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Chapter 5 • Applying Newton’s Laws Equilibrium An object is in equilibrium when the net force acting on it is zero. In component form, this is The net force on each man in the tower is zero. Slide 5-13 Equilibrium A hanging street sign with more than one force acting on it 1 2 3 1 + 2 + 3 =0 1 + 2 + 3 =0 Equilibrium What are the components of the forces? 1 1 1 3 2 2 0 1 + 2 + 3 = 0 3 0 1 + 2 + 3 =0 Slide 5-14 Example Problem A 100 kg block with a weight of 980 N hangs on a rope. Find the tension in the rope if A. the block is stationary. B. it’s moving upward at a steady speed of 5 m/s. C. it’s accelerating upward at 5 m/s2. Slide 5-15 Example Problem A 100 kg block with a weight of 980 N hangs on a rope. Find the tension in the rope if A. the block is stationary. = 0 = + = 0 100kg + = 0 m = 100kg ∙ 9.8 2 = 980N s Slide 5-15 A 100 kg block with a weight of 980 N hangs on a rope. Find the tension in the rope if A. B. C. the block is stationary. it’s moving upward at a steady speed of 5 m/s. it’s accelerating upward at 5 m/s2. = 0 = + = 0 100kg + = 0 m = 100kg ∙ 9.8 2 = 980N s Slide 5-15 A 100 kg block with a weight of 980 N hangs on a rope. Find the tension in the rope if A. B. C. the block is stationary. it’s moving upward at a steady speed of 5 m/s. it’s accelerating upward at 5 m/s2. = + = 100kg = − = − m m = 100kg 5 2 − −9.8 2 s s = 1480N Slide 5-15 Example Problem A wooden box, with a mass of 22 kg, is pulled at a constant speed with a rope that makes an angle of 25° with the wooden floor. If the coefficient of kinetic friction is = .5, what is the tension in the rope? Slide 5-16 A wooden box, with a mass of 22 kg, is pulled at a constant speed with a rope that makes an angle of 25° with the wooden floor. If the coefficient of kinetic friction is = .5, what is the tension in the rope? = + + = 0 = + = 0 = − − = − 25° Slide 5-16 A wooden box, with a mass of 22 kg, is pulled at a constant speed with a rope that makes an angle of 25° with the wooden floor. If the coefficient of kinetic friction is = .5, what is the tension in the rope? = − = − − cos = − sin = − − cos sin = − + cos = − 25° Slide 5-16 A wooden box, with a mass of 22 kg, is pulled at a constant speed with a rope that makes an angle of 25° with the wooden floor. If the coefficient of kinetic friction is = .5, what is the tension in the rope? cos sin = − + cos sin − = − cos sin − = − − = cos sin − Example Problem A ball weighing 50 N is pulled back by a rope to an angle of 20°. What is the tension in the pulling rope? Slide 5-19 Example Problem A ball weighing 50 N is pulled back by a rope to an angle of 20°. What is the tension in the pulling rope? Slide 5-19 A ball weighing 50 N is pulled back by a rope to an angle of 20°. What is the tension in the pulling rope? = + = 0 = − ° cos 20 = = + = 0 = − = 50N Slide 5-19 Using Newton’s Second Law Slide 5-20 Dynamics with the 2nd law s Σ = s + = Σ = s + = Example Problem A sled with a mass of 20 kg slides along frictionless ice at 4.5 m/s. It then crosses a rough patch of snow which exerts a friction force of -12 N. How far does it slide on the snow before coming to rest? Slide 5-21 A sled with a mass of 20 kg slides along frictionless ice at 4.5 m/s. It then crosses a rough patch of snow which exerts a friction force of -12 N. How far does it slide on the snow before coming to rest? Slide 5-21 A sled with a mass of 20 kg slides along frictionless ice at 4.5 m/s. It then crosses a rough patch of snow which exerts a friction force of -12 N. How far does it slide on the snow before coming to rest? Slide 5-21 A sled with a mass of 20 kg slides along frictionless ice at 4.5 m/s. It then crosses a rough patch of snow which exerts a friction force of -12 N. How far does it slide on the snow before coming to rest? 2 = 2 − 2 = + 2∆ =2 ∆ −2 ∆ = 2 ∆ = Slide 5-21 Mass and Weight • Mass and weight are not the same = m The moon’s gravity • The moon has about 1/6 of the gravity of earth m = 1 6 m Weightlessness • Falling doesn’t mean you have no weight • But this is the term we use anyway Mass and Weight –w = may = m(–g) w = mg Slide 5-23 Which of the following statements about mass and weight is correct? . m ea s sa m e ur e of .. ev ... .. a th e is w ei gh t Yo ur w ei gh t Yo ur is s m as s Yo ur is am e ea s m is a m as s Yo ur ev er yw he of th ... 25% 25% 25% 25% ur e A. Your mass is a measure of the force gravity exerts on you B. Your mass is same everywhere in the universe C. Your weight is the same everywhere in the universe D. Your weight is a measure of your resistance of being accelerated 30 Apparent Weight Slide 5-24 Example Problem A 50 kg student gets in a 1000 kg elevator at rest. As the elevator begins to move, she has an apparent weight of 600 N for the first 3 s. How far has the elevator moved, and in which direction, at the end of 3 s? Slide 5-25 A 50 kg student gets in an elevator at rest. As the elevator begins to move, she has an apparent weight of 600 N for the first 3 s. How far has the elevator moved, and in which direction, at the end of 3 s? = + = = 600N + = = 1 2 1 2 ∆ = + → ∆ = 2 2 0 1 + ∆ = 2 2 Slide 5-25 Normal Forces N m N = Normal Forces + N + − cos − sin Normal Forces = N + + Σ = − sin = Σ = − cos + N = − cos + N = 0 N = cos − cos − sin Static Friction fs max = µsn Slide 5-30 Kinetic Friction fk = µkn Slide 5-31 Coefficients of static friction are typically greater than coefficients of kinetic friction. 50% 50% Fa lse Tr u e A. True B. False Include static friction f = N N + + − cos − sin Kinetic and Rolling friction f = N f = N f f Working with Friction Forces Slide 5-32 Drag An object moving in a gas or liquid experiences a drag force Drag coefficient. Depends on details of the object’s shape. “Streamlining” reduces drag by making CD smaller. For a typical object, CD 0.5. Density of gas or liquid. Air has a density of 1.29 kg/m3. A is the object’s cross section area when facing into the wind. Drag depends on the square of the speed. This is a really important factor that limits the top speed of cars and bicycles. Going twice as fast requires 4 times as much force and, as we’ll see later, 8 times as much power. Slide 5-34 Drag = 1 2 2 Direction opposite of motion Drag = 1 2 4 is the fluid density A is the cross-sectional area is the drag coefficient(most objects have = 1/2) Terminal Speed Sky divers go no faster than this It’s about 150 mph for average objects At this speed… Σ = 0 Example Problem A car traveling at 20 m/s stops in a distance of 50 m. Assume that the deceleration is constant. The coefficients of friction between a passenger and the seat are μs = 0.5 and μk = 0.3. Could a 70 kg passenger slide off the seat if not wearing a seat belt? Slide 5-33 Example Problem A car traveling at 20 m/s stops in a distance of 50 m. Assume that the deceleration is constant. The coefficients of friction between a passenger and the seat are μs = 0.5 and μk = 0.3. Could a 70 kg passenger slide off the seat if not wearing a seat belt? m = 20 s ∆ = 50m Slide 5-33 Example Problem A car traveling at 20 m/s stops in a distance of 50 m. Assume that the deceleration is constant. The coefficients of friction between a passenger and the seat are μs = 0.5 and μk = 0.3. Could a 70 kg passenger slide off the seat if not wearing a seat belt? m = 20 s ∆ = 50m 0 2 = 2 + 2∆ = 2 − 2∆ = = ≷ Slide 5-33 Cross-Section Area Slide 5-35 Terminal Speed A falling object speeds up until reaching terminal speed, then falls at that speed without further change. If two objects have the same size and shape, the more massive object has a larger terminal speed. At terminal speed, the net force is zero and the object falls at constant speed with zero acceleration. Slide 5-36 Summer 2014 Beginning of lecture Applying Newton’s Third Law: Interacting Objects Slide 5-37 Example Problem Block A has a mass of 1 kg; block B’s mass is 4 kg. They are pushed with a force of magnitude 10 N across a frictionless surface. a. What is the acceleration of the blocks? b. With what force does A push on B? B push on A? Slide 5-39 Block A has a mass of 1 kg; block B’s mass is 4 kg. They are pushed with a force of magnitude 10 N across a frictionless surface. a. What is the acceleration of the blocks? b. With what force does A push on B? B push on A? = 10N = 1kg + 4kg 10N m = =2 2 5kg s 10N Slide 5-39 Block A has a mass of 1 kg; block B’s mass is 4 kg. They are pushed with a force of magnitude 10 N across a frictionless surface. a. What is the acceleration of the blocks? b. With what force does B push on A? A push on B? Force from block A on hand Force from hand on block A Slide 5-39 Block A has a mass of 1 kg; block B’s mass is 4 kg. They are pushed with a force of magnitude 10 N across a frictionless surface. a. What is the acceleration of the blocks? b. With what force does A push on B? B push on A? = ℎ + = 1kg m 2 2 s 10N + = 2N Force from hand on block A = −8N Force from block B on block A Slide 5-39 Block A has a mass of 1 kg; block B’s mass is 4 kg. They are pushed with a force of magnitude 10 N across a frictionless surface. a. What is the acceleration of the blocks? b. With what force does A push on B? B push on A? = = From Newton’s 3rd law = − − = − −8N = 8N = = 4kg m 2 2 s Force from block B on block A Force from block A on block B Slide 5-39 Which pair of forces is a “third law pair”? The string tension and the friction force acting on A The normal force on A due to B and the weight of A The normal force on A due to B and the weight of B The friction force acting on A and the friction force acting on B Th e st rin gt en sio n an d no th rm e fri al ... fo rc e Th o n en A or du m e al to fo .. rc e Th on ef A r ic du t io e to n fo .. rc e ac tin go n ... 25% 25% 25% 25% Th e A. B. C. D. Example Problem What is the acceleration of block B? Slide 5-42 Sum the forces on block B? = 20N + = = = = 20N + m = = 5.764 2 s Slide 5-42 Ropes and Pulleys All ropes are assumed massless with uniform tension All pulleys are assumed frictionless Example Problem Block A, with mass 4.0 kg, sits on a frictionless table. Block B, with mass 2.0 kg, hangs from a rope connected through a pulley to block A. What is the acceleration of block A? Slide 5-44 Block A, with mass 4.0 kg, sits on a frictionless table. Block B, with mass 2.0 kg, hangs from a rope connected through a pulley to block A. What is the acceleration of block A? Same string, same tension?? = = 0 ∙ Slide 5-44 Block A, with mass 4.0 kg, sits on a frictionless table. Block B, with mass 2.0 kg, hangs from a rope connected through a pulley to block A. What is the acceleration of block A? SUM THE FORCES on each mass separately = = = + = Note that the motions of these two masses are coupled Slide 5-44 Block A, with mass 4.0 kg, sits on a frictionless table. Block B, with mass 2.0 kg, hangs from a rope connected through a pulley to block A. What is the acceleration of block A? = + = − + = = − Newton’s Third But… = So just call them both − + = Now solve for 2.0kg ∙ 1 = = = + 6.0kg 3 Slide 5-44 Summary Slide 5-45 Summary Slide 5-46