2.3. Applying Newton s Laws of Motion. Objects in Equilibrium
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1 Appling Newton s Laws of Motion As ou read in Section 2.2, Newton s laws of motion describe ow objects move as a result of different forces. In tis section, ou will appl Newton s laws to objects subjected to various forces in two dimensions, as well as objects tat are accelerating. For eample, Figure 1 sows a skier moving downill. You can draw an FBD of all te forces acting on te skier. Eart s gravit acts directl downward and as components parallel and perpendicular to te slope. Te normal force acts perpendicular to te ill and cancels te component of gravit perpendicular to te ill. Finall, friction acts parallel to te ill, opposing te skier s motion. You can use te sum of tese forces and Newton s laws to learn about te motion of te skier. 2.3 F f Figure 1 Te forces on tis skier are gravit, te normal force, and friction. Compared to te oter forces acting on te skier, air resistance is negligible ere. Tese forces can be broken into components parallel and perpendicular to te illside to analze te motion of te skier. Objects in Equilibrium Wen te net force on an object is zero, tat object is said to be in equilibrium. As discussed in Section 2.2, an object wit no net force acting on it will not accelerate. So, an object in equilibrium will remain at rest or remain moving at a constant velocit until a force acts on it. Matematicall, an object is in equilibrium wen SF > 5 0, or, wen ou break te forces down into teir components, bot SF 5 0 and SF 5 0. Wen solving problems involving objects in equilibrium, ou can set te positive -ais in an direction, but ou sould draw te FBD first and ten pick te most convenient direction. B convenient we mean te direction tat will give ou te fewest components. Te Tutorial on te net page sows ou ow to solve problems wen an object is in equilibrium. equilibrium a state in wic an object as no net force acting on it Investigation Static Equilibrium of Forces (page 95) In tis investigation, ou will analze te conditions for equilibrium using vector components. 2.3 Appling Newton s Laws of Motion 77
2 Tutorial 1 Solving Problems for Objects Tat Are in Equilibrium Tis Tutorial sows ow to solve problems for objects in equilibrium wen acted on b two-dimensional forces. Sample Problem 1: Calculating Tension and te Normal Force A sled as a mass of 14 kg and is on a ill tat is inclined 258 to te orizontal, as sown in Figure 2(a). Te ill is ver ic (negligible friction), and te sled is eld at rest b a rope attaced to a post. Te rope is parallel to te ill as sown. Figure 2 sows te FBD. (a) Calculate te magnitude of te tension in te rope. Calculate te magnitude of te normal force acting on te sled. 25 Solution (a) Given: m 5 14 kg; u Required: Analsis: Tension is parallel to te illside, and te normal force is perpendicular to te illside. So let te positive -ais point up te illside as sown in Figure 2. Determine te components of gravit, and resolve te forces in te -direction to solve for tension. Te sled is in equilibrium, so te net force is zero. Solution: SF SF mg sin u mg sin u 5 mg sin u kg m/s 2 2 sin N Statement: Te magnitude of te tension in te rope is 58 N. Given: m 5 14 kg; u Required: Analsis: Resolve te forces in te -direction to solve for te normal force. Te sled is in equilibrium, so te net force is zero. Solution: SF SF mg cos u mg cos u 5 mg cos u kg m/s 2 2 cos N Statement: Te magnitude of te normal force on te sled is N. (a) 25 mg cos mg mg sin Figure 2 Sample Problem 2: Force Applied at an Angle Your car is stuck in te mud, and ou ask a friend to elp ou pull it free using a rope. You tie one end of te rope to our car and ten pull on te oter end wit a force of 10 3 N. Unfortunatel, te car does not move. Your friend ten suggests tat ou make a knot in te middle of te rope, tie te oter end of te rope to a tree, and ten pull on te knot. Altoug ou are skeptical tat our friend s idea will elp, ou tr it anwa. You make a knot in te middle of te rope. You leave one end of te rope attaced to te car and tie te oter end to a tree at an angle u Ten ou and our friend pull on te knot in te direction indicated b F > a in Figure 3(a). Figure 3 sows te FBD wit te forces acting on te knot at point O. You discover tat wen a 10 3 N force is applied to te knot in te middle of te rope in te direction sown in Figure 3(a), ou are just able to free te car at a slow constant velocit. W does tis work? 78 Capter 2 Dnamics
3 (a) left O F a rigt tree 10 FT left sin FT rigt sin left O FT left cos F cos knot in rope Figure 3 F a T rigt rigt Given: F a N; angle, u, of te rope to te -ais is 108 Required: Analsis: Calculate te magnitude of te tension in te rope given te 10 3 N force eerted b ou and our friend at te point were te car as just started to move at a slow constant velocit. Since te car is just on te verge of moving, ou can appl te conditions for equilibrium to tis situation. Consider te forces acting on te rope at point O (te point at wic ou and our friend eert our force) to determine te tension in terms of te applied force. Appl te conditions for static equilibrium to te rope at point O. Use te coordinate sstem to calculate te components of te tree forces along and, and ten appl te condition for equilibrium along. Te rope is continuous and te angles on te two sides are equal, so te tensions in te left and rigt portions of te rope are te same, rigt 5 left 5. Solution: SF 5 1 rigt sin u 1 left sin u 2 F sin u 1 sin u 2 F F 5 2 sin u 5 F 2 sin u N 2 sin N Statement: Tis arrangement multiplies te applied force. Te tension in te rope is able to pull te car out because it is 3 times te applied force (3F a ). Practice 1. Te static friction on one block is olding anoter block up, as sown in Figure 4. Block A as a weigt of 6.5 N, sits on a table, and is connected to a wall b a string. Block B as a weigt of 2.8 N, is attaced to a string, and is connected to block A s string at point P. Te string from block A to point P is orizontal. Te magnitude of te force of friction on block A is 1.4 N. K/U T/I C A (a) Draw an FBD for block B. Determine te magnitude of te tension in te vertical rope. [ans: 2.8 N] Draw an FBD for block A. Determine te magnitude of te tension in te orizontal rope and te magnitude of te normal force acting on block A. [ans: 1.4 N; 6.5 N] (c) Draw an FBD of point P. Calculate te tension (te magnitude and te angle u) in te tird rope. [ans: 3.1 N [rigt 638 up]] 2. A 62 kg rock climber is attaced to a rope tat is allowing im to ang orizontall wit is feet against te wall. Te tension in te rope is N, and te rope makes an angle of 328 wit te orizontal. Determine te force eerted b te wall on te climber s feet. K/U T/I A [ans: N [left 218 up]] 3. Te tree forces sown in Figure 5 act on an object. Te object is in equilibrium. Calculate te magnitude of te force F 3 and te angle u 3. T/I [ans: 78 N; W 9.88 S] A P u B Figure N 3 F 3 Figure N 2.3 Appling Newton s Laws of Motion 79
4 Unit TASK BOOKMARK You can appl wat ou ave learned about forces and acceleration to te Unit Task on page 146. Accelerating Objects If an object is not in equilibrium, ten it is accelerating in some direction. You can use Newton s second law, SF > 5 ma >, to determine te acceleration from te net force on te object, SF >. Wen solving problems tat involve accelerating objects, set te positive -ais in te direction of te net force (acceleration). Tis will ensure tat te net force as no additional -component, wic will simplif te solution. If ou do not know te direction of te net force, ten just set te positive -ais in te direction tat is te most convenient to solve te problem. In te following Tutorial, ou will use Newton s second law of motion to calculate velocit, acceleration, and tension for objects acted on b two-dimensional forces. Tutorial 2 Solving Problems for Objects Tat Are Accelerating Te Sample Problems model ow to calculate velocit, acceleration, and tension for objects tat are accelerating wen acted on b two-dimensional forces. Sample Problem 1: Velocit Due to Acceleration A sled is at te top of a ill, wic makes an angle of 188 wit te orizontal, as sown in Figure 6(a). Figure 6 sows te FBD for te sled. Te eigt of te ill is 25 m. Calculate te speed of te sled as it reaces te bottom of te ill. Assume tat no friction acts on te sled. (a) mg cos 18 mg sin sin 18 Given: Dd 5 25 m; u 5 188, v i 5 0 Required: v f Analsis: Te sled will accelerate down te ill, so te net force is down te ill according to Newton s second law. Terefore, make te positive -ais down te ill. Tis means tat te normal force is in te direction of te positive -ais. Determine te components of te force of gravit in te - and -directions as defined b te coordinate aes in Figure 6. Use Newton s second law of motion to determine te acceleration along ; ten appl v 2 f 5 v 2 i 1 2a Dd to calculate te final speed. Solution: SF 5 mg sin u ma 5 mg sin u a 5 g sin u v 2 f 5 v 2 i 1 2a Dd Dd 5 sin u v 2 f g sin u2 a v f 5 "2g sin u b 5 " m/s m2 v f 5 22 m/s Statement: Te speed of te sled at te bottom of te ill is 22 m/s. Figure 6 80 Capter 2 Dnamics
5 Sample Problem 2: Acceleration and Tension A crate wit a mass of 32.5 kg sits on a frictionless surface and is connected to a second crate b a string tat passes over a pulle, as sown in Figure 7(a). Te second crate as a mass of 40.0 kg. Te pulle is frictionless and as no mass. Te string also as no mass. FBDs are sown in Figure 7. Determine te acceleration of te sstem of crates and te magnitude of te tension in te string. m 1 m 1 g (a) Figure 7 m 2 g m 2 m 1 m 2 m 1 g m 2 g Given: m kg; m kg Required: te acceleration of te crates, a; te magnitude of te tension in te string, Analsis: Appl Newton s laws to determine te acceleration of te crates, considering all te forces acting on tem. Te FBDs in Figure 7 sow all tese forces. Te positive -direction for eac FBD is determined b te direction of te acceleration of eac mass: rigt for mass 1 and down for mass 2. Write Newton s second law for eac crate, and solve for te unknown values. To determine te magnitude of te tension, use te FBD for te crate on te surface. Te accelerations of bot masses are equal because te are tied togeter and te string does not stretc. Solution: SF 5 1 1For crate 12 m 1 a 5 1Equation 12 SF 5 m 2 g 2 1For crate 22 m 2 a 5 m 2 g 2 1Equation 22 m 1 a 1 m 2 a m 2 g 2 1Equation 1 1 Equation 22 m 1 a 1 m 2 a 5 m 2 g a 5 m 2g m 1 1 m kg 19.8 m/s kg kg a m/s 2 1one etra digit carried2 SF 5 1 m 1 a kg m/s N Statement: Te acceleration of te crates is 5.4 m/s 2, and te magnitude of te tension in te string is N. Practice 1. Two blocks are fastened onto strings inside an elevator, as sown in Figure 8. Te mass of te top block is 1.2 kg, and te mass of te bottom block is 1.8 kg. Te elevator is accelerating up at 1.2 m/s 2. K/U T/I A 1.2 kg a 1.8 kg Figure 8 (a) Calculate te tension in eac string. [ans: top string: 33 N; bottom string: 20 N] Te maimum tension te strings can witstand is 38 N. Determine te maimum acceleration of te elevator tat will not break te strings. [ans: 2.9 m/s 2 [up]] 2.3 Appling Newton s Laws of Motion 81
6 2. A skier wit a mass of 63 kg glides wit negligible friction down a ill covered wit ard-packed snow. Te ill is inclined at an angle of 148 above te orizontal. K/U T/I A (a) Determine te magnitude of te normal force on te skier. [ans: N] Determine te magnitude of te skier s acceleration. (Hint: Remember to coose te 1-direction as te direction of te acceleration, parallel to te illside.) [ans: 2.4 m/s 2 ] 3. A cild on a toboggan slides down a ill wit an acceleration of magnitude 1.9 m/s 2. Friction is negligible. Determine te angle between te ill and te orizontal. K/U T/I A [ans: 118] 4. You pull a desk across a orizontal fl oor b eerting a force of 82 N, at an angle of 178 above te orizontal. Te normal force eerted b te fl oor on te desk is 213 N. Te acceleration of te desk across te fl oor is 0.15 m/s 2. K/U T/I A (a) Determine te mass of te desk. [ans: 24 kg] Determine te magnitude of te friction force on te desk. [ans: 75 N] 5. A store clerk pulls tree loaded sopping carts connected wit two orizontal cords to elp customers load teir cars (Figure 9). Cart 1 as a mass of 9.1 kg, cart 2 as a mass of 12 kg, and cart 3 as a mass of 8.7 kg. Friction is negligible. A tird cord, wic pulls on cart 1 and is at an angle of 238 above te orizontal, as a tension of magnitude 29 N. K/U T/I A 29 N m 3 m 2 m 1 23 cart 3 cart 2 cart 1 Figure 9 (a) Determine te magnitude of te acceleration of te carts. [ans: 0.90 m/s 2 ] Determine te magnitude of te tension in te cord between m 3 and m 2. [ans: 7.8 N] (c) Determine te magnitude of te tension in te cord between m 2 and m 1. [ans: 19 N] 6. Block A, wit a mass of 4.2 kg, is suspended from a vertical string as sown in Figure 10. Te string passes over a pulle and is attaced to block B. Te mass of block B is 1.8 kg. Te pulle and te surface of te ramp are essentiall frictionless. Calculate (a) te acceleration of te blocks and te tension in te string. T/I [ans: (a) 5.3 m/s2 ; 19 N] block B block A Figure Capter 2 Dnamics
7 2.3 Review Summar An object is in equilibrium wen te net force on it is zero. For objects eperiencing forces in two dimensions, break te motion into perpendicular components, wic can be analzed independentl. Once ou ave determined te net force using components, use Newton s second law to determine te acceleration. Questions 1. In Figure 11, two ropes are pulling on a skater, and te eert forces on er as sown in te figure. Calculate te magnitude and direction of te total force eerted b te ropes on te skater. T/I top view 50 Figure N 40 N 2. Determine te tensions in all tree cables in Figure 12. T/I m 45 kg Figure A flag of mass 2.5 kg is supported b a single rope as sown in Figure 13. A strong orizontal wind eerts a force of 12 N on te flag. Calculate te tension in te rope and te angle, u, te rope makes wit te orizontal. T/I wind Figure A car is parked on a slipper ill (Figure 14). Te ill is at an angle of 158 to te orizontal. To keep it from sliding down te ill, te owner attaces a cable at te back of te car and to a post. Te mass of te car is kg. K/U T/I C cable 15 Figure 14 (a) Draw an FBD sowing te forces on te car. Write te equations for te conditions for static equilibrium along te orizontal and vertical directions. (c) Calculate te tension in te cable. Assume tere is no friction between te road and te tires. 5. A student puses on a lawn mower from rest parallel to te andle of te mower. Te student puses wit a force of magnitude 42 N. Te andle makes an angle of 358 to te orizontal. Te mower accelerates across a level drivewa wit negligible friction on te mower toward te lawn, 5.0 m awa. Te mass of te lawn mower is 18 kg. K/U T/I C (a) Draw te FBD of te mower. Calculate te acceleration of te mower. (c) Calculate te normal force acting on te mower. (d) Calculate te velocit of te mower wen it reaces te lawn. 6. In a psics eperiment, a 1.3 kg dnamics cart is placed on a ramp inclined at 258 to te orizontal. Te cart is initiall at rest but is ten pulled up te ramp wit a force sensor. Te force sensor eerts a force on te cart parallel to te ramp. Negligible friction acts on te cart. T/I (a) Wat force is required to pull te cart up te ramp at a constant velocit? Wat force is required to pull te cart up te ramp at an acceleration of 2.2 m/s 2? 2.3 Appling Newton s Laws of Motion 83
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