1. Mechanics 105 Forces of friction (static, kinetic) Uniform circular motion Nonuniform circular motion Velocity dependent forces Numerical methods Fundamental forces Gravitational field More applications of Newton’s laws (chapter five)

2. Mechanics 105 Friction Force acting parallel to an interface that opposes the relative motion Static – frictional force opposite to applied force - magnitude f s where  s is the coefficient of static friction and n is the magnitude of the normal forces between the surfaces the equality holds just as the object starts to slip

3. Mechanics 105 Friction Kinetic – frictional force opposite to relative motion – magnitude f k where  k is the coefficient of static friction and n is the magnitude of the normal forces between the surfaces the kinetic frictional force is constant  s and  k are constants that depend on the nature of the surfaces Usually,  s >  k

4. Mechanics 105 Friction Note: Static friction is not constant – it is whatever is needed to match the applied force, up to the limit of  S n As the applied force increases, the static frictional force also increases, until the limit, then the object begins to slide, and the frictional force goes to a constant value

5. Mechanics 105 Friction ConcepTest Examples Demo

6. Mechanics 105 Example m1m1 m2m2 m1m1 +y +x m2m2

7. Mechanics 105 Example

8. Mechanics 105 Question What do you call a broken boomerang?

9. Mechanics 105 Question What do you call a broken boomerang? Answer: A stick.

10. Mechanics 105 Newton’s 2 nd law applied to uniform circular motion A mass in uniform circular motion (speed v) accelerates according to This acceleration must be caused by some force along a direction towards the center of the radius of curvature (r)

11. Mechanics 105 Example: conical pendulum  L r

12. Mechanics 105 Uniform circular motion ConcepTest

13. Mechanics 105 Nonuniform circular motion If an object changes its speed while in circular motion, there is both a radial and a tangential component to the acceleration, therefore, there will be a radial and tangential force applied. Example: mass moving in a vertical circle R 

14. Mechanics 105 Words of wisdom "If I had only known, I would have been a locksmith." -Albert Einstein "There is no clearer manifestation of pure evil than teachers giving assignments over holiday breaks." -James Halloran

15. Mechanics 105 Velocity dependent forces Two models: 1. Force proprtional to the velocity (viscous, low speed) b is a constant that depends on the object size and shape and the medium 2. Force proportional to the square of the magnitude of the velocity (air, high speed) D: drag coefficient  : density of air A: cross sectional area of object

16. Mechanics 105 Velocity dependent forces 1. Force proprtional to the velocity

17. Mechanics 105 Velocity dependent forces Can solve differential equation where  = m/b is a time constant related to the motion Or, just find terminal speed (a=0)

18. Mechanics 105 Velocity dependent forces Force proportional to the square of the magnitude of the velocity Nonlinear differential equation Terminal speed:

19. Mechanics 105 Words of wisdom "I love deadlines. I like the whooshing sound they make as they fly by." -Douglas Adams "In a survey taken several years ago, all incoming freshman at MIT were asked if they expected to graduate in the top half of their class. Ninety-seven percent responded that they did." -??? "We made too many wrong mistakes." -Yogi Bera

20. Mechanics 105 Numerical representations of particle dynamics Euler method

21. Mechanics 105 Fundamental forces of nature Gravitational: force between any two objects where G is the universal gravitational constant Electromagnetic: force between two charged objects (q) where k e is the Coulomb constant Nuclear (strong) – short range Weak – short range

22. Mechanics 105 Gravitational field Field: the effect in a region of space that induces a force on an object e.g the field (created by a mass) exerts the force on the other masses

23. Mechanics 105 Last bad joke for this chapter An atom walking down the street says to its friend “I think I lost an electron. The friend asks “Are you sure?” to which the first atom repiles “Yea, I’m positive.”