2. 7/10/06ISP 209 - 2A1 The Laws of Motion …or, Newtonian mechanics

3. 7/10/06ISP 209 - 2A2 The Laws of Motion …or, Newtonian mechanics

4. 7/10/06ISP 209 - 2A3 Isaac Newton Newton believed that mathematics can describe nature, accurately. He solved the premier scientific problem of his day – to explain the motion of the planets. Where is this sculpture from?

5. 7/10/06ISP 209 - 2A4 1.The law of inertia. An object in motion remains in motion with constant velocity if the net force on the object is 0. 2.Force and acceleration. If the net force acting on an object of mass m is F, then the acceleration of the object is a = F/m. Or, F = ma. 3.Action and reaction. For every action there is an equal but opposite reaction. Newton’s Laws of Motion

6. 7/10/06ISP 209 - 2A5 Atwood machine with equal weights Net force = 0 implies constant velocity.

7. 7/10/06ISP 209 - 2A6 Newton’s second law of motion F = m a F = force acting on the moving object, a vector m = mass of the moving object a = acceleration of the moving object, a vector If the force is known, Newton’s second law states how the object (on which the force is acting) accelerates:

8. 7/10/06ISP 209 - 2A7 Understanding vectors A vector is a mathematical quantity that has both a magnitude and a direction. F = m a an equation of vectors

9. 7/10/06ISP 209 - 2A8 Three kinds of acceleration Example: Circular motion has centripetal acceleration and centripetal force; F = ma. m slows down m speeds up m changes direction

10. 7/10/06ISP 209 - 2A9 Quantitative applications of F = ma

11. 7/10/06ISP 209 - 2A10 Mass is a property of matter = quantitative measure of inertia. The standard unit of mass is the kilogram (kg). A gram is 0.001 kg. This picture shows the international prototype 1 kg mass standard. It is a platinum-iridium cylinder kept in Sevres, France. Any physical object has a mass m, which could be measured against a standard, e.g., using a balance.

12. 7/10/06ISP 209 - 2A11 Force = a push or a pull exerted on one object by another object. The standard unit of force is the newton (N). The newton is defined in terms of more fundamental units by 1 N = 1 kg m/s 2 (consistent with F = m a)

13. 7/10/06ISP 209 - 2A12 Acceleration and Force Given the mass of the object, and given the force that is acting on it, the acceleration is If necessary, identify a and F as vectors and take into account their direction (both in the same direction!)

14. 7/10/06ISP 209 - 2A13 Example. Playing catch with a softball The trajectory has 3 parts. the throw the catch free fall What is the force acting on the ball, during each part of the trajectory?

15. 7/10/06ISP 209 - 2A14 Example Problem A dragster accelerates from 0 to 60 mi/hr in 6 seconds. Calculate the force on the car if the mass is 10 3 kg. Answer: 4.47 x 10 3 N

16. 7/10/06ISP 209 - 2A15 Two examples of forces weight (the force of gravity on an object) string tension (the force exerted by a taut string)

17. 7/10/06ISP 209 - 2A16 Gravity What is the force acting on the mass m due to the Earth’s gravity? If released, the acceleration of m would be … By Newton’s second law the force on m must be … By the way, what is the reaction force? Solution That is, the magnitude, or strength, is mg and the direction is downward. g = 9.81 m/s 2 F

18. 7/10/06ISP 209 - 2A17 Weight = the force of gravity The weight of an object is, by definition, the strength of the force of gravity pulling the object downward. force of gravity W = m g newtons kg

19. 7/10/06ISP 209 - 2A18 Example: What is the weight of a 1 kg mass near the Earth’s surface? W = mg = 9.81 N Or, W = 2.21 pounds

20. 7/10/06ISP 209 - 2A19 Mass and Weight  different! ► Mass is an intrinsic property of an object. It is completely determined by the number and type of atoms that make up the object. It does not depend on the environment in which the object is located. ► But weight is different. Weight depends on both the object itself, and on some other object that exerts the gravitational force. So, for example, the mass of an object would be the same on the moon or the Earth; but the weight would be different.

21. 7/10/06ISP 209 - 2A20 An astronaut would not need a car jack to change a flat tire on the Moon Buggy.

22. 7/10/06ISP 209 - 2A21 Analyze free fall including air friction The aerodynamic drag force (C) depends on the size, shape and surface roughness; it is about the same for both balls. The gravitational acceleration (g) is independent of mass. Effect of air is inversely proportional to mass: heavy --- small effect of air light --- large effect of air

23. 7/10/06ISP 209 - 2A22 String tension Example. Suppose a string can withstand string tension 500 N without breaking. What is the maximum mass M that it can hold suspended in Earth’s gravity? Answer: 51.0 kg The force exerted by the string, at either end: direction is parallel to the string magnitude (same at both ends) is called the tension

24. 7/10/06ISP 209 - 2A23 Analyze the motion of a pendulum. Solve by calculus.

25. 7/10/06ISP 209 - 2A24 More example problems 3. A gymnast weighs 100 pounds. (a) What is her weight in newtons? (b) What is her mass? 4. A car of 2000 pounds moving 30 mi/hr crashes into a brick wall. The collision lasts 0.3 seconds. Calculate the force acting on the car during the collision. Express the answer in both newtons and pounds. 45.3 kg 445 N 4.05 x 10 4 N or 9,100 pounds

26. 7/10/06ISP 209 - 2A25 Newton’s second law and calculus differential equation Isaac Newton invented calculus to solve the equations of motion; i.e., to calculate motion for the force that is acting. Generally, calculus is the mathematics that describes continuous change.

27. 7/10/06ISP 209 - 2A26

28. 7/10/06ISP 209 - 2A27 Quiz Question A car accelerates away from a stop sign with acceleration 0.1 g ( = 0.981 m/s 2 ). The mass of the driver is 50 kg. What is the force on the driver? (Be sure to include the unit of measurement!)