Vector analysis homework

1. Introduction

A vector is a quantity that has both a magnitude and a direction. This is different from a scalar

quantity, which is just a single number. In this lab we will learn to accurately describe and ma-

nipulate vectors, including resolving them into components and adding them together. We will

add vectors graphically, in terms of components, and also experimentally using the force table

apparatus.

2. Key Concepts

• Vectors and vector addition

• Free body diagrams

• Forces

• Equilibrium

3. Theory

3.1 Vectors

When completely describing a physical quantity, sometimes it is necessary to assign it a mag-

nitude and direction rather than using only a magnitude. Quantities needing only a single value

are scalars, and quantities requiring a magnitude and a direction are vectors. Vectors are usually

written with an arrow on top, like ⃗⃗ , while scalars are without the arrow, like .

To identify and use a vector, first establish a coordinate system, or set of axes, that is appro-

priate for the system you are trying to describe. This will give you a frame of reference from

which to make your measurements. An example is labeling North on a map. In this lab manual,

we will use the – and -axes. All angles, as denoted by , are measured counterclockwise from

the -axis (see Figure 1). Remember, some problems are made easier by a proper choice of axes.

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Vectors can be described in two ways: the

magnitude–direction method or the compo-

nent method. The magnitude–direction

method describes the vector with a given

magnitude and direction, but one needs to be

careful to include enough information in giv-

ing the direction. For example, one has to

mention clearly what the direction is with re-

spect to. This is very intuitive to graph but

mathematically more difficult to use. The component form of a vector breaks up the magnitude

and direction into how much along the -axis and how much along the -axis (see Figure 1).

Vectors in component form look like

⃗⃗ ̂ ̂

Here ̂ and ̂ are unit vectors in the and directions, respectively. We will use this notation

in the manual.

To find the components of vectors, you need to use the trig identities

Which equation one needs to use depends on the coordinate system. For the angles used in

Figure 1, for example, if ⃗⃗ describes a force of 5 N (Newton) @ 30 degrees, can be computed

from cosine and from sine,

⃗⃗ ̂ ̂

If the component form is given, the angle can be solved for by

and the magnitude

of the vector is solved by | ⃗⃗ | √ ( )

.

3.2 Adding Vectors

Adding two vectors results in a third called the resultant vector. Now that a vector is broken

up into its components, adding multiple vectors is simply adding up all of the components to

find the component of the resultant vector and adding all of the components to find the

component of the resultant vector. For example, if ⃗⃗ ̂ ̂ and ⃗⃗ ̂ ̂ then

⃗⃗ ⃗⃗ ̂ ̂ ̂ ̂. Notice that you need to be careful to take the sign of

the components into account when adding vectors.

To graphically find the resultant vector, use the ‘tail to tip’ method. Once a coordinate axis is

drawn, the ‘tail’ of the first vector begins at the origin. Measure the appropriate angle using a

protractor, and draw the length of the vector to match the vector’s magnitude. The ending point

�⃗⃗�

�⃗⃗� �̂� �̂�

Figure 1: The relation between the components of a vector and the angle measured counter- clockwise from the -axis.

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of the first vector is known as the ‘tip,’ and an arrow is drawn at this end. From the ‘tip’ of the

first vector, start the ‘tail’ of the next vector. To measure the angle for the second vector, hold

the protractor parallel to the original axis. Be sure to use the same scale as the first vector in

drawing the length of the second vector. Continue this way until all vectors to be added have

been drawn. The resultant vector begins at the origin and ends at the tip of the last vector add-

ed.

To experimentally find the resultant vector, it is easiest to first find the equilibrant vector,

and the resultant is then equal in magnitude but in opposite direction to the equilibrant. Oppo-

site direction for component form amounts to changing the sign of each component. When an

angle and magnitude are given, the opposite direction is given by adding to the angle. The

equilibrant vector is the vector that balances out the system. It brings the system to equilibrium

and therefore keeps the system stationary.

For more information on vectors, please see the sections on vector components and vector

addition in your textbook. There are also some great figures to help you understand the tech-

niques.

When you add two or more vectors, it can be very helpful to make a table for organizing the

data. An example is given in Table 1. Another thing that is sometimes helpful in picturing these

vectors and their components is to draw them all on the same coordinate system with their tails

all at the origin.

Vector -component -component Magnitude Angle

⃗⃗ √ ( )

⃗⃗ √ ( )

⃗⃗ √ ( )

⃗⃗

⃗⃗ ⃗⃗ ⃗⃗

√

( )

Table 1: Table that is helpful in organizing data to add multiple vectors.

3.3 Free Body Diagrams

It is often beneficial in a problem to be able to draw a picture. A free body diagram is a repre-

sentation of all the vectors of the same type that are acting on the system. In this lab, these vec-

tors will be forces. If the free body is stationary, that means the body is in equilibrium; the sum

of all the forces acting on the body is zero. For simplicity, assume all objects are a single point at

the center. In a free body diagram, all the ‘tails’ of the vectors acting on the body are located at

this center point. Be sure to label each vector and that all vectors represent the same type of

quantity. A simple check can be done with unit analysis. If all the vectors must add, they must

have all the same units. Like the graphical method of finding vectors, each vector length must

represent the magnitude of the vector, and must be labeled with its angle if it is not on an axis.

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�⃗⃗�

�⃗⃗�

�⃗⃗�

Figure 4: Vectors on the x-y plane.

Figure 2 shows the free body diagram of a golf ball at impact. The ball starts to move because

the sum of the forces from the club, the tee, and the Earth is not zero. For more information,

please see the section in your textbook on free body diagrams.

Question (write down the answers in Pre-lab)

(1) As shown in Figure 3, six vectors with unit length ( ) form a regular hexagon. Please de- scribe the vectors using the magnitude–direction method and the component method by filling in the table. (Note that the angles in the magnitude–direction method are measured counterclockwise from the -axis).

(2) Consider the vector forces in Figure 4. Please draw forces ⃗⃗⃗⃗ ,

⃗⃗ , ⃗⃗ , and ⃗⃗ , such that

i. ⃗⃗⃗⃗ has twice the strength and is opposite to ⃗⃗ .

ii. ⃗⃗ is the resultant force of ⃗⃗ and ⃗⃗ .

iii. ⃗⃗ is the resultant force of ⃗⃗ and ⃗⃗ .

iv. ⃗⃗ is the equilibrant force of ⃗⃗ , ⃗⃗ , and ⃗⃗ .

(3) Figure 5 is an inclined-plane system that will be stud- ied in the first part of this experiment. As labelled in the figure, the ̂ ̂ direction is parallel (perpen- dicular) to the inclined plane, and the gravitational acceleration is toward the ground. If the hanging mass is too small, the block mass on the in- clined plane slides down. When the hanging mass is gradually increased to a lower-bound value such that the block mass just stops sliding down, the forces on satisfy the equilibrium conditions:

{ ̂

̂

Here is a friction force and is a normal force. Both of them are acted by the inclined plane. If we keep increasing the hanging mass to a higher-bound value

�⃗⃗�

�⃗⃗�

�⃗⃗�

�⃗⃗�

�⃗⃗�

�⃗⃗�

Figure 3: Six vectors form- ing a regular hexagon.

Gravitational force: Earth on Ball

Normal force: Tee on Ball

Normal force: Club on Ball

Figure 2: The cartoon (left) and free body diagram (right) of a golf ball at im- pact.

�̂�

Figure 5: Inclined-plane system.

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such that the block mass is just about to slide up (but not moving yet), the forces on sat- isfy the equilibrium conditions (here we assume that the magnitudes of and remain the same):

{ ̂

̂

i. Use 4 vectors , , , and to draw the free-body diagram for mass in the case corresponding to Eq. . How does mass move if there is no friction ( )?

ii. Use the same 4 vectors draw the free-body diagram in the case corresponding to Eq. . How does mass move if there is no friction ( )?

iii. Combing the ̂-direction conditions in Eqs. and , please show that

We will experimentally examine Eq. in the lab. Note that to verify Eq. we do not need to know the details about gravitational acceleration or friction in the system.

4. Experiment The first part of the experiment studies force vectors acting on a metal roller on an inclined

plane. The system looks similar to Figure 5. Since the roller will move only along the inclined

plane, its motion and the forces that cause it to move are one-dimensional. The forces are pro-

vided by a string (attached to a mass hanger) pulling at the end of the inclined plane as well as

by the ̂ component of the roller’s weight. By adjusting the mass on the hanger, you will be de-

termining experimentally what force is needed to bring the roller to equilibrium and verifying

the relation of Eq. you have shown in Pre-lab. In the second part of the experiment, you will

be balancing forces on a ring, but this time in two dimensions using the force table. You will ana-

lyze the equilibrium of forces by decomposing each force into ̂ and ̂ components.

4.1 Equipment • Triple beam balance

• Set of masses

• Mass hangers (3)

• Short red cable

• Inclined plane

• Metal roller

• Force table with ring, pulleys (3), and

black cables

4.2 Procedure

Forces in 1D

1. Measure the masses of the metal roller and the hanger. Record the measurements on Re- port Sheets. Note that for only one measurement, the error is the measurement error.

2. Adjust the angle of the inclined plane to , connect the red cable to the roller and hanger, and put them on the inclined plane, as in Figure 6. Make sure that the cable is par- allel to the inclined plane when the roller is pulled by the cable (you could drag the hanger a little bit to stretch the cable).

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3. Gradually add masses on the hanger until the roller can maintain equilibrium at the middle of inclined plane. Because the smallest mass is 5g, you need to find mass such that when is added on the hanger, the roller is at rest, and when 5g is taken off, i.e. on the hanger, the roller starts to move down. Record on Table R1 on Report Sheets.

4. Keep adding masses until the roller starts to move up. You need to find mass such that when is added on the hanger, the roller is at rest, and when additional 5g is put on, i.e. on the hanger, the roller starts to move up. Record on Table R1.

5. Repeat 3 and 4 for different inclined angles and .

Check point 1

Ask your TA to check your data recorded and get their initials on Check Box 1 on Report Sheets.

Forces in 2D

1. Measure the mass of three hangers used with the force table. Record this measurement on Report Sheets. In all the work below, you will need to use the total mass (mass of the hanger plus the added mass) in your calculations.

2. Place the ring around the center peg. Make sure that all three cords are parallel to the force table—you might need to adjust the heights of the pulleys to achieve this.

3. Place the first two pulleys at the angles given for hanger 1 and hanger 2 in Trial 1 of the ta- ble on Table R2 on Report Sheets, and add the specified masses to the hangers at each an- gle.

4. Place the third pulley where you would guess the equilibrant vector to be, and hang a trial mass. (Note that you can put the cords of more than one hanger over the same pulley if you need to.) Adjust both the angle and mass of hanger 3 until the ring appears centered around the peg.

5. Test the equilibrium position by giving the system a small pull along one of the vectors. If the ring oscillates slightly but remains centered, you have found the correct equilibrant force. That is, you have found the force that balances out the net effect of the first two.

6. Record the mass added on hanger 3 and the angle of hanger 3 on Table R2.

7. Repeat 3-6 with different configurations to complete Table R2.

Check point 2

Ask your TA to check your data recorded and get their initials on Check Box 2 on Report Sheets.

Figure 6: Roller on the inclined plane.

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4.3 Analyses

I. Forces in 1D (write on Report Sheets)

1) Analyze the measurements by examining Eq. (6). 2) Make your conclusion, answer questions, and discuss possible sources of errors.

II. Forces in 2D (write on Report Sheets)

1) Compute each force and the resultant force the force table. 2) Compare the experimental relation between the forces with the theory and make a con-

clusion.