Chapter 4: Newton s Laws of Motion

Transcription

1 Chapter 4: Newton s Laws of Motion We will study classical motion: No quantum mechanics No relativity We introduce the concept of force and define it in terms of the acceleration of a standard d body

2 Intuitively, we know that force is a push or pull. Forces come in different classes (types): Contact Field (or action at a distance) Force F is a vector quantity F

3 In 1686, Newton presented The Laws of Motion: Newton s First Law Every body continues in its state of rest or of uniform speed in a straight line as long as no net force acts on it.

4 Examples 1. An object that is moving and that continues to move with constant velocity when the summation of the forces acting on it is zero. 2. An object at rest that remains at rest. 3. What about pushing a chair? 4. What happens when you turn a corner quickly in your car?

5 We know from experience that different objects resist a change in motion differently. Example: push a door push a train Not the same response!

6 Mass The tendency of an object to resist a change in its velocity is called inertia. The measure of inertia is mass. SI units of mass is the kilogram (kg). Newton s First Law tells us about motion if F = 0. What if F 0?

7 Newton s Second Law The acceleration of an object is directly proportional to the resultant force acting on it and inversely proportional to its mass. The direction of the acceleration is the direction of the resultant force. F = ma The resultant force acting on the mass m.

8 The Vector Nature of Forces In the formula F = ma, F is the total (net or resultant) t) force acting on the object. We must consider the vector sum of all forces acting on an object. We can also consider each dimension separately: F = ma F F x y = ma = ma x y F z ma z

9 Example An object of mass 5 kg undergoes an acceleration of 8 m/s 2 ŷ. What is the force on that object? F = ma = (5 kg)(8 m/s 2 ) = 40 kg m/s 2 ŷ The force is in the same direction as the acceleration.

10 Units The SI unit of Force is the Newton defined as: 1 N = 1 kg m/s 2 For comparison: 1 lb. = N

11 Example A catcher stops a 92 mi/h pitch in his glove, bringing it to rest in 0.15 m. If the force exerted by the catcher is 803 N, what is the mass of the ball?

12 Newton s Third Law action and re-action If object 1 exerts a force F on object 2, then object 2 exerts a force F on object 1. Forces come in pairs. The force pairs act on different objects. The forces have the same magnitude and opposite direction. Example: I push on the wall with a force of 20 N. The wall pushes back on me with a force of 20 N in the opposite direction.

13 Weight The weight of any object on the Earth is the gravitational force exerted on it by the Earth: W = mg Note: Weight is a force (and therefore a vector). Weight is not equivalent to mass, but it is proportional. Can a person s weight be zero? When we say we want to lose weight, what do we really mean?

14 W = mg Notice the direction of the weight! Mass m W = mg

15 Problem Solving 1. Draw a sketch of the situation. 2. Consider only one body at a time and draw a free body diagram for it, showing all of the forces acting on it. 3. Resolve all of the vectors into x and y components. 4. Apply Newton s laws to each component separately. For example, evaluate the net force in the x direction. 5. Solve the equation for the unknown.

16 Free-Body Diagrams Drawing a picture always helps us understand what is going gon! Consider a tug of war between two companies of rats Echo Alpha What is the net force on the rope? F E F r Since F E is larger than F A, Echo company wins. F A

17 Free-Body Diagrams Continued

18 What about a box sitting on the floor? The normal force, F N, is the force acting perpendicular to the surface to the surface the box is sitting on. F N Fx = W + FN = mg+ FN = = mg FN 0 W=mg

19 Example What is the net force on a 10 kg box that is being pushed across the floor at a constant velocity of 2.5 m/s? Draw a free-body diagram.

20 What about more complicated systems? Consider two forces acting on a ball. y F 1 F 2 x These arrows represent force vectors. All you have to do is add the vectors. Graphically, y F 1 F 2 F NET x

21 In the picture we just drew, assume that force 1 has a magnitude of 10 N directed at 230 o relative to the x axis and force 2 has a magnitude of 2 N at an angle of 0 o relative to the x axis. Calculated the resultant vector. F 1 F 2 x

22 Calculate the vector components, F 1x = 10N cos 230 o = -6.4N F 1y = 10N sin 230 o = -7.7N F 2x = 2N cos 0 o = 2N F 2y = 2N sin 0 o = 0N So the resultant vector is F = Niˆ N ˆ j r Magnitude, Direction, θ = 2 2 F r = ( 4.4N) + ( 7.7N) = 8. 9N tan 1 F F ry rx = tan 1 7.7N 4.4N = 60 o -x θ -y

23 Example 34 A 7500 kg helicopter accelerates upward at 0 52 m/s A 7500 kg helicopter accelerates upward at 0.52 m/s while lifting a 1200 kg car. (a) What is the lift force exerted by the rotors on the air? (b) What is the tension in the cable that connects the car to the helicopter?

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25 Example A box of mass m slides down a frictionless inclined plane with an angle θ relative to the horizontal. (a) Draw a free body diagram. (b) What is the acceleration of the box down the ramp? (c) Does the acceleration depend on mass? (d) If θ = 45, what is the acceleration of the box?

26 Example: A farm tractor tows a 4300-kg trailer up a 26 incline at a steady speed of 3.0 m/s. What force does the tractor exert on the trailer? (Ignore friction.)

27 Since the trailer is not accelerating, the sum of the forces along the incline = zero. F -mgsinθ = 0 tractor F tractor = sin 26 = N 18 kn

28 Example 19 The cable s pporting a 2100 kg ele ator has a ma im m 19. The cable supporting a 2100 kg elevator has a maximum strength of 21,750 N. What maximum upward acceleration can it give the elevator without breaking?

29 Example 23. An exceptional standing jump would raise a person 0.80 m off the ground. To do this, what force must a 61 kg person exert against the ground? Assume the person crouches a distance of 0.20 m prior to jumping, and thus the upward force has this distance to act over before he leaves the ground.

30 Example 11. A fisherman yanks a fish out of the water with an acceleration of 3.5 m/s 2 using very light fishing line that has a breaking strength of 25 N. The fisherman unfortunately loses the fish as the line snaps. What can you say about the mass of the fish?

31 Example 31. The two forces F 1 and F 2 shown in the figures below act on a 29.0 kg object on a frictionless tabletop. If F 1 =20.2N 2N and F 2 =26.0N, find the net force on the object and its acceleration for each situation a and b.

32 Example 6. What average force is required to stop a 1050 kg car in 7.0 s if it is traveling at 90 km/h.

33 Example 41. A block is given an initial speed of 4.0 m/s up the 22 plane shown in the figure below. (a) How far up the plane will it go? (b) how much time will have elapsed before it returns to its starting position? Ignore friction.