Chapter 4 Dynamics: Newton s Laws of Motion

Transcription

1 Chapter 4 Dynamics: Newton s Laws of Motion

2 Units of Chapter 4 Force Newton s First Law of Motion Mass Newton s Second Law of Motion Newton s Third Law of Motion Weight the Force of Gravity; and the NormalForce Solving Problems with Newton s Laws: Free Body Diagrams Problem Solving A General Approach

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4 4 1 Force A force is a push or pull An object at rest needs a force to get it A force is a push or pull. An object at rest needs a force to get it moving; a moving object needs a force to change its velocity.

5 4 1 Force Force is a vector, having both magnitude and direction. The magnitude of a force can be measured using a spring scale.

6 4 2 Newton s First Law of Motion It may seem as though it takes a force to keep an object moving. Push your book across a table when you stop pushing, it stops moving. But now, throw a ball across the room. The ball keeps moving after you let it go, even though you are not pushing it any more. Why? It doesn t take a force to keep an object moving in a straight line it takes a force to change its motion. Your book stops because the force of friction stops it.

7 4 2 Newton s First Law of Motion This is Newton s first tlaw, which h is often called the law of inertia: Every object continues in its state of rest, or of uniform velocity in a straight line, as long as no net force acts on it.

8 4 2 Newton s First Law of Motion Conceptual lexample 4 1: Newton s first law. A school bus comes to a sudden stop, and all of the backpacks on the floor start to slide forward. What force causes them to do that?

9 4 3 Mass Mass is the measure of inertia of an object, sometimes understood as the quantity of matter in the object. In the SI system, mass is measured in kilograms. Mass is not weight. Mass is a property of an object. Weight is the force exerted on that object by gravity. If you go to the Moon, whose gravitational acceleration is about 1/6 g, you will weigh much less. Your mass, however, will be the same.

10 4 4 Newton s Second Law of Motion Newton s second law is the relation between acceleration and force. Acceleration is proportional to force and inversely proportional to mass. It takes a force to change either the direction or the speed of an object. More force means more acceleration; the same force exerted on a more massive object will yield less acceleration.

11 4 4 Newton s Second Law of Motion Force is a vector, so is true along each coordinate axis. The unit of force in the SI system is the newton (N). Note that the pound is a unit of force, not of mass, and can therefore be equated to newtons but notto to kilograms.

12 4 4 Newton s Second Law of Motion Example 4 2: Force to accelerate a fast car. Estimate the net force needed to accelerate (a) a 1000 kg car at ½ g; (b) a 200 g apple at the same rate. Example 4 3: Force to stop a car. What average net force is required to bring a 1500 kg car to rest from a speed of 100 km/h within a distance of 55 m?

13 4 5 Newton s Third Law of Motion Any time a force is exerted on an object, that force is caused by another object. Newton s third law: Whenever one object exerts a force on a second object, the second exerts an equal force in the opposite direction on the first.

14 4 5 Newton s Third Law of Motion A key to the correct application of the third law is that the forces are exerted on different objects. Make sure you don t use them as if they were acting on the same object.

15 4 5 Newton s Third Law of Motion Rocket propulsion can also be explained using Newton s third law: hot gases from combustion spew out of the tail of the rocket at high speeds. The reaction force is what propels the rocket. Note that the rocket does not need anything to push against.

16 4 5 Newton s Third Law of Motion Conceptual Example 4 4: What exerts the force to move a car? Response: A common answer is that the engine makes the car move forward. But it is not so simple. The engine makes the wheels go around. But tifth the tires are on slick ice or deep mud, they just spin. Friction is needed. On firm ground, the tires push backward against the ground because of friction. By Newton s third law, the ground pushes on the tires in the opposite direction, accelerating the car forward.

17 4 5 Newton s Third Law of Motion Helpful notation: the firstsubscript is the object that the force is being exerted on;the Helpful notation: the first subscript is the object that the force is being exerted on; the second is the source.

18 4 5 Newton s Third Law of Motion Conceptual Example 4 5: Third law clarification. Michelangelo s assistant has been assigned the task of moving a block of marble using a sled. He says to his boss, When I exert a forward force on the sled, the sled exerts an equal and opposite force backward. So how can I ever start it moving? No matter how hard I pull, the backward reaction force always equalsmy forward force, so the net force must be zero. I ll never be able to move this load. Is he correct?

19 4 6 Weight the Force of Gravity; and the Normal Force Weight is the force exerted on an object by gravity. Close to the surface of the Earth, where the gravitational force is nearly constant, the weight of an object of mass m is: where

20 4 6 Weight the Force of Gravity; and the Normal Force An object at rest must have no net force on it. If it is sitting on a table, the force of gravity is still there; what other force is there? The force exerted perpendicular to a surface is called the normal force. It is exactly as large as needed to balance the force from the object. (If the required force gets too big, something breaks!)

21 4 6 Weight the Force of Gravity; and the Normal Force Example 4 6: Weight, normal force, and a box. A friend has given you a special gift, a box of mass kg with a mystery surprise inside. The box is resting on the smooth (frictionless) horizontal surface of a tbl table. (a) Determine the weight of the box and the normal force exerted onitby the table. (b) Now your friend pushes down on the box with a force of 40.0N. Again determine the normalforce exerted on the box by the table. (c) If your friend pulls upward on the box with a force () y p p of 40.0 N, what now is the normal force exerted on the box by the table?

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25 4 6 Weight the Force of Gravity; and the Normal Force Example 4 7: Accelerating the box. What happens when a person pulls upward on the box in the previous example with a force greater than the box s weight, say N?

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27 4 6 Weight the Force of Gravity; and the Normal Force Example 4 8: Apparent weight loss. A 65 kg woman descends in an elevator that briefly accelerates at 0.20gg downward. She stands on a scale that reads in kg. (a) During this acceleration, what is her weight and what does the scale read? (b) What does the scale read when the elevator descends at a (b) What does the scale read when the elevator descends at a constant speed of 2.0 m/s?

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29 4 7 Solving Problems with Newton s Laws: Free Body Diagrams 1. Draw a sketch. 2. For one object, draw a free body diagram, showing all the forces acting on the object. Make the magnitudes and directions as accurate as you can. Label each force. If there are multiple objects, draw a separate diagram for each one. 3. Resolve vectors into components. 4. Apply Newton s second law to each component. 5. Solve.

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31 4 7 Solving Problems with Newton s Laws: Free Body Diagrams Example 4 11: Pulling the mystery box. Suppose a friend asks to examine the 10.0 kg box you were given previously, hoping to guess what is inside; and you respond, Sure, pull the box over to you. She then pulls the box by the attached cord along the smooth surface of the table. The magnitude of the force exerted tdby the person is F P = N, and it is exerted at a 30.0 angle as shown. Calculate (a) the acceleration of the box, and (b) the magnitude of the upward force F N exerted by the table on the box.

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34 4 7 Solving Problems with Newton s Laws: Free Body Diagrams Example 4 12: Two boxes connected by a cord. Two boxes, A and B, are connected by a lightweight g cord and are resting on a smooth table. The boxes have masses of 12.0 kg and 10.0 kg. A horizontal force of N is applied to the 10.0 kg 0 box. Find (a) () the acceleration of each box, and (b) the tension in the cord connecting the boxes.

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36 4 7 Solving Problems with Newton s Laws: Free Body Diagrams Example 4 13: Elevator and counterweight (Atwood s machine). A system of two objects suspended over a pulley by a flexible cable is sometimes referred to as an Atwood s machine. Here, let the mass of the counterweight be 1000 kg. Assume the mass of the empty elevator is 850 kg, and its mass when carrying four passengers is 1150 kg. For the latter case calculate (a) the acceleration of the elevator and (b) the tension in the cable.

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38 4 7 Solving Problems with Newton s Laws: Free Body Diagrams Conceptual Example 4 14: The advantage of a pulley. A mover is trying to lift a piano (slowly) up to a second story apartment. He is using a rope looped over two pulleys as shown. What force must he exert on the rope to slowly lift the piano s 2000 N weight?

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40 4 7 Solving Problems with Newton s Laws: Free Body Diagrams Example 4 15: Accelerometer. A small mass m hangs from a thin string and 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 constant a = 1.20 m/s 2, and (b) when the car moves at constant velocity, v = 90 km/h?

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42 4 7 Solving Problems with Newton s Laws: Free Body Diagrams Example 4 16: Box slides down an incline. A box of mass m is placed on a smooth incline that A box of mass m is placed on a smooth incline that makes an angle θ with the horizontal. (a) Determine the normal force on the box. (b) Determine the box s acceleration. (c) Evaluate for a mass m = 10 kg and an incline of θ = 30.

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44 4 8 Problem Solving A General Approach 1. Read the problem carefully; then read it again. 2. Draw a sketch, and then a free body diagram. 3. Choose a convenient coordinate system. 4. List the known and unknown quantities; find relationships bt between the knowns and the unknowns. 5. Estimate the answer. 6. Solve the problem without putting in any numbers (algebraically); once you are satisfied, put the numbers in. 7. Keep track of dimensions. 8. Make sure your answer is reasonable.

45 Summary of Chapter 4 Newton s first 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 second law: Newton s third law: Weight is the gravitational force on an object. Free body diagrams are essential for problem solving. Do one object at a time, make 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 and find the force components, and apply Newton s second law along each axis.

46 5 1 Applications of Newton s Laws Involving Friction Friction is always present when two solid surfaces slide along each other. The microscopic details are not yet fully understood.

47 5 1 Applications of Newton s Laws Involving Friction Slidingfrictioniscalled is called kineticfriction friction. Approximation of the frictional force: F fr = μ k F N. Here, F N is the normal force, and μ k is the coefficient of kinetic friction, which is different for each pair of surfaces.

48 5 1 Applications of Newton s Laws Involving Friction Static friction applies when two surfaces are at rest with respect to each other (such as a book sitting on a table). The static tti fiti frictional lforce is as big as it needs to be to prevent slipping, i up to a maximum value. F fr μ s F N. Usually it is easier to keep an object sliding than it is to get it started.

49 5 1 Applications of Newton s Laws Involving Friction Note that, in general, μ s > μ k.

50 5 1 Applications of Newton s Laws Involving Friction Example 5 1: Friction: static and kinetic. Our 10.0 kg mystery box rests on a horizontal floor. The coefficient of static friction is 0.40 and the coefficient of kinetic friction is Determine the force of friction acting on the box if a horizontal external applied force is exerted on it of magnitude: (a) 0, (b) 10 N, (c) 20 N, (d) 38 N, and (e) 40 N.

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52 5 1 Applications of Newton s Laws Involving Friction Conceptual Example 5 2: A box against a wall. You can hold a box against a rough wall and prevent it from slipping down by pressing hard horizontally. How does the application of a horizontal force keep an object from moving vertically?

53 5 1 Applications of Newton s Laws Involving Friction Example 5 3: Pulling against friction. A 10.0 kg box is pulled along a horizontal surface by a force of 40.0 N applied at a 30.0 angle above horizontal. The coefficient of kinetic friction is Calculate the acceleration.

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55 5 1 Applications of Newton s Laws Involving Friction Conceptual Example 5 4: To push or to pull a sled? Your little sister wants a ride on her sled. If you are on flat ground, will you exert less force if you push her or pull her? Assume the same angle θ in each case.

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57 5 1 Applications of Newton s Laws Involving Friction Example 5 5: Two boxes and a pulley. Two boxes are connected by a cord running over a pulley. The coefficient of kinetic friction between box A and the table is We ignore the mass of the cord and pulley and any friction in the pulley, which means we can assume that a force applied to one end of the cord will have the same magnitude at the other end. We wish to find the acceleration, a, of the system, which will have the same magnitude for both boxes assuming the cord doesn t stretch. As box B moves down, box A moves to the right.

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59 5 1 Applications of Newton s Laws Involving Friction Example 5 6: The skier. This skier is descending a 30 slope, at constant g p, speed. What can you say about the coefficient of kinetic friction?

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61 5 1 Applications of Newton s Laws Involving Friction Example 5 7: A ramp, a pulley, and two boxes. Box A, of mass kg, rests on a surface inclined at 37 to the horizontal. It is connected by a lightweight cord, which passes over a massless and frictionless pulley, to a second box B, which hangs freely as shown. (a) If the coefficient ofstatic friction is 0.40, determine what range ofvalues for mass B will keep the system at rest. (b) If the coefficient of kinetic friction is 0.30, and m B = 10.0 kg, determine the acceleration of the system.

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63 4. circular motion

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65 10 1 Angular Quantities In purely rotational motion, all points on the object move in circles around the axis of rotation ( O ). The radius of the circle is R. All points on a straight line drawn through the axis move through the same angle in the same time. The angle θ in radians is defined: l, R where l is the arc length.

66 The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again Angular Quantities Example 10 1: Birds of prey in radians. A particular bird s eye can just distinguish objects that subtend an angle no smaller than about 3 x 10 4 rad. (a) How many degrees is this? (b) (b) How small an object can the bird just distinguish when flying at a height of 100 m?

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68 10 1 Angular Quantities Angular displacement: The average angular velocity is dfi defined das the total t angular displacement divided by time: The instantaneous angular velocity:

69 10 1 Angular Quantities The angular acceleration is the rate at which the angular velocity changes with time: The instantaneous acceleration:

70 10 1 Angular Quantities Every point on a rotating body has an angular velocity ω and a linear velocity v. They are related:

71 10 1 Angular Quantities Conceptual Example 10 2: Is the lion faster than the horse? On a rotating carousel or merry go round, one child sits on a horse near the outer edge and another child sits on a lion halfway out from the center. (a) () Which child has the greater linear velocity? (b) Which child has the greater angular velocity?

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73 10 1 Angular Quantities Objects farther from the axis of rotation will move faster.

74 10 1 Angular Quantities If the angular velocity of a rotating object changes, it has a tangential acceleration: Even if the angular velocity is constant, each point on the object has a centripetal acceleration:

75 10 1 Angular Quantities Here is the correspondence between linear and rotational quantities:

76 10 1 Angular Quantities Example 10 3: Angular and linear velocities and accelerations. A carousel is initially at rest. At t = 0 it is given a constant angular acceleration α = rad/s 2,which increases its angular velocity for 8.0 s. At t = 8.0 s, determine the magnitude of the following quantities: (a) the angular velocity of the carousel; (b) the linear velocity of a child located 2.5 m from the center; (c) (d) (e) the tangential (linear) acceleration of that child; the centripetal acceleration of the child; and the total linear acceleration of the child.

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79 10 1 Angular Quantities The frequency is the number of complete revolutions per second: Frequencies are measured in hertz: The period is the time one revolution takes:

80 10 1 Angular Quantities Example 10 4: Hard drive. The platter of the hard drive of a computer rotates at 7200 rpm (rpm = revolutions per minute = rev/min). (a) What is the angular velocity (rad/s) of the platter? (b) If the reading head of the drive is located 3.00 cm from the rotation axis, what is the linear speed of the point on the platter just below it? (c) If a single bit requires 0.50 μmof length along the direction of motion, how many bits per second can the writing head write when it is 3.00 cm from the axis?

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82 Angular Quantities Example 10 5: Given ω as function of time. A disk of radius R = 3.0 m rotates at an angular velocity ω = ( t) rad/s, where t is in seconds. At the instant t = 2.0 s, determine (a) the angular acceleration, and (b) (b) h d d h f h l (b) (b) the speed v and the components of the acceleration a of a point on the edge of the disk.

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84 5 2 Uniform Circular Motion Kinematics Uniform circular motion: motion in a circle of constant radius at constant speed Instantaneous velocity is always tangent to the circle.

85 5 2 Uniform Circular Motion Kinematics Looking at the change in velocity in the limit that the time interval becomes infinitesimally small, we see that.

86 5 2 Uniform Circular Motion Kinematics This acceleration is called the centripetal, or radial, acceleration, and it points toward the center of the circle.

87 5 2 Uniform Circular Motion Kinematics Example 5 8: Acceleration of a revolving ball. A 150 g ball at the end of a string is revolving uniformly in a horizontal circle of radius m. The ball makes 2.00 revolutions in a second. What is itscentripetal acceleration?

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90 5 2 Uniform Circular Motion Kinematics Example 5 9: Moon s centripetal acceleration. The Moon s nearly circular orbit about the Earth has a radius of about 384,000 km and a period T of 27.3 days. Determine the acceleration of the Moon toward the Earth.

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92 5 2 Uniform Circular Motion Kinematics A centrifuge works by spinning very fast. This means there must be a very large centripetal force. The object at A would go in a straight line but for this force; as it is, it winds up at B.

93 5 2 Uniform Circular Motion Kinematics Example 5 10: Ultracentrifuge. The rotor of an ultracentrifuge rotates at 50,000 rpm (revolutions per minute). A particle at the top of a test tube is 6.00 cm from the rotation axis. Calculate its centripetal acceleration, in g s.

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