Vector Analysis (Includes Pre Lab Assignment)

Objectives: To understand the difference between vector and scalar quantities, learn the mathematics of vectors and get comfortable using vector algebra. To intuitively understand the idea of Force and equilibrium. Please note that this lab has an assignment that needs to be turned in to the TA at the start of the lab. Apparatus: Plastic Vectors, Ruler, Force Table, Spring Balance.

Figure 1. (a) Experimental set up with 3 forces applied to the spring balances attached to the protractor disc. The reed is shown in red. (b) Spring balance used for the application of the force.

Introduction: A quantity that is fully described by a single number (with units) is called a “scalar” quantity. Examples are money (such as in “5 dollars”), temperature (such as in “the temperature is 77 oF”), mass (such as “I can lift a 50 kg mass”), volume, density, etc. There are physical parameters, however, which require both a number (magnitude) and a direction to be completely sensible. Such quantities are called “vectors”. Examples are position (such as “ I am lost 10 miles North-East of Riverside”) and velocity (such as “I am driving South at 60 mph”), acceleration, etc. A single number cannot describe a vector quantity. Arrows are used to represent vectors. The arrow points in the direction of the vector and the length of the arrow corresponds to its magnitude. For example, to represent your lost position of 10 miles approximately North-East (lets say at an angle θ = 36.9o) of Riverside by the vector

r A , you would have the following picture in which the

magnitude of vector r A is given by the length of 10 miles (scale is 10 miles to an inch). The vector points in a North-

East direction.

(b)(a)

North

West East

South

r A θ = 36.9°

Figure 2

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The North-South axis is the y-axis and the East-West axis is the x-axis. Mathematically we do not use North, South East and West to represent directions, however. East is represented by , West, which is the opposite direction of East, is . North is represented by and South as the opposite direction to North is − . The third set of directions pointing above the plane of the paper would be given by and the direction pointing below the plane of the paper would be given by − .

ˆ i −i ˆ j j

k ˆ k

The vector A in terms of the two relevant directions, North and East, would be written as

r A = 8 + 6 . This tells

you that you can get to your position through an alternate route by first going 8 miles East and then going 6 miles North, where the ‘8’ is called the x-component represented as ‘Ax’ and ‘6’ is referred to as the y-component represented by ‘Ay’. The x-component, y-component and the vector A make a right triangle. From the Pythagoras

theorem of right angles,

ˆ i ˆ j

r A = Ax( )2 + Ay( )2

. This is more elegant, as now one can use trigonometry to represent

the vector r A . The x component is given by Ax =

r A cos θ (which in this case is 10 cos 36.9o) and the y-component

by Ay =

r A sin θ. The angle θ is always measured in the anti-clockwise direction from the x-axis. Also, from

trigonometry tan θ = Ay

Ax and. Note that only the magnitude and direction is specified for vectors, and the starting

position is not given. This means that vectors can be moved to any starting point and they do not have to necessarily start at the origin. Addition of two vectors: Vectors and scalars add differently; scalars add algebraically while vectors add geometrically. For example in the case of scalars, 3 plus 4 dollars always equals 7 dollars because currency does not have direction. On the other hand, when adding vectors their direction, as well as their magnitude, must to be taken into account. Let’s say that after reaching a certain point by following the vector

r A , you drive for another 18 miles

at a different angle φ = 110° with respect to the x-axis. If this new displacement is given by the vector , your total displacement

r from Riverside is given by the sum of

r B

T r A and

r B , i.e., Total Displacement =

r =

r T A +

r . Graphically

this is represented as shown in Figure 2. B

North

Figure 3

(a) (b) Note the method of addition which follows from the example. You drive

r B after

r A , so you first get to the head of

r A and place the tail of

r on the head of

r B A and the total sum is given by the vector

r T which starts at the tail of

r A

and goes to the head of as shown in Figure 3b. The components of r B

r B are Bx = 18 cos 110° = -6.16 miles and By

=18 sin 110° = +16.91 miles. In terms of the components, r B = -6.16 + 16.91 . Note the minus sign in front of Bx,

tells you that you drove in the “- ” direction, i.e., West, given that “+ ” is the East direction. Note the important point that a minus sign in front of a vector means that the vector is pointing in the opposite direction to that with a positive sign. Now the resultant vector

r is given by:

ˆ i ˆ j ˆ i ˆ i

T

West East

South

θ φ

r A

T

T

r A

BB

2

r

T = r A + = Ax + Ay + Bx + By = (Ax + Bx) + (Ay + By )

r B ˆ i ˆ j ˆ i ˆ j ˆ i j

Therefore, Tx = (Ax + Bx) and Ty = (Ay + By) are the x- and y-components of the total displacement respectively. Note that it makes sense to add all of the East-West components separately to get the total East-West component, and all of the North-South components separately to get the total North-South displacement. From geometry of

right angle triangles, the angle γ that makes with the x-axis is given by tan γ = r

T Ty

Tx.

Lets say you want to add more than two vectors, for example you drive further by a vector after r C

r A +

r B . Then

graphically after doing the procedure shown in above, you would place the tail of r C on the head of

r . The total

vector would start at the tail of r

BA (your starting point) and go to the head of

r C (your final destination). In terms of

the components the total vector would be given by Tx = (Ax + Bx + Cx) and the y-component by Ty = (Ay + By + Cy). Subtraction of Vectors: For the same two vectors

r A and

r B , if you wish to subtract from

r B

r A , then the

resultant vector r =

r R A -

r . This is the same as B

r R =

r A + (-

r B ), which tells you that to subtract these two vectors

you reverse the direction of r (because it has a minus sign in front) and then add it to B

r A .

North B

Figure 4

In terms of the components:

r R =

r A - = Ax + Ay - Bx - By = (Ax - Bx) + (Ay - By)

r B ˆ i ˆ j ˆ i ˆ j ˆ i j

The x- and y-components are Rx = (Ax - Bx) and Ry = (Ay - By), respectively. Experiment I: Vector algebra [2.0 pts.] You are given three colored vectors, Green (

r A ), blue (

r B ) and black (

r C ). Measure the length (magnitude) of each

vector with the ruler, and record it in your notebook. Assume that the vectors

r A , , and make angles of 20o, 100o and 165o degrees, respectively, with respect to the x-

xis.

r B

r C

a (a) Find the total vector, =

r T

r A + + , graphically by tracing the arrows in your notebook and drawing the total

vector. Measure the magnitude and direction of the total vector using your ruler and protractor.

r B

r C

( b) Then, perform the addition mathematically in terms of the magnitudes and directions of the components.

(c) Is the graphical total vector consistent with the calculated r

T ? (b) Repeat for the vector =

r R

r A - - both graphically and mathematically.

r B

r C

West East

South

r A

−B

R

r A

−B

R

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Experiment II: Measurement of Force Equilibrium The purpose of this experiment is to use the force table and the idea of force equilibrium to demonstrate the addition of vectors. As you will learn this quarter, force is a vector, i.e., it has both magnitude and direction. You need to specify how hard you are pulling and in what direction you are pulling. In this experiment you and your partner will apply force on a point in 3 different directions. If the three applied forces all balance out then the point at which all three are applied will not be displaced. The point is said to be in “equilibrium”. The apparatus for doing this is called a force table and is shown in Figure 1 above. 1. Calibrate each of the spring balances individually to read zero with no applied force using the screw top. Note

that these balances provide a direct measure of force in units of Newtons (N). When pulled, the scales measure forces up to 5 N in magnitude.

2. Connect the S-hook of the spring balances to the three string loops on the disc protractor. Strings attached to the

disc pull it in different directions. The magnitude (strength) of each pull, and its direction, can be varied. 3. Thread the reed through the hole in the bottom plate and through the center of the disc. If the forces applied to

the disc are not balanced, the reed will bend. 4. Raise the protractor disc with the spring balances and apply forces to the 3 spring scales. This requires three

hands (two people). Adjust the forces such that the reed in the middle is standing straight up, i.e., it does not bend. A straight reed signifies zero net displacement of the center of the protractor, and thus equilibrium. At equilibrium the net force acting on the reed is zero. Measurements that should be recorded are angles obtained from the protractor disc and force readings from the spring balance when the system is at equilibrium.

Determining the equilibrium forces [2.0 pts.]

Direction (Angles)

Forces

θa

r A

θb

r B

θc

r C

• Choose 3 angles along which to apply forces, and record them in a table (as shown to the right).

• Lay out the strings at the chosen angles and attach the spring balances to the strings.

• Apply forces on the spring scales keeping the angles fixed. Adjust the forces so that the reed remains straight.

• Note the magnitude of the applied forces

r A ,

r B and

r C from the

spring balances, and enter them into the table. Component Analysis: From the angles and the magnitude of the applied forces, find the components Ax, Bx, Cx and Ay, By, Cy. Show that Ax + Bx + Cx = 0 and Ay + By + Cy = 0. This is required for equilibrium as no net force

as applied on the reed. w Graphical Method: From the angles and the magnitudes of the vectors

r A ,

r B , and

r C , represent them graphically in

your lab notebook. Show that r A +

r +

r = 0 as required for the case of equilibrium. B C

Repeat the above experiment for another set of 3 angles. Predicting forces needed for equilibrium [2.0 pts.] The first true test of any scientific theory is whether or not it can be used to make accurate predictions. This is the reverse of the above experiment. Please select any 2 angles and 2 forces. Using the component method described above find the third force necessary to maintain equilibrium. Experimentally verify your prediction with the apparatus. Is the measured third force the same as you predicted? What is the percentage error? Discuss the reasons for any deviation between the predicted and the measured values. Repeat for another set of 2 angles and 2 forces. Additional points will be awarded for your statements of Purpose [0.5], your written Conclusions [0.5], and for the Quiz and/or overall neatness and organization of your report [1.0].

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Prelab Assignment [2.0 points]: The vector r A has a magnitude of 5 mph and points along the x-axis,

vector r

has a magnitude of 8 mph and makes an angle of 150o, and vector B r C has a magnitude of 10 mph

and an angle of 255o. Please draw the vectors on paper and complete the sections (a), (b), and (c). Note the unit mph stands for miles per hour. Explain the scale in your drawing and label lengths and angles.

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