Exploring Newton's Laws in Motion Dynamics

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

2. Friction • Force acting parallel to an interface that opposes the relative motion • Static – frictional force opposite to applied force - magnitude fs • 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. Friction • Kinetic – frictional force opposite to relative motion – magnitude fk • 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. Friction • Note: Static friction is not constant – it is whatever is needed to match the applied force, up to the limit of Sn • 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. Friction • ConcepTest • Examples • Demo

6. +y +x m1 m1 m2 m2 Example

7. Example

8. Question • What do you call a broken boomerang?

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

10. Newton’s 2nd 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. L  r Example: conical pendulum

12. Uniform circular motion • ConcepTest

13. R  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

14. 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. 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. Velocity dependent forces 1. Force proprtional to the velocity

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

19. 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. Numerical representations of particle dynamics Euler method

21. 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 ke is the Coulomb constant Nuclear (strong) – short range Weak – short range

22. 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. 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.”