Graphical Vector Addition and Components

1. Recognize the relationship between vectors that are directly proportional to each other. The direction of each is always the same.

Examples:

Vectors	Relationship
Force and acceleration	F = ma
Velocity and Displacement	V = (Displacement) / (time increments

2. Sketch 2 ways to graphically add vectors. Vectors are represented as arrows. The length of the arrow represents the magnitude and the way the arrow points represents the direction. For addition purposes, arrows can be moved around as long as the angle they make with respect to the vertical and horizontal dimensions is not changed. The sum of 2 or more vectors is called the resultant.

• head-to-tail or heal to toe
• parallelogram method

 

Vector Components

3. Define vector components. A vector's components are the x and y coordinates of a vector, assuming that the vector's tail is at the origin. Components can also be considered the or x and y parts of a vector that added together will equal the original vector.

   Given vector components find the resultant vector (A) and its angle (theta). A = (Ax2 + Ay2)1/2 q = arctan Ay / Ax Given a vector (A) find its components Ax and Ay. Ax = A cos q Ay = A sin q

The Engineering Connection: One of the first steps in designing complex structures such a bridges and buildings consists of analyzing the components of the various complex forces in the structure. From this, it is possible to correctly size the parts of the structure that use standardized items such as i-beams. This process is called statics and is one of the first courses almost all college engineering majors take. It is particularly of interest to civil or structural engineers

Homefun (formative/summative assessment): Read sections 5.1

Formative Assessment: Physics Investigation

Follow up Questions

Mathematical Vector Addition

6. State the relationship between the magnitudes of vector components that exist in different dimensions (such as the x and y dimensions). They are totally independent of each other. Different dimensions are like separate worlds.

7. State the relationship between the direction of vector components in different dimensions. They are at a 90 degree angle with each other.

8. Add vectors together using the component method.

   1. Make a vector diagram showing all the vectors with their tails at the origin of an x and y axis.

   2. Break all the vectors into components.

   3. Sum the x-components and sum the y-components.

   4. Vectorially add the x-components and sum the y-components.

9. Solve problems involving adding or subtracting 2 vector components.

   Formative assessments:

• Swimmer problems -- Running Bear (1960, Johnny Preston. The song was #1 for three weeks in January 1960 on the Billboard Hot 100 in the United States. The song also reached #1 in the UK in 1960.)

• Airplane problems -- Meet me in St. Louis (Judy Garland, 1944 movie)

• Displacement problems

Homefun (formative/summative assessment):

Read sections 4.2,

do Practice Problems 5, 7, and 9 on page 125.

Write a paragraph discussing the following question: Before the advent of modern navigation techniques such as GPS, why would having a compass not be enough to successfully navigate the oceans and actually arrive at a given destination?

The 3 Models for Friction

10. Describe static friction.

    • Prevents sliding between surfaces
    • Variable - adjusts to match the force which would otherwise cause sliding (the parallel force).

     

11. Correctly use the model for calculating static friction.

F_s = F_p

static friction = parallel force

12. Correctly use the model for calculating the transition point between static and dynamic friction.

F_smax = m_sF_n

m_s= static COF or static coefficient of friction, an experimentally determined or measured constant for given surfaces

13. Describe the relationship of normal force to transition point between static and dynamic friction and describe how this knowledge is used with fasteners. Fasteners, such as screws, are a form of inclined plane and have very high mechanical advantages that can produce extremely high normal forces resulting in extremely high friction forces that resist "unscrewing".

14. Describe dynamic or sliding  friction.

    • Resists sliding between surfaces

    • Constant for a give normal force

15. Correctly use the model for calculating the dynamic friction.

Weight Force Components on a Slope

19. Find the normal (F_wn) and parallel (F_wp) components of the weight force (F_w) for objects on a slope that makes an angle of with the horizon β.

    Note: F_w= mg

F_wn = F_w cosβ, this is the component of the weight force that acts perpendicular to the slope.

F_wp = F_w sinβ, this is the component of the weight force that can make an object slide down a slope.

20. Find the angle of the slope where the normal component of weight exceeds the parallel component.

21. Solve for acceleration of objects on a slope with zero friction.

    Formative assessments

    An ice road truck sliding down an icy slope

The Engineering Connection: Forces on slopes are a major issue to traffic and civil engineers. If the the slope is two steep in either the upward or downward directions, the result can be serious safety or traffic flow problems.

Homefun (formative/summative assessment): problems 33, 35, 37 page 133

Review of Objectives 1- 13 (1-3 days)

Formative Assessments:

1. Work review problems at the board

2. Work practice problems.

Metacognition Problem Solving Question: Can I still work the problems done in class, several hours or days later? Some amount of repetition on the exact same problems is necessary to lock in learning. It is often better to thoroughly understand a single example of a problem type than to work example after example understanding none of them completely.

Relevance: Good test preparation is essential to performance in physics class.

Homefun (formative/summative assessment): problems 67, 75, 77, 89 (add the vectors mathematically), 85, and 99 , page 141-143; problems turn in on the day stapled to the back of the test.

Summative Assessment: Unit exam objectives 1-16