1. Forces in Two Dimensions
Chapter 5
Forces
in Two Dimensions

2. Vector vs. Scalar Review

3. Vector vs. Scalar Review
All physical quantities encountered in this text will be either a _____ or a ______.
A vector quantity has both ________ (size) and ________.
Add/subtract vectors require special rules
A scalar is completely specified by only a _________ (size).
Add/subtract vectors require no special rules

4. Vector Notation When handwritten, use an arrow:
When printed, will be in bold print with an arrow:
When dealing with just the magnitude of a vector in print, an italic letter will be used: A
Italics will also be used to represent scalars

5. Properties of Vectors Equality of Two Vectors
Two vectors are ________if they have the same magnitude and the same direction.
These are parallel rays.
Movement of vectors in a diagram
Any vector can be moved _________ to itself without being affected.

6. More Properties of Vectors
Negative Vectors
Two vectors are negative if they have the same _______ but are _________(opposite directions).
Resultant Vector
The ________ vector is the sum of a given set of vectors.

7. Parallel Vectors A B These two vectors are equal. A B
These two vectors are not equal.

8. Adding Vectors
When adding vectors, their ________ must be taken into account.
Units must be the same.
Geometric Methods
Use scale drawings
Algebraic Methods
More convenient
More on this later

9. Adding Vectors Geometrically (Triangle Method)
Choose a scale
Draw the first vector with the appropriate length and in the direction specified, with respect to a coordinate system
Draw the next vector using the same scale with the appropriate length and in the direction specified, with respect to a coordinate system whose origin is the end of vector and parallel to the coordinate system used for .

10. Graphically Adding Vectors
Continue drawing the vectors “________”
The resultant is drawn from the origin of to the end of the last vector
Measure the length of and its angle
Use the scale factor to convert length to actual magnitude.

11. Graphically Adding Vectors
When you have many vectors, just keep repeating the process until all are included.
The resultant is still drawn from the origin of the first vector to the end of the last vector.

12. Sample Problem
Find the magnitude of the sum of a 15 km displacement and a 5 km displacement when the angle between them is 90o and when the angle between them is 135o. Use the graphical method of addition.

13. Notes about Vector Addition
Vectors obey the _______________ Law of Addition
The order in which the vectors are added doesn’t affect the result

14. Sample Problem
A car travels 20 km due north and then 35 km in a direction 60o west of north. Find the magnitude and direction of a single vector that gives the net effect of the car’s trip. This vector is called the car’s resultant displacement.

15. Vector Subtraction Special case of vector addition:
Add the negative of the subtracted vector
Continue with standard vector addition procedure

16. Components of a Vector A component is a _______.
It is useful to use rectangular components
These are the projections of the vector along the x- and y-axes.

17. Components of a Vector
The x-component of a vector is the projection along the ________:
The y-component of a vector is the projection along the _________:
Then,

18. More About Components of a Vector
The previous equations are valid only if ____ is measured with respect to the _______.
The components can be positive or negative and will have the same units as the original vector.
We can use various trig. functions to solve for the length and angle of the resultant.

19. More About Components
The components are __________________ whose hypotenuse is
May still have to find θ with respect to the positive x- axis.
The value will be correct only if the angle lies in the first or fourth quadrant.

20. Trig. and Vectors Trig. Ratios: Law of Cosines: Law of Sines:
R2 = A2 + B2 – 2ABcosθ
R =resultant vector
A/B = two vectors added together
Θ = angle between A and B
Law of Sines:

21. Adding Vectors Algebraically
Choose a coordinate system and sketch the vectors.
Find the x- and y-components of all the vectors.
Add all the x-components
This gives Rx:

22. Adding Vectors Algebraically
Add all the y-components
This gives Ry:
Use the _____________ to find the magnitude of the resultant:
Use the ________________to find the direction of R:

23. Sample Problem
A GPS receiver indicates that your home is km and 40.0o north of west, but the only path through the woods leads directly north. If you follow the path 5.0 km before it opens into a field, how far, and in what direction, would you walk to reach your home?

24. Forces of Friction
When an object is in motion on a _______ or through a _____________, there will be a resistance to the motion.
This is due to the interactions between the object and its environment.
The “viscous” mediums include water or air.
This is resistance is called __________.
Static friction
Kinetic friction
Friction plays an important role in our daily lives.
Friction allows us to run and walk and allows the wheels on our cars to move the car.

25. More About Friction Friction is ______________ to the _____________.
The force of static friction is generally ________ than the force of kinetic friction.
The direction of the frictional force is __________ the direction of motion.
The _______________(µ) depends on the surfaces in contact.
The coefficients of friction are nearly __________ of the _____________.

26. Static Friction, ƒs
Static friction acts to keep the object from moving.
As long as F = ƒs no motion occurs.
When F >ƒs motion occurs.
ƒs arises from the nature of the two surfaces in contact with each other.
µs accounts for this
ƒs  µs n
This gives us the magnitude of the force of static friction.
Use = sign for impending motion only.

27. Static Friction, ƒs If F increases, so does ƒs
If F decreases, so does ƒs
ƒs, max is the maximum limit to the force of static friction.
When an object is in motion the friction force is less than the ƒs, max

28. Kinetic Friction, ƒk
The force of kinetic friction acts when the object is in motion.
Fnet = F – ƒk
At constant velocity (a = 0):
F = ƒk
If F = 0
ƒk will provide an acceleration in the direction opposite of the movement and eventually bring the object to rest.
ƒk = µk n
Variations of the coefficient with speed will be ignored

29. Sample Problem
You push a 25.0 kg wood box across a wooden floor at a constant speed of 1.0 m/s. How much force do you exert on the box? Use Table 5.1 (p.129) to help you solve this problem.

30. Some Coefficients of Friction

31. Friction and the Motion of a Car
Friction plays an important role in the movement of a car.
Friction between the moving car’s wheels and the road is static friction.
This moves the car forward.
Unless the car is skidding.
Also have the air resistance,
This slows the car down

32. Equilibrium
An object either at rest or moving with a constant velocity is said to be in ____________.
The ________acting on the object is zero (since the acceleration is zero.)

33. Equilibrium
Easier to work with the above equation in terms of its components:
This could be extended to three dimensions.
We would also have to consider ΣFz = 0
A zero net force does not mean the object is not moving, but that it is not accelerating.

34. Inclined Planes Problems
These are important and common types of problems here and elsewhere.
Choose the coordinate system with x along the incline and y perpendicular to the incline
Replace the force of gravity with its components.
Fg itself must remain perpendicular to the ground. x y n W Wx Wy FG

35. Sample Problem
A crate weighing 562 N is resting on a plane inclined 30.0o above the horizontal. Find the component of the weight forces that are parallel and perpendicular to the plane.

36. Forces in Two Dimensions
Chapter 5
THE END
Forces
in Two Dimensions