1. Two Dimensional Forces
Chapter 5
Two Dimensional Forces

2. 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)

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

4. 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

5. 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

6. 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

7. 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.

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

9. 4-8 Applications Involving Friction, Inclines

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

11. Applications Involving Friction, Inclines
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.

12. 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

13. Atwood Machine

14. Atwood Machine
Let’s take a look at the forces on each mass…

15. Atwood Machine

16. Atwood Machine

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

18. Hanging sign f.b.d. Free Body Diagram T22 T2
Since the sign is not accelerating in any direction, it’s in equilibrium. Since it’s not moving either, we call it Static Equilibrium. 1 T1 mg
Thus, red + green + black = 0.

19. Components & Scalar Equationsmg T1 1 2
T1 sin 1
(y component)
T2 sin 2
(Y component)
T1 cos 1
(x component)
T2 cos 2
(X component)
If in Equilibrium……..the following would be true
Vertical:
T1 sin 1 + T2 sin 2 = mg
T1 cos 1 = T2 cos 2
Horizontal:

20. 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)
What is the tension in each chain holding the swing seat and the daughter?
A. 703N

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

22. Equilibrium Example – Free Body Diagrams