graphs and equations.pdf

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Laboratory Manual n College Physics (Volume 1 - Mechanics) Graphs and Equations

Laboratory Group # & NameDate PerformedCourseCode SectionGroup Members

OBJECTIVES:.:. To apply the rules in plotting the numerical results o an experiment..:. To linearize parabolic and hyperbolic graphs which will verifY the actual relationship

between two physical quantities :. To interpret the graphs and determine the relationship between two physical quantities.:. Formulate an equation relating two or three quantities based on the data and thegraphs.

THEORY:A graphical presentation is often used as an effective tool to show explicitly how one

variable varies with another. By plotting the numerical results o an experiment andobserving the shape o the resulting graph, a relationship between two quantities can beestablished. The shape o the graph gives us a clue o the relationship o the variablesinvolved. Some o the common ones are the following:* A straight line graph indicates linear or direct relationship between two quantities.* A hyperbolic graph indicates an inverse relationship.* A parabolic graph tells us o a specific kind oflinear or direct relationship.

The specific equation relating the two variables o the graph can only be formulatedwhen the graph is linearized. We will see how this can be done in the succeedingdiscussion.


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Laboratory Manual in College Physics (Volume 1 - Mechanics) Graphs and Equations

A. Straight Line raphsA. 1 inear Relationship

Figure 1 shows a straight-line graph that does not pass through the origin. This is alinear graph. t shows a linear relationship between the two variables. t means that thereis a first-degree relationship between the Celsius readings and the Fahrenheit readings. Thegeneral equation for a linear graph is

y mx bEquation (1)

where m and b are constants; m is the slope of the line and b is the y-intercept. They-intercept of the line is the value o y when x is zero. Ifwe take = 68 , x = 20, and b= 32 in graph 1, the slope can be obtained using Eq. (1):

Equation (2)

Fahrenheit CelciusOF) COG32 o68 20104 40140 60176 80

Substituting the value of the slope obtained in Eq. (2) to Eq. (1) and considering that they-axis is O and the x-axis is c the equation relating Fahrenheit reading and Celsiusreading is therefore:Equation (3)

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We can also extrapolate values from the graph. f we extend the line downward until thetemperature is OaF we get the corresponding value in Celsius which is 17.8 Dc. Byinterpolation we get values within the line such as 50C for the corresponding Fahrenheitreading of 122F.

A.2 Direct ProportionalityGraph 2 shows a straight line passing through the origin. The zero values for

both variables simultaneously occur. When time is doubled the distance is also doubled.In this case, we say that distance is directly proportional to time. In general, when twovariables x and y re directly proportional to each other, the equation relating them is:

y a x y = kx or k y Equation (4)xwhere k is the constant of proportionality. This equation shows that the quotient of the

two variables is always equal to a constant.

Distance(m)o20406080100120

Times)o123456

140120- 100EW 80u

I :C ll 600 40

20

Distance VS. Time

_.

In graph 2, the physical slope represents the constant k:slope = 4Y = Y2 - yl tJ d Equation (5)Ax X2-X l M

The physical slope is always our concern in graphical analysis. The value is independentof the choice of scales and it expresses a significant fact about the relationship between the

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plotted variables. For example, the slope of the distance vs time graph represents theaverage speed of the object.

On the other hand, the geometrical slope which is defined to be tan e (where eisthe angle between the straight line connecting the points and the x-axis) depends on theinclination of the line and hence, on the choice of scales.

B Parabolic raphsIn general, a parabolic graph passing through the origin can be obtained for the

quantities x and y obeying the following equations:y = ld y = x3 Y = x4 .... Y = kxn Equation (6)

The relationship between x and y can be expressed as y a xn . Rewriting Eq. (6),Ln = k constant)xThe ratio ofy and xn is a constant. To verifY the actual relationship, one has to linearizethe graph, i.e., p ot y vs xn where n= 2 3 4...

250

T

.I .200

: :. , J-. 150

c.2-Q) 100:I:0

lo o 5 10 15

Time s)

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Graph # 3 shows a parabolic graph. From Eq. (6), the value of n determines thespecific equation relating x and y. By inspection, squaring the time in the data yields adirect square relationship between height and time. Thus we say, "height is directlyproportional to the square of time". To verify this relationship, plot height vs. square oftime. The result is shown in Graph #4.

Height(y) Time(t) Time squared(f)(m) (5) 52)o o o1 1 14 2 49 3 916 4 1625 5 2536 6 3649 7 96 8 6481 9 81100 10 100121 11 121144 12 144169 13 169196 14 196225 15 225

Height(y) vs Square oflime t2)

250 ~ ~ T

I 150 ~ ~ ~ ~ W 2 S ~ f f i ~ s::liii 100 ~ ~ ~ 1 , ~ ~ :J:

O ~ = ~ = = ~ = = ~ = = f = = ~ o 50 100 150 200 250Time Squared

In general, if one quantity (y) varies directly with the square of another quantity (x") wewrite, y x 2 . In this case n = 2. Thus the equation that correctly expresses therelationship of height (h) and time t) in the data is:

h k() h _ t 22 = constant =>twhere the constant k represents the slope ofheight vs. time squared graph.

C. Hyperbolic GraphsHyperbolic graphs can be obtained for quantities obeying the following equations:

y = k/x, Y = klx", Y = k/x3 Y = k/X'. Equation (7)A hyperbolic graph indicates an inverse relationship between two quantities i.e.,y a _1_n . The specific equation can be verified by determining the value ofn For n = 1,xthe equation is y k/x.

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y X200 1100 267 350 440 533 6

o 2 4 6 8x~ ~ ~ ~ ~ ~

C l Inverse ProportionalityGraph # 5 shows a hyperbolic graph. To linearize it, try n = 1 such that y = IxPlotting Ys iiX yields a straight-line graph as shown in Graph # 6 Hence is directly

proportional to 1 x or y is inversely proportional to x In equation form:1 ky a-=> y =- => or k = xy Equation (8)x xwhere k is a constant which is equal to the slope ofy Ys i/x graph.

y x 1/x200 1 1100 2 0.567 3 0.3350 4 0.1540 5 0.233 6 0.16

y vs 1/x2:l

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0.00 0.20 0.40 0.60 0.80 1.00 1.201/x

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C.2 Inverse Square ProportionalitySometimes, plotting y vs. l x will not yield a straight line but plotting y vs. l r will

yield one. This kind of relationship is called inverse square proportionality The variabley) is inversely proportional to the square ofx Graph 7 illustrates such a case.y V5 X

: . .t ~ y , x1fi:7" 6:6i

: 9 : ~ f O : ~ L 2 r 1 : 2 ;- ..1:4,. .. 2.1,

; , - .0.8 2.80 5 3 4,,(,-.-4 -. -. : : 3 ~ 9 6:3, ..4.4

_x

The linearized graph is shown in Graph 8 This can only be obtained if n = 2 suchkthat y 2x

y x16.7 0.69.4 0.84.2 1.21.4 2.10.8 2.80.5 3.40.4 3.90.3 4.40.2 4.8

1/x?2.781.560.690.230.130.090.070.050.04

yvs:2

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0 00 0.50 1 00 1.50 2 00 2.50 3.001/x2-- .J

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Laboratory Manual in College Physics Volume 1 - Mechanics) Graphs and Equations

D Method o Least SquaresThe method of least squares is a statistical way of determining the best-fittingcurve for a given set of data. If the set of data given does not yield any of the givenrelationships above, then the best way to plot the results would be through the application

of the method ofleast squares.The method of least squares usually yields a straight line whose slope and whose y-

intercepts can be solved by applying the following equations:The slope m) is

n{ X,y;)} x;) y;)m = ~ - - - - - - - - - - - - - - - - - - - - - - - - - - Equation 9)n{ } x,) x,)

The y-intercept isb= {t, X,x,l}nv,l- t, X;)t, X,y;)

Equation 10){ t x x } ~ t, x;)t, X,l

where n represents the number of samples. After determining the slope m) and theintercept, the equation for the best line is determined by

y =mx bIt is important that experimental data be plotted correctly for accurate graphicalinterpretation. To achieve this, the rules enumerated below can be of help. For Microsoft

Excel users, the software provides almost all the necessary tools. All you need to do isenter the values needed and select the appropriate command.

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1 Determination o CoordinatesDetermine which of the quantities to be graphed is the dependent variable andwhich one is the independent variable. The independent variable is the quantity, which

controls or causes a change in the other quantity (dependent variable) whenever it isincreased or decreased. By convention, plot the independent variable along the x axis anddependent variable on the y axis.2 Labeling the axes.

Label each axis with the name of the quantity being plotted and its corresponding unit.Abbreviate all units in standard form.3 Choosing the Scale.Choose scales that are easy to plot and read. In general, choose scales for thecoordinate axes so that the curve extends over most of the graph sheet. The same scaleneed not be used for both axes. In many cases it is not necessary that the intersection ofthe two axes represent the zero values of both variables. The number should increase fromleft to right and from bottom to top. In cases where the values to be plotted areexceptionally large or small, rewrite the numbers in scientific notation. Place thecoefficients on the coordinate scale and the multiplying factor beside the unit used.4 Location of Points.

Encircle each point plotted on the graph to indicate that the value lies anywhere closeto that point. Draw the curve up to the circle on one side. If several curves appear on thesame sheet and the points might interfere, use squares and triangles to surround the dotsof the second and third curves, respectively.5 Drawing the curve.

When the points are plotted, draw a smooth line connecting the points; ignore anypoints that are obviously erratic. Smooth suggests that the line does not have to passexactly through each point but connects the general areas of significance. If there is a cluethat the quantities are linear, then a straight line representing an average value should beused. There should be more or less equal number of points above and below the line. Fornonlinear curves, points should be connected with a smooth curve so that the pointsaverage around the line. For Microsoft Excel users, this procedure is automatically doneby a specific command.6 Title of the Graph

At an open space near the top of the paper, state the title of the graph in the form ofthe dependent variable 6J vs. the independent variable x).

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PPR TUS M TERIALS

Graphing paper, pencil and pen, ruler, Computer with :tvficrosoft Excel

EXERCISES1. The following data were obtained in an experiment relating time t) (the independent

variable) to the speed (v) of an accelerating object.

! ~ ~ S ~ ~ O = = = = = I = ~ ~ o = = = = = = = ~ = ~ = = = = = = I = ~ ~ o = = = = = = I = ~ = ~ = = = = I _ ~ _ ~ o _ __ Plot these data on rectangular coordinate paper. For those with computers, useMcrosoft Excel.(a) Determine the slope of the graph(b) What physical quantity does the slope represent?(c) Determine the y-intercept of the graph. What does it represent?(d) What is the equation of the curve?

2. The heating effect of an electric current in a rheostat is found to vary directly withthe square of the current. What type of graph is obtained when the heat is plotted asa function of current? ow could the variables be adjusted so that a linear relationwould be obtained?

3. The current in a variable resistor to which a given voltage is applied is found to varyinversely with the resistance. What is the shape of the current resistance curve?How could these variables be changed in order for a straight-line graph to beobtained?

Do the following for exercises #4 to #8:(a) Plot the given values y vs. x . Select proper coordinate scales, label plotpoints, draw a smooth curve through points.(b) Linearize the graph. If necessary, compute different powers of variables andplot until you get a straight line.(c) Determine the equation o the line obtained. Indicate the value of n, k, andother constants or intercepts present in the graph.

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4. The data below shows how the electric field E) due to a point charge varies withdistance r).

5. The following values represent a particle with an x-coordinate that varies in time.3

654 I 5 I 6 7

-120 I -425 I -880 I -1515 I

6. The following values represent the motion of a particle with a y-coordinate thatvaries in time.

7. Potential energy Us) as a function ofx-coordinate for the mass-spring system.

8. The values below are unknown variables x and y with a characteristic behavior.

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Laboratory Manual College Physics Volume Mechanics) Graphs and Equations

9 Detennine the equation, which will represent the best line for the following set of dataand plot the graph of the equation.Method ofLeast Squares

1 o 4 6 o2 1 7.1 1 7.13 2 9.5 4 19.14 3 11.5 9 34.55 4 13.7 16 54.86 5 15.9 25 79.87 6 18.6 36 111.68 7 20.9 49 146.39 8 23.5 64 188.010 9 25.4 81 228.6

n = number of samples)nL x.x.)1

The slope is:

The y-intercept is:

nL X) -jnL xy)i l

nL yJ = ~ _

The equation of the best line for the data is: y=mx+b=_._x

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graphs and equations.pdf

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