2. Chapter 5 Two Dimensional Forces

3. Equilibrium An object either at rest or moving with a constant velocity is said to be in equilibrium The net force acting on the object is zero (since the acceleration is zero)

4. Equilibrium cont. Easier to work with the equation in terms of its components:

5. Inclined Planes Choose the coordinate system with x along the incline and y perpendicular to the incline Replace the force of gravity with its components

6. Forces of Friction When an object is in motion on a surface or through a viscous medium, there will be a resistance to the motion This is due to the interactions between the object and its environment This is resistance is called friction

7. More About Friction Friction is proportional to the normal force The force of static friction is generally greater than the force of kinetic friction The coefficient of friction (µ) depends on the surfaces in contact The direction of the frictional force is opposite the direction of motion The coefficients of friction are nearly independent of the area of contact

8. Applications Involving Friction, Inclines On a microscopic scale, most surfaces are rough. The exact details are not yet known, but the force can be modeled in a simple way. For kinetic – sliding – friction, we write: is the coefficient of kinetic friction, and is different for every pair of surfaces.

9. Kinetic Friction, ƒ k The force of kinetic friction acts when the object is in motion ƒ k = µ n Variations of the coefficient with speed will be ignored

10. 4-8 Applications Involving Friction, Inclines

11. Static Friction, ƒ s Static friction acts to keep the object from moving If F increases, so does ƒ s If F decreases, so does ƒ s ƒ s  µ n

12. The static frictional force increases as the applied force increases, until it reaches its maximum. Then the object starts to move, and the kinetic frictional force takes over. Applications Involving Friction, Inclines

13. Block on a Ramp, Example Axes are rotated as usual on an incline The direction of impending motion would be down the plane Friction acts up the plane Opposes the motion Apply Newton’s Laws and solve equations

14. Atwood Machine

15. Let’s take a look at the forces on each mass…

16. Atwood Machine

17. Atwood Machine

18. Static Equilibrium Condition of an object when net forces equal zero Object is motionless

19. mg 1 1 T1T1 2 2 T2T2 Since the sign is not accelerating in any direction, it’s in equilibrium. Since it’s not moving either, we call it Static Equilibrium. Thus, red + green + black = 0. Hanging sign f.b.d. Free Body Diagram

20. T 2 mg T 1 11 22 T 1 cos  1 (x component) T 2 cos  2 (X component) T 1 sin  1 (y component) T 2 sin  2 (Y component) T 1 cos  1 = T 2 cos  2 Horizontal: Vertical: T 1 sin  1 + T 2 sin  2 = mg Components & Scalar Equations If in Equilibrium……..the following would be true

21. Sample Problem A mother and daughter are outside playing on the swings. The mother pulls the daughter and swing (total mass 55.0 kg) back so that the swing makes an angle of 40.0° with the vertical (50.0 ° from horizontal) A. 703N What is the tension in each chain holding the swing seat and the daughter?

22. Sample Problem… Is this a case of equilibrium? Calculate the magnitude of the net force 45° 40 ° 150N 160N 75N

23. Equilibrium Example – Free Body Diagrams