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Section 4-7 Solving Problems with Newton’s Laws; Free Body Diagrams

Section 4-7 Solving Problems with Newton’s Laws; Free Body Diagrams. “It sounds like an implosion!”. Forces are VECTORS !! Newton’s 2 nd Law: ∑ F = ma ∑ F = VECTOR SUM of all forces on mass m  VECTOR addition is needed add forces in the 2 nd Law ! Forces add according to

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Section 4-7 Solving Problems with Newton’s Laws; Free Body Diagrams

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  1. Section 4-7Solving Problems with Newton’s Laws; Free Body Diagrams “It sounds like an implosion!”

  2. Forces are VECTORS!! • Newton’s 2nd Law: ∑F = ma ∑F =VECTORSUMof all forces on mass m  VECTORaddition is needed add forces in the 2nd Law! • Forces add according to the rules of VECTOR ADDITION! (Ch. 3)

  3. Problem Solving Procedures • Draw a sketch. For each object separately, sketch a free-body diagram, showing allthe forces acting on that object. Make the magnitudes & directions as accurate as you can. Label each force. • Resolve vectors into components. • Apply Newton’s 2nd Law separately to each object & for eachvector component. • Solve for the unknowns. • Note that this often requires algebra, like solving 2 linear equations in 2 unknowns!

  4. Example Find the resultant force FR

  5. Example Find the resultant force FR FR = [(F1)2 + (F2)2](½) = 141 N tanθ = (F2/F1) = 1, θ = 45º

  6. Example 4-9 Find the resultant force FR If the boat moves with acceleration a, ∑F = FR = ma FRx = max, FRy = may

  7. Example 4-9 • Illustrates the procedures for Newton’s 2nd Law problems: STEP 1:Sketchthe situation!! • Sketch a “Free Body” diagram for EACHbody in problem & draw ALL forces acting on it. • Part of your grade on exam & quiz problems! STEP 2:Resolvethe forces on each body into components • Use a convenient choice of x,y axes • Use the rules for finding vector components from Ch. 3. STEP 3:Apply N’s 2nd Law to EACH OBJECT SEPARATELY ∑F = ma, Note: This is theLASTstep, NOTthe first! We NEED A SEPARATE equation like this for each object! Resolved into components: ∑Fx = max, ∑Fy = may

  8. Conceptual Example 4-10 Moving atCONSTANT v, withNO friction, which free body diagram is correct?

  9. Example 4-11 A box of mass m = 10 kg is pulled by an attached cord along a horizontal smooth (frictionless!) surface of a table. The force exerted is FP = 40.0 N at a 30.0° angle as shown. Calculate: a. The acceleration of the box. b. The magnitude of the upward normal force FN exerted by the table on the box. Free Body Diagram The normal force, FN is NOT always equal & opposite to the weight!!

  10. Example 4-12 Two boxes are connected by a lightweight (massless!) cord & are resting on a smooth (frictionless!) table. The masses are mA = 10 kg & mB = 12 kg. A horizontal force FP = 40 N is applied to mA. Calculate:a. The acceleration of the boxes. b. The tension in the cord connecting the boxes. Free Body Diagrams

  11. Example 4-13 (“Atwood’s Machine”) Two masses suspended over a (massless frictionless) pulley by a flexible (massless) cable is an “Atwood’s machine”. Example: elevator & counterweight. Figure: Counterweight mC = 1000 kg. Elevator mE = 1150 kg. Calculate a. The elevator’s acceleration. b. The tension in the cable. a   a aE = - a Free Body Diagrams aC = a

  12. Conceptual Example 4-14 Advantage of a Pulley A mover is trying to lift a piano (slowly) up to a second-story apartment. He uses a rope looped over 2 pulleys. What force must he exert on the rope to slowly lift the piano’s mg = 2000 N weight? mg = 2000 N Free Body Diagram

  13. Example 4-15: Accelerometer A small mass m hangs from a thin string & can swing like a pendulum. You attach it above the window of your car as shown. What angle does the string make a.When the car accelerates at a constanta = 1.20 m/s2? b.When the car moves at constant velocity,v = 90 km/h? Free Body Diagram

  14. Inclined Plane Problems The tilted coordinate System isconvenient, but not necessary. Engineers & scientists MUSTunderstand these! Understand∑F = ma& how to resolve it into x,y components in the tilted coordinate system!! a

  15. Example 4-1b: Sliding Down An Incline A box of mass m is placed on a smooth (frictionless!) incline that makes an angle θwith the horizontal. Calculate: a. The normal force on the box. b. The box’s acceleration. c. Evaluate both for m = 10 kg & θ = 30º Free Body Diagram

  16. Example = 300 N Free Body Diagram FT1x = -FTcosθ FT1y = -FTsinθ FT2x = FTcosθ FT2y = -FTsinθ

  17. Problem 32 Newton’s 2nd Law FT   FT Take up as positive! m= 65 kg mg = 637 N FT + FT - mg = ma 2FT -mg = ma FP = - FT ∑F = ma (y direction) on the woman + the bucket!  FP  a Free Body Diagram Newton’s 3rd Law!!  mg

  18. Problem 32 Solution Newton’s 2nd Law FT   FT Take up as positive! m= 65 kg mg = 637 N FT + FT - mg = ma 2FT -mg = ma FP = - FT ∑F = ma (y direction) on the woman + the bucket!  FP  a Free Body Diagram Newton’s 3rd Law!!  mg

  19. Problem 32 Solution The window washer pulls down on the rope with a tension force so the rope pulls up on her hands with a tension force The tension in The rope is also applied at the other end of the rope, where it attaches to the bucket. So there is another force pulling up on the bucket. The bucket-washer combination has a net force upwards. The free body diagram shows only forces on the bucket-washer, not forces exerted by them (the pull down on the rope by the person) or internal forces (normal force of bucket on person). (a) Write Newton’s second law in the vertical direction, with up as positive. The net force must be zero if the bucket and washer have constant speed. (b) Now the force is increased by 15%, so again write Newton’s second law, but with a non-zero acceleration.

  20. We draw free-body diagrams for each bucket. a.Since the buckets are at rest, their acceleration is 0. Write Newton’s 2nd Law for each bucket, calling UP the positive direction. Problem 33  FT1  a m1g  FT2 b.Now repeat the analysis, but with a non-zero acceleration. The free-body diagrams are unchanged.  FT2  a  m2g

  21. General Approach to Problem Solving • Read the problem carefully; then read it again. • Draw a sketch, then a free-body diagram. • Choose a convenient coordinate system. • List the known & unknown quantities; find relationships between the knowns & the unknowns. • Estimate the answer. • Solve the problem without putting in any numbers (algebraically); once you are satisfied, put the numbers in. • Keep track of dimensions. • Make sure your answer is REASONABLE!

  22. Chapter 4 Summary • Newton’s 1st Law: If the net force on an object is zero, it will remain either at rest or moving in a straight line at constant speed. • Newton’s 2nd Law: • Newton’s 3rd Law: • Weight is the gravitational force on an object. • Free-body diagrams are essential for problem-solving. Do one object at a time, make sure you have all the forces, pick a coordinate system & find the force components, & apply Newton’s 2nd Law along each axis.

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