Lab 3 Vector Lab Simulation PDF

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Physics 110 Lab 3 Vector Lab Simulation...

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INTRODUCTORY PHYSICS

HUNTER COLLEGE

VECTORS Abstract: A Vector is a quantity dependent on both magnitude and direction. Vectors can be represented in two ways by arrows, graphically on a coordinate plane, and in a more abstract sense that does not require coordinate plane. Vectors are composed of two components (x and y) that allow for calculations to be done through trigonometric functions and Pythagorean theory to relay important values like magnitude and direction of a vector. We used simulations to confirm that magnitude of vectors do not change when direction changes, and direction does not change when magnitude of vectors change. Pythagorean calculations with trigonometric calculations were performed to confirm observations from simulation. Purpose of Exercise: Students will be able to calculate the resultant vectors’ magnitude and direction by ● ● ● ●

Resolving vectors into their components Using the rules of vector addition Graphing the vectors on a cartesian coordinate system Using basic trigonometry to determine the angle of the resultant vector

Background Information ● Vector quantities are any quantity that has a size (magnitude) and direction. Such quantities are displacement, velocity, acceleration, and force. ● A scalar quantity is any quantity with a magnitude only. Velocity without its direction would be the scalar quantity we call speed. Distance, time, temperature, and mass are examples of scalar quantities. Graphically Adding Components There are graphical methods to add vectors. The tip to tail and parallelogram methods both are visual aids to adding vectors. With proper scaling and a protractor, you can measure the magnitude of the resultant vector with a ruler and determine its angle with a protractor. Both methods are displayed in the figure below:

https://reepingdavid.weebly.com/uploads/2/7/2/3/27230509/11299_orig.png

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Adding Vectors Using Components Any vector can be expressed as the sum of two other vectors, which are called its components. This is useful and a more practical means of adding vectors. Determining the components of a vector is to resolve a vector into its components. The figure shows vector V resolved (left) into its components V x and Vy (right)

Resolving vectors into its components is useful when adding vectors that are not one dimensional and are in different directions. For example, you are walking North 4 miles per hour while the wind is blowing 3 miles per hour. You are blown off course and your resultant velocity is 5 mph at 53.1⁰, above the horizontal axis, or 53.1⁰North of East. Steps to adding vectors using components: 1.

Draw a diagram; add the vectors graphically.

2.

Choose x and y axes.

3.

Resolve each vector into x and y components.

4.

Calculate each component using sines and cosines.

5.

Add the components in each direction.

6.

To find the length (or magnitude of the vector) use the Phythagorean Theorem, or distance formula,

7. To find the direction of the vector use, use:

Pre-Lab Questions 1. Resolve a vector A of the 5.0 unit at 53.1⁰ into its components. Ax =

Ay =

2. Vector B has components, Bx = -3.0 units and By = +4.0 units, calculate the magnitude and direction of B. 3. Although vector A has the same magnitude as vector B, why are they not the same vector?

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Procedures: Vector Addition Part I: Explore 2D 1.

Open the vector simulation: https://phet.colorado.edu/sims/html/vector-addition/latest/vectoraddition_en.html

2. Open Explore 2D by clicking on the icon above. Your screen will open the simulation like the one in the figure below.

4. Re-orient the origin on the graph. Go to the origin (0,0). Click and drag it to the center of the

graph:

Your graph should look like this

.

5. Add vector a by clicking and dragging vector a from the lower right panel on your screen 6. Place the tail of a at the origin.

.

7. Put your curser on the tip of the vector arrow; the hand cursor will appear. 8. Click and drag the tip of the vector to change its magnitude and direction. 9. Set a to have a magnitude of 9 units and direction of 0⁰, so that its coordinates are (9,0). Page 3 of 8

10. Now add vector b to the graph. Put the tail at the zero. Set its magnitude to 12 units and its direction at 90 ⁰, so that its coordinates are (0,12).

11. Use the Pythagorean Theorem to determine the magnitude and the angle, relative to the x-axis. C2= 122+ 132 c= 15 units Sin-1 (12/15)= 53.13 degrees

12. If you calculated correctly, you should have 15 units at 53.1⁰ (above the positive x-axis). Confirm this by checking the Sum and Values options in the upper-right corner on your screen

.

13. Record your vector measurements and calculation in Table 1 below. Explore the various ways you can read vector values in the simulation. Checking the values option in the upper right corner is one way. Another way is to click a vector and read off the values in the grey rectangle above the graph.

14. Now, uncheck the Sum option.

15. Click on the vectors a and b to drag and move them around the coordinate system (graph). Be careful to only click and drag the body of the arrow, not the tip. Clicking the tip will change the vector value.

16. Table one has specified values for you to assign vectors a and b. For each trial fill in the table with the information requested in the top row of the table. In other words, complete the table Page 4 of 8

by “resolving” the vectors into their components then sum to get the resultant s vector’s magnitude and determine its direction. Describe the quadrant where s is located according to its components and calculated angle. The first entry has been completed for you, as a guide. Table 1

Vector Label

Vector Magnitud e [units U]

Vector Directio n

Horizontal Componen t

Vertical Componen t

[units U]

[units U]

ϴv

Trial 1

9

0⁰

ax= 9

ay= 0

b

12

90⁰

b x= 0

by= 12

s x = ax+ b x

sy = ay+ by

sx = 9

sy = 12

Trial 2

Calculate ->

a

9

0⁰

a x= 9

ay= 0

b

12

270⁰

b x= 0

by= -12

s x = ax+ b x

sy = ay+ by

sx =9

sy =-12

s

Trial 3

Calculate ->

a

9

180⁰

ax= -9

ay= 0

b

12

270⁰

b x= 0

by=-12

s x = ax+ b x

sy = ay+ by

sx =-9

sy =-12

s

Trial 4

Calculate ->

a

9

180⁰

ax= -9

ay= 0

b

12

90⁰

b x= 0

by= 12

s x = ax+ b x

sy = ay+ by

sx =-9

sy = 12

s

Calculate ->

Magnitude

Vector Direction

[units U]

ϴs = tan-1(sy / sx )

In what quadrant is the Resultant S?

53.1⁰

I

Resultant

S = √( ax 2+ bx 2 )

a

s

Resultant Vector S

15

North of east

15

53.1⁰

IV

south of east

15

53.1⁰ south of west

III

15

53.1⁰ North of west

I

Question 1: How did the magnitudes and direction of s change as you varied vectors a and b? The magnitudes all stayed the same because the length of the vector components stayed the same. Since we changed direction of each vector only direction changed when the vectors were found in different quadrants of the graph.

Part III: Vector Equations Page 5 of 8

1. Click on the home tab

at the bottom center of the screen.

2. Click on the Lab tab, , to open the lab simulation. 3. Create two vectors, a = 6 units at 0⁰ and vector b = 7 units at -90⁰, by clicking the base vector tab in on the lower right of the screen and manually entering the component values of a and b. 4. Click the a + b = c equation at the top left tab above the graph. 5. Record the values of vectors and record them in the table below. 6. Repeat step 5 for the other two equation options.

7. 8. 9. 10.

Calculate c for each equation. Show your work. Record your calculated values in the table below. Now, check the sum option on the simulation. Compare your results with the simulation’s calculation for each value of c determined from the 3 equations.

Table 3: Vector Equations Equation

c=a+b

x – component

cx = ax + bx = 6

y-component

cy = ay + by = -7

calculated

Calculated

Magnitude of c

Direction of c θc

c= 9.21

√ 62±7 2

=

c=a-b

cx = ax - bx = 6

cy = ay - by = 7

c= √ 62 + 72 = 9.21

c+a+b=0

cx =(ax + bx ) =

cy = (ay + by)=

c=-ax-bx= -6

c=-ay-by=7

c= √ −62 +72 = 9.21

tan

tan

tan

-1

(-7/6)= -49.4

-1

(7/6) = 49.4

-1

(7/-6) = -49.4

180-49.4= 130.6 Page 6 of 8

11. Show your calculations for each calculation of c, using the equations in Table 3 and compare your results with the simulation values for c. Simulation value for |C|= 9.2 Calculated value is 9.21. Very accurate from simulation and calculation 12. How would vector c change if you multiplied vector a and b by 2? Change the magnitude and direction of vector c for: a. 2a + 2b, c = 9.2 x 2= 18.4, angle= -49.4 b. 2a – 2b, c = 9.2 x 2 =18.4, angle= 49.4 c. 2a + 2b + 2c = 0, c= 9.2 x 2 = 18.4 angle, -49.4 or 130.6 The vector’s magnitude will change by a magnitude of 2 (double).

13. In your own words how would you describe what happens to a vector when you multiply it by a scalar? When multiplied by a positive scalar only the magnitude of the vector changes while the direction stays absolutely the same. If multiplied by a negative scalar, magnitude stays the same while the direction is inverted Post Lab Questions 1. You are in the middle of a large field. You walk in a straight line for 100 m, then turn left and walk 100 m more in a straight line before stopping. When you stop, you are 100 m from your starting point. By how many degrees did you turn? (a) 90°. (b) 120°. (c) 30°. (d) 180°. (e) This is impossible. You cannot walk 200 m and be only 100 m away from where you started.

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