething breaks!) Actin-Reactin Pairs N is the nrmal frce, the frce the table exerts n the TV. N is always perpendicular t the surface F g is the frce the Earth exerts n the bject. F g is always straight dwn regardless f the angle f the bject. 7

8 The nrmal frce may be equal t, greater than, r less than the weight. Nrmal Frces The nrmal frce is always perpendicular t the surface. Free Bdy Diagram Must identify all the frces acting n the bject f interest Chse an apprpriate crdinate system If the free bdy diagram is incrrect, the slutin will likely be incrrect The frce T is the tensin acting n the bx The tensin is the same at all pints alng the rpe n and F g are the frces exerted by the earth and the grund Only frces acting directly n the bject are included in the free bdy diagram Reactin frces act n ther bjects and s are nt included The reactin frces d nt directly influence the bject s mtin Slving Newtn s Secnd Law Prblems Read the prblem at least nce Draw a picture f the system Identify the bject f primary interest Indicate frces with arrws Label each frce Use labels that bring t mind the physical quantity invlved 8

9 Draw a free bdy diagram If additinal bjects are invlved, draw separate free bdy diagrams fr each bject Chse a cnvenient crdinate system fr each bject Apply Newtn s Secnd Law The x- and y-cmpnents shuld be taken frm the vectr equatin and written separately Slve fr the unknwn(s) Equilibrium An bject either at rest r mving with a cnstant velcity is said t be in equilibrium The net frce acting n the bject is zer (since the acceleratin is zer) Easier t wrk with the equatin in terms f its cmpnents: F 0 and F 0 x This culd be extended t three dimensins y Example 6: A traffic light weighing 100N hangs frm a vertical cable tied t tw ther cables that are fastened t a supprt. The upper cables make angles f 37.0 and 53.0 with the hrizntal. Find the tensin in each f the three cables. T3 Fg 100N T reslve all three tensin frces int cmpnents develp the fllwing table: Frce x-cmpnent y-cmpnent T T T 1 3 T1 cs 37.0 T1 T cs T sin 37.0 sin N Apply the cnditins fr equilibrium t the knt, using the cmpnents f the table: F T T X 9 1 cs 37.0 cs F T sin 37.0 T sin N 0 y 1 There are tw equatins and tw remaining unknwns. Slve equatin 1 fr : cs T T1 T1 1.33T 1 cs

10 Substitute the result frt int equatin : T sin 37.0 (1.33 T )(sin 53.0 ) 100N T 60.1N 1 T 1.33T 1.33(60.0 N) 79.9N 1 Inclined Planes Chse the crdinate system with x alng the incline and y perpendicular t the incline Replace the frce f gravity with its cmpnents Example 7: A sled is held at rest n a frictinless, snwcvered hill. If the sled weighs 77.0 N, find the frce exerted by the rpe n the sled and the magnitude f the frce n exerted by the hill n the sled. Apply Newtn s nd law t the sled, with a = 0: Find the x-cmpnent, nrmal frce = 0 F T n F g 0 F T 0 mg sin T (77.0 N) sin X Find the y-cmpnent, there is a nrmal frce T 38.5N FY 0 n mg cs n (77.0 N )(cs 30.0 ) 0 n 66.7N When yu have mre than ne bject, the prblem-slving strategy is applied t each bject Draw free bdy diagrams fr each bject Apply Newtn s Laws t each bject Slve the equatins Multiple Objects Example 10

11 Frces f Frictin When an bject is in mtin n a surface r thrugh a viscus medium, there will be a resistance t the mtin This is due t the interactins between the bject and its envirnment This is resistance is called frictin Frictin is prprtinal t the nrmal frce The frce f static frictin is generally greater than the frce f kinetic frictin The cefficient f frictin (µ) depends n the surfaces in cntact The directin f the frictinal frce is ppsite the directin f mtin The cefficients f frictin are nearly independent f the area f cntact Static Frictin, ƒ s Static frictin acts t keep the bject frm mving If F increases, s des ƒ s If F decreases, s des ƒ s ƒ s µ n Kinetic Frictin, ƒ k The frce f kinetic frictin acts when the bject is in mtin ƒ k = µ n Variatins f the cefficient with speed will be ignred 11

12 Blck n a Ramp, Example Axes are rtated as usual n an incline The directin f impending mtin wuld be dwn the plane Frictin acts up the plane Oppses the mtin Apply Newtn s Laws and slve equatins Example 8: A blck is launched up a frictinless incline in the diagram t the right with an initial velcity f 5.5 m/s. What is the maximum displacement, d, f the blck up the incline? v (5.5 m/ s) D.4m g sin (9.80 m / s )(sin 40 ) Example 9: The blck shwn n the right remains at rest. What is the frictin frce acting n the blck? F ma F (0.50 kg)(9.80 m / s ) 4.9N F 4.9N sin N fr Example 10: A hckey puck is struck by a hckey stick and given an initial speed f 0.0 m/s. The puck remains n the ice and slides 10 m, slwing dwn steadily until it cmes t a rest. Determine the cefficient f kinetic frictin between the puck and the ice. v v ax 0 (0.0 m/ s) a (10 m) 1.67 m / s a 1.67 m / s k g 9.80 m / s 1

13 Cnnected Objects Apply Newtn s Laws separately t each bject The magnitude f the acceleratin f bth bjects will be the same The tensin is the same in each diagram Slve the simultaneus equatins Example 11: A blck with a mass f 4.00kg and a ball with a mass f 7.00 kg are cnnected by a light string that passes ver a frictinless pulley. The frictin between the blck and the surface is k (a) Find the acceleratin f the tw bjects and the tensin n the string. Find the acceleratin by finding the cmpnents f m 1. F T f m a F n m g X k 1 1 Y 1 0 The equatin fr the y-cmpnent gives n m1g. Substitute this value fr n and fk kn int the equatin fr the x-cmpnent. T m g m a k Apply Newtn s secnd law t the ball: F m g T m a m a Y 1 Subtract the secnd equatin frm the first, eliminating T and slve fr a. mg km1g mg km1 g ( m1 m ) a1 a1 m m a 1 1 (7.00 kg)(9.80 m / s ) (0.300)(4.00 kg)(9.80 m / s ) (4.00kg 7.00 kg) 5.17 m / s Substitute the value fr a int the first equatin t find T. T m g m a k (0.300)(4.00 )(9.80 / ) (4.00 )(5.17 / ) T 3.45 T kg m s kg m s N 13

14 Example 1: A flatbed truck slwly tilts its bed upward t dispse f a 95.0 kg crate. Fr small angles f tilt the crate des nt mve, but when the tilt angle exceeds 3., the crate begins t slide. What is the cefficient f static frictin between the bed f the truck and the crate? W mg sin W mg cs x F N f W N 0 mg cs ma 0 y y s, y y y N mg cs F N f W ma 0 x x s, x x x 0 N mg sin 0 mg cs mg sin mg cs mg sin s s s y mg sin tan tan mg cs Example 13: Atwd s machine cnsists f tw masses cnnected by a string that passes ver a pulley. Find the acceleratin f the masses fr general m1 and m, and evaluate fr the specific case m1 = 3.1kg, m = 4.4kg. F T m g m g m a 1x F m g T m a x T m g m a 1 1 m g T m a 1 ( m m ) g ( m m ) a 1 1 m m 4.4kg 3.1kg m1m 3.1kg 4.4kg 1 a g (9.81 m / s ) 1.7 m / s 14

15 Example 14: Tw blcks are cnnected by a string n a smth inclined surface that makes an angle f 4 with the hrizntal. The blck n the incline has a mass f 6.7kg. Find the mass f the hanging blck that will cause the system t be in equilibrium. F T Mg sin 0 x T mg Mg sin m M sin 6.7 kg sin kg Example 15: Tw crates f masses 10.0kg and 5.0kg are cnnected by a light string that passes ver a frictinless pulley. The 5.0kg crate lies n a smth incline f angle 40. Find the acceleratin f the 5.0kg crate and the tensin n the string. Let m kg, m 5.00 kg, and Applying the secnd law t each bject gives m1a m1 g T (1) and ma T m g sin () Adding these equatins yields m1 msin a g m1 m, r 10.0 kg 5.00 kg sin 40.0 a 9.80 m s 4.43 m s 15.0 kg Then, Equatin (1) yields T m1 g a 10.0 kg m s 53.7 N 15

16 CHAPTER 4 LAWS OF MOTION CONCEPTS 1. As the vectr sum f all the frces acting n an bject increases, the acceleratin f the bject increases.. Zer acceleratin is an essential characteristic f an bject in equilibrium. 3. An 800 N persn is standing in an elevatr. If the upward frce f the elevatr n the persn is 600 N, the persn is accelerating dwnward. 4. A baseball bat mving at high velcity strikes a feather. If air resistance is neglected cmpared t the frce exerted by the bat n the feather, the frce exerted by the feather n the bat will be the same. 5. A cart is unifrmly accelerating frm rest. The net frce acting n the cart is cnstant. 6. In the diagram t the right, the numbers 1,, 3, and 4 represent pssible directins in which a frce culd be applied t a cart. If the frce applied in each directin has the same magnitude, the directin where the vertical cmpnent f the frce will be the least is A lawnmwer is pushed with a cnstant frce F, as shwn t the right. As the angle between the lawnmwer handle and the hrizntal increases the hrizntal cmpnent f F decreases. 8. In the diagram t the right, bx M is n a frictinless table with frces F1 and F as shwn. If the magnitude f F1 is greater than the magnitude f F, then the bx is accelerating in the directin f F1. 9. The diagram t the right represents a car resting n a hill. The vectr that best represents the weight f the car is B. 16

17 10. A cart rlls dwn an inclined plane with cnstant speed as shwn t the right. The arrw D represents the directin f the frictinal frce. 11. As shwn t the right, an inflated balln released frm rest mves hrizntally with velcity v. The velcity f the balln is mst likely caused by actinreactin. 1. If yu blw up a balln, and then release it, the balln will fly away. This is an illustratin f Newtn s Third Law. 13. An bject is acted upn by a cnstant unbalanced frce. The graph C best represents the mtin f the bject. 14. The graph C best represents the mtin f a mving bject with n unbalanced frce acting n it. 15. Yu are standing n a mving bus, facing frward, and yu suddenly fall backwards. Yu can imply frm this that the bus s velcity has increased. 16. A packing crate slides dwn an inclined ramp at cnstant velcity. We can deduce that a frictinal frce is acting n it. 17. It is mre difficult t start mving a heavy cartn frm rest than it is t keep pushing it with cnstant velcity because kinetic frictin is less than static frictin. 17

18 18. The number f frces acting n a car parked n a hill is three. 19. When yu sit n a chair the resultant frce n yu is zer. 0. If tw identical masses are attached by a massless crd passing ver a massless, frictinless pulley f an Atwd s machine, but at different heights, and then released, the masses will nt mve. 1. When the rcket engines n a starship are suddenly turned ff, while traveling in empty space, the starship will mve with cnstant speed.. In the absence f an external frce, a mving bject will mve with cnstant speed. 3. If yu exert a frce F n an bject, the frce which the bject exerts n yu will be F in all cases. 4. An arrw is sht straight up. At the tp f its path, the net frce acting n it is equal t its weight. 5. Tw identical masses are attached by a light string that passes ver a small pulley, as shwn. The table and the pulley are frictinless. The system is mving with acceleratin less than g. 6. A bx f mass M is resting n a flat bard. One end f the bard is lifted up until the bx just starts t slide. The angle θ that the bard makes with the hrizntal, fr this t ccur depends n the cefficient f static frictin. 7. A 0-kg fish is weighed with tw spring scales, each f negligible weight, as shwn t the right. The readings n each scale will read 0-kg. 18

19 8. A brick and a feather fall t the earth at their respective terminal velcities. The bject that experiences the greater frce f air frictin is the brick. 9. A push r pull best expresses the meaning f the wrd frce. 30. A rcket mves thrugh empty space in a straight line with cnstant speed. It is far frm the gravitatinal effect f a star r planet. Under these cnditins, the frce that must be applied t the rcket in rder t sustain its mtin is zer. 31. The frce that keeps yu frm sliding n the sidewalk is static frictin. 3. A mass rests n tp f a frictinless inclined plane. The nrmal frce acting n the mass decreases as the angle f elevatin increases. 19