Lecture 4 Dynamics: Newtons Laws of Motion

1. Lecture 4 Dynamics: Newtons Laws of Motion Objects undergo motion and accelerations. This is caused by an interaction between bodies. Such interactions are called forces. Recall most of our forces are contact forces Newton’s laws of motion: Published Newton’s Principia (1642)

2. Examples of Contact Forces: • A push or pull can be a force • Normal force • Tension in a string • Friction • Two balls colliding • Example of Noncontact forces • Gravitational force • Electric force

3. NEWTON’S FIRST LAW • An object continues in a state of rest or of motion at constant speed in a straight line unless acted upon by a net force. • If you push it and let it go, it will move at constant velocity. Example : Hockey puck of mass m on ice Block of Ice

4. Newton’s First Law of Motion Conceptual Example : 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?

5. NEWTON’S FIRST LAW • An object continues in a state of rest or of motion at constant speed in a straight line unless acted upon by a net force. • If you don’t push it, it won’t move. A body has inertia. Example : Hockey puck of mass m on ice Block of Ice

6. Inertia Another way of understanding the First Law Inertia is a bodies resistance to change due to forces. Inertia is related to mass. • Some examples of inertia • Hit a nail in a piece of wood on an anvil sitting • on your head • Mass on string (Demo)

7. Two different ways of breaking the string: Inertia and Tension Upper string breaks when you pull slowly because tension is greater Lower string breaks when you pull quickly because of inertia First pull fast see where it breaks Then pull slowly see where it breaks Explain

8. NEWTON’S SECOND LAW Sum of all external forces acting on the body = net force on body of mass m System Mass Acceleration Force SI kg m/s2 Newton (N) CGS g cm/s2 dyne (dyn) BE slug (sl) ft/s2 pound (lb)

9. Because force is a vector we have 3 equations relating force and acceleration each corresponding To a different coordinate

10. 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

11. Example Force to stop a car. What average net force is required to bring a 1500-kg car to rest from a speed of 30 m/s within a distance of 50 m?

12. NEWTON’S THIRD LAW When two bodies interact, the forces on the bodies due to each other are always equal in magnitude and opposite in direction. M Table

13. 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.

14. Newton’s Third Law of Motion Helpful notation: the first subscript is the object that the force is being exerted on; the second is the source.

15. Rules for drawing free body diagrams. The purpose is to Isolate the forces acting on one body • Draw a diagram • Represent the body by a point. • Each force acting on the body is represented by a vector with tail at the point and the length of vector indicating the approximate magnitude of the force. • It may be convenient to resolve the forces into components • Apply Newton’s second law and solve for the unknowns

16. Weight and the force of gravity • Falling objects accelerate at the rate of 9.81 m/s2 • or g. • From Newtons 2nd law, we know F= ma where a =g. • If a table or floor is in the way gravity is still acting • and trying to accelerate the object. This produces the • gravitational force acting on the object called weight. • But the object is not accelerating. Isn’t this a violation • of Newtons 2nd law • NO. Because the table or floor exerts an upward force • back on the object so that the net force is 0. This • upward force is called the Normal force.

17. What is the free body diagram of a block at rest on the table? N W M Table

18. Book leaning against a crate on a table at rest. What are the action –reaction pairs because of Newtons 3rd Law? Table T

19. Draw a free body diagram of the forces acting • on the crate NT B mg 2) Does the crate C accelerate?

20. N T f A crate of mass 310 kg is being pulled by a man as shown in the figure. What is the acceleration of the crate along the x direction? Man does not move. +y N +x W • First draw a free body diagram of the forces acting on the crate

21. Draw a free body diagram of the forces acting • on the crate y is vertical x is horizontal N T f W Now we want to apply Newton’s second Law to the x and y components independently.

22. N T f W A crate of mass 310 kg is being pulled by a man as shown in the figure. What is the acceleration of the crate along the x direction? Man does not move. +y N +x x component of forces in free body diagram

23. What is the normal force assuming there is no acceleration in the y direction? N T f W N y component of forces in free body diagram

24. Problem: What is the acceleration of the system of the two blocks and the contact force between the blocks? What is the net force on Block B? B A B A 65 N 24 kg 31 kg 65 N 24

25. Problem: What is the acceleration of the system of the two blocks and the contact force between the blocks? What is the net force on Block B? B A B A 65 N 24 kg 31 kg 65 N Student Version 25

26. Now lets look at tension in a string Tension in the string is equal to the weight = 10 N The scale reads the tension in the string

27. Is the tension in the string any different when I have weights pulling it down on both sides? 10N 10 N

28. a) b) c) Problem 12 kg 24 kg 31 kg T3=65 N kg 31 kg 24 kg 12 kg 65 N What is the acceleration of the system? Find T1 Find T2

29. 1 2 Rev George Atwood’s machine 1746 -1807Tutor Trinity College, Cambridge

30. Set up Coordinate system • Assume acceleration in some direction • Draw free body diagram for each body • Apply Newtons second law to each body 1) Assume mass 1 or M is going down T T T a 1 2 Mg mg +y +x

31. Free body diagram for each body and apply Newtons second law 1. 2. T T T a 1 2 Mg mg +y Solve eq 1 and 2 for T and a +x

32. Free body diagram for each body and apply Newtons second law 1. 2. T T T a 1 2 Mg mg +y +x

33. 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.

34. T Another example. Find tension T in the cord and the downward acceleration a. Draw Free Body Diagram of each body M and m assuming m is accelerating down: Frictionless pulley T

35. T Set up Coordinate systemAssume acceleration in some directionDraw free body diagram for each bodyApply Newtons second law to each body Frictionless pulley +y +x a a

36. +y +x

38. 2.25 m f Therefore, The smaller the angle the larger the magnification If then Tug-of-war demo illustrates how a small sideways force can produce a large horizontal force Suppose two guys in the tug of war are at 4.5 meters apart and I pull the rope out 0.15 meters. Then φ =4 degrees