Experiment 5: Vector Addition Part 2

Figure 5.1: Force Table

EQUIPMENT

Force Table(4) Pulleys(4) Mass HangersMassesLevel (TA’s Table)

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2 Experiment 5: Vector Addition Part 2

Advance Reading

Vectors and vector addition (Serway and Vuille 1.9-1.10)

Objective

The objective of this lab is to add vectors using thecomponent method and to verify the results using aforce table.

Theory

Vectors are quantities that have both magnitude anddirection. When vector quantities are added together,the result is called the resultant.

The figure illustrates how to add vector �A and vector�B, also known as finding the resultant vector �R, usingthe component method. We take each vector anddecompose it (break it down) into its x-component andy-component. The x-component of a vector is its pro-jection onto the x-axis. We can find the x-componentof vector �A either by: 1) using right-triangle trigonom-etry (SOHCAHTOA) and including a positive or neg-ative sign based on the direction of the vector OR 2)using Ax = AcosθA where A is the magnitude of vec-tor �A and θA is the direction of vector �A given as a

standard position angle.

Similarly, we can find the y-component by trigonome-try or by using Ay = AsinθA where A is the magnitude

of vector �A and θA is the direction of vector �A givenas a standard position angle. Standard position anglesare measured counter-clockwise from the positive hor-izontal axis (usually +x). And do the same for vector�B and find Bx = BcosθB and By = BsinθB .

Once every vector is written in terms of its compo-nents, we can find the overall x-component and y-component, Rx = Ax +Bx and Ry = Ay +By. Those

are the components of the resultant �R. If we leave �Rwritten in terms of its components, we say that �R isin component form or vector form. Sometimes thatis written as �R = Rxî + Ry ĵ or �R = Rxx̂ + Ry ŷ

or �R = �Rx, Ry�. But usually we prefer to writephysics quantities in polar form, which means giv-ing the magnitude and direction.

Looking at the largest right triangle in the figure, wecan see that the magnitude of vector �R is the hy-

potenuse of that right triangle and so R =�R2x +R

2y.

The direction of vector �R is the angle θ which we canfind using tanθ =

RyRx

.

Figure 5.2: Vector Addition

Experiment 5: Vector Addition Part 2 3

Be careful when using the inverse tangent function,called tan−1 or arctan. This function only ever re-turns values in quadrants 1 and 4, but we want to beable to find resultant vectors in all four quadrants. Wecan use the property that the inverse tangent functionrepeats every 180◦. If tan−1 gives a value that is inthe wrong quadrant, add 180◦ to get the correct angle.You will know if tan−1 gave you an answer that is inthe wrong quadrant by looking at the signs of Rx andRy.

Force Tables

We will use a force table to verify our results of vectoraddition and gain a hands-on perspective. The forcetable is a circular steel disc with angles 0◦ to 360◦ in-scribed on the edge. There is a pin in the center of theforce table and a metal ring around the pin.

If you havent seen forces in lecture yet, it will beenough to know, for now, that forces are pushes orpulls on an object. We will cause forces to be exertedon the center ring by hanging masses from strings thatare tied to the center ring. The force exerted on thecenter ring is equal to the weight of the hanging mass.The magnitude of this force is:

W = mg

where W is the weight in Newtons (N), m is the mass inkilograms (kg), and g is the magnitude of gravitationalacceleration on Earth’s surface (9.80 m/s2, though forthis experiment you may approximate to 10.00 m/s2).The direction of this force is controlled by moving thepulleys that are clamped to the force table.

When all the forces are balanced, they add to zero.The ring will be centered around the pin, not touchingit, and the system is in equilibrium. If there are mul-tiple forces acting on an object, their combined effectis the vector sum of the forces the resultant. Theforce that would balance out the resultant and placethe system in equilibrium is called the equilibrant.

Percent Difference vs Percent Error

When we compare a measured value to a calculatedor theoretical value, we use a quantity called percenterror to quantify how far off the measurement is fromwhat we expected to get. The definition of percenterror is:

% error =measuredvalue− expectedvalue

expectedvalue× 100

(5.1)If the percent error is positive, that means the mea-sured value is above the expected value. The percenterror is negative when the measured value is below theexpected value.

Often though, we won’t have a theoretical or calcu-lated value. Instead, we will compare two measuredresults to see how different they are from one other,as a percentage of the mean of the two measurements.This quantity is called percent difference and is givenby:

% difference =|value1 − value2|

value1+value22

× 100 (5.2)

4 Prelab 5: Vector Addition Part 2

Name:

1. What is the objective of this lab?

2. A vector is given in terms of its x-component Dx = -3.4 m and its y-component Dy = +2.1 m.

a) What is this vector in polar form?

b) Which quadrant is this vector in?

3. A vector is given in terms of its magnitude, v = 8.7 m/s and its direction, 256◦ standard position.

a) What is this vector in component form?

b) Which quadrant is this vector in?

Give the angle shown in the diagram as a:

a) positive standard position angle

b) negative standard position angle

c) ◦ east of south

d) ◦ south of east

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Name: Section: Date:

Worksheet - Exp 3: Vector Addition Part 2

Objective: The objective of this lab is to add vectors using the component method and to verify the results usinga force table.

Procedure:

Part 1: Adding 2 Vectors

1. Given �A = 2.5N at 40 degrees and �B = 3.2N at 250 degrees, record them in the table and use componentaddition to find the resultant vector �R=�A+�B. Show your work in the space below. (10 points table, 5 pointsmath)

Vector Magnitude (N) Direction, θ x-component (N) y-component (N)

�A

�B

Resultant

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Part 2: Adding 3 Vectors

2. Given that �C = 1.5N at 345 degrees, record the vectors in the table and use component addition to find theresultant vector �R=�A+�B+�C . Show your work in the space below. (10 points table, 5 points math)

Vector Magnitude (N) Direction, θ x-component (N) y-component (N)

�A

�B

�C

Resultant

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Part 3: Verify Result from Part 2

3. Use the level to make sure the surface of the force table is horizontal.

4. Use the force table to determine the equilibrant vector and test whether the resultant �R is equivalent to �A+�B+�C.(10 points)

Vector Magnitude (N) Direction, θ

�A

�B

�C

�E

�R

5. Compare your computation of the resultant vector (Part 2) with the force table measurement of the resultantvector (Part 3). Show your calculations for percent error in the space below. (15 points)

Part 2 Part 3 Percent Error

Magnitude of �R

Direction of �R

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6. What sources of error could be contributing to:

a) Head-to-tail method of vector addition (5 points)

b) Component method of vector addition (5 points)

c) Experimental, force-table method of vector addition (5 points)

7. Four vectors are all in the same quadrant (ex: quadrant 1). Can the sum of these vectors equal zero. Explainwhy or why not using components. (5 points)

8. Four vectors are all in the same two adjacent quadrants (ex: quadrants 1 and 2). Can the sum of these vectorsequal zero? Explain why or why not using components. (quadrants do not contain the adjacent axis, a vectorlying on the positive x axis is not in quadrant 1) (5 points)

9. Describe what properties of the four vectors would be required for the sum of the four vectors to equal zero. (5points)

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10. Design an experiment on the force table that would show that breaking a vector into its x- and y-components isvalid. (15 points)

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