A graph is a picture of ordered pairs of numbers. Thus, to draw a graph we...

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A graph is a picture of ordered pairs of numbers. Thus, to draw a graph we must use graph paper and a set of ordered pairs of numbers. Graph paper is provided at the end of this experiment. You must cal culate or determine experimentally the ordered pairs of numbers. In this experiment, calculate the or- dered pairs using the mathematical equation or formula for the circumference of a circle as a function of the diameter of the circle (Eq. 1.1), then record the values in Data Table 1.1. Values for the dir- cumference should be rounded off to two decimal places. The circumference of a circle equals the value pi (r) times the diameter of the circle. This can be written in symbol notation as C = nd (11) where C the circumference, d - the diameter, -3.14. म Data Table 1.1 Diameter (d) centimeters (cm) Independent variable 1 2 3 4 5 6 7 Circumference (C) centimeters (cm) Dependent variable C=314² C=628 C=942 C-125 C÷15-7 C=18.84-718.8 C=21.98=722.0 Average value of C/d Ratio C/d 1. Make the proper choice of graph paper. 2. Use a pen or a sharp pencil. 3. Write neatly and legibly. 3-11 3.14 3.14 3.14 3.14 3.14 3.14 3.14 c=14 व gel & fun Plot a graph (Graph number one] of the circumference of a circle as a function of the diameter using the information in Data Table 1.1. The following guidelines should be used when plotting graphs: 2r=4 G isa Funcru OFF 4. Choose scales so that the major portion of the graph paper is used. 5. Choose scales for the x and y axes that are easy to read and plot. 6. Plot the independent variable on the horizontal or r axis and the dependent variable on the vertical or y axis, The two quantities (circumference and diameter) plotted for this experiment are called variables. For the data given, the diameter of the circle is the independent variable and the circumference the de- pendent variable. d 7. Plot each ordered pair of numbers as a single point (d₁, C₁), (d2, C₂), etc. 8. Plot each point clearly using a dot surrounded by a small circle Ⓒ. 9a. Label the xr and y axes with the quantity plotted. Examples: distance, time, area, volume, etc. 10. Draw a smooth line connecting the plotted points. "Smooth" suggests that the line does not have to 9b. State the units of each quantity plotted. Examples: centimeters, seconds, cm², cm³, etc, pass exactly through each point but connects the general areas of significance. 11. Give the graph a title and place the title on the graph (usually upper center of graph). The name of the graph is taken from the labels of the x and y axes plus reference to the object to which the data refer. Examples: volume versus radius for a sphere; period versus length for a simple pendulum. 12. Place your name and the date on the graph (usually lower right side of graph). See Fig. 1.1 for an example of a graph plotted according to the above instructions. 13. Determine the slope of the curve. The slope of a curve (for this graph the curve is a straight line) is defined as the change in the y or vertical values divided by the change in the x or horizontal values. This can be stated in symbol notation as Ay/Ar. Read this as "delta y over delta .x" (delta means "change in"). Using m as the symbol for the slope we can write Ay y2-yi Slope Ar x2-x₂ If the straight line passes through the origin (ordered pair, 0, 0) then we can write 32-0 y2 X2-0 X2 Slope or = M = y2-y₁ X2-X1 y 771=- x 2 M y = mx This is the general equation for a straight line that passes through the origin. Remember, the slope (m) of a straight line has a constant value. Therefore, the values (V2. x2) can be any ordered pair of points on the plotted graph. The point where the curve crosses the y axis is known as the y intercept. In the equation above, when x is zero, y is zero. Thus the curve crosses the y axis at the origin. Later we shall have curves that do not pass through the origin, and they intercept will have a numerical value. PROCEDURE 2 The relationship between the area of a circle and the radius of the circle can be stated as follows: The area of a circle equals the constant pi (r) times the radius squared, or written in symbol notation, or where A= the arca in square meters, R the radius in meters, T -3.14. Using Eq: 1.2, calculate ordered pairs of numbers for seven values of the radius. Record the cal- culated values in Data Table 1.2. Plot a graph [Graph number two] of the area (y axis) as a function of the radius (r axis). Label the graph completely. The equation y - k is the general equation for a curve called a parabola. In this equation y is a function of x squared (²) where, as in the equation for a straight line (v mux), y is a function of the linear value of x. When the relationship between pressure (p) and volume (V) for an ideal gas (Boyle's law) is studied, you will be asked to plot the equation pl k. This is the equation for a hyperbola. Stated in terms of x and y, the equation is y= k/x. See Part 2, Section 8.4 in the Study Guide for more information on the parabola and the hyperbola. Data Table 1.2 Radius (R) meters 1 PROCEDURE 3 Many times the data plotted from an experiment in the laboratory yield a curve whose true nature is difficult to observe because of limited data. The true nature (the equation of the curve) can be deter- mined when the data are extended and replotted in order to obtain a straight line. The third column in Data Table 1.2 is labeled (R2). Calculate the seven values for the radius squared and record them in Data Table 1.2. Plot a graph (Graph number threel with the area as a function of the radius squared. Label the graph completely and calculate the slope. 2 2 3 Area = x(radius)² Á TRỊ 4 5 6 7 Area (4) (meters) 3.14 13.0 28.2 50.2 22=4 3²=9 4²=16 78.552=25 113. 154 314x R 1st graph R² (meters)² (1.2) 6²=36 72:49 2nd graph A(R)² y (x) QUESTIONS 1. Describe the method you used to determine the magnitude of the divisions for the x and y axes in order to plot a graph that uses the major portion of the graph paper. 2. Distinguish between dependent and independent variables. 3. Name the dependent and independent variables in Procedure 2. 4. How is the name for the graph selected? 5. How are the units for the slope determined? 6. What is the relationship between the units for the x and y axes and the units for the slope? 7. Is the relationship between the circumference and the diameter of a circle a linear relationship? Explain your answer. 8. What is the relationship between the slope of the curve in Procedure 1 and the ratio of C/d? 9. When the value for x increases faster than the values for y, which way (toward or away from) will the graph curve with respect to the y axis? 10. State in words the relationship between the area of a circle and the radius². A graph is a picture of ordered pairs of numbers. Thus, to draw a graph we must use graph paper and a set of ordered pairs of numbers. Graph paper is provided at the end of this experiment. You must cal culate or determine experimentally the ordered pairs of numbers. In this experiment, calculate the or- dered pairs using the mathematical equation or formula for the circumference of a circle as a function of the diameter of the circle (Eq. 1.1), then record the values in Data Table 1.1. Values for the dir- cumference should be rounded off to two decimal places. The circumference of a circle equals the value pi (r) times the diameter of the circle. This can be written in symbol notation as C = nd (11) where C the circumference, d - the diameter, -3.14. म Data Table 1.1 Diameter (d) centimeters (cm) Independent variable 1 2 3 4 5 6 7 Circumference (C) centimeters (cm) Dependent variable C=314² C=628 C=942 C-125 C÷15-7 C=18.84-718.8 C=21.98=722.0 Average value of C/d Ratio C/d 1. Make the proper choice of graph paper. 2. Use a pen or a sharp pencil. 3. Write neatly and legibly. 3-11 3.14 3.14 3.14 3.14 3.14 3.14 3.14 c=14 व gel & fun Plot a graph (Graph number one] of the circumference of a circle as a function of the diameter using the information in Data Table 1.1. The following guidelines should be used when plotting graphs: 2r=4 G isa Funcru OFF 4. Choose scales so that the major portion of the graph paper is used. 5. Choose scales for the x and y axes that are easy to read and plot. 6. Plot the independent variable on the horizontal or r axis and the dependent variable on the vertical or y axis, The two quantities (circumference and diameter) plotted for this experiment are called variables. For the data given, the diameter of the circle is the independent variable and the circumference the de- pendent variable. d 7. Plot each ordered pair of numbers as a single point (d₁, C₁), (d2, C₂), etc. 8. Plot each point clearly using a dot surrounded by a small circle Ⓒ. 9a. Label the xr and y axes with the quantity plotted. Examples: distance, time, area, volume, etc. 10. Draw a smooth line connecting the plotted points. "Smooth" suggests that the line does not have to 9b. State the units of each quantity plotted. Examples: centimeters, seconds, cm², cm³, etc, pass exactly through each point but connects the general areas of significance. 11. Give the graph a title and place the title on the graph (usually upper center of graph). The name of the graph is taken from the labels of the x and y axes plus reference to the object to which the data refer. Examples: volume versus radius for a sphere; period versus length for a simple pendulum. 12. Place your name and the date on the graph (usually lower right side of graph). See Fig. 1.1 for an example of a graph plotted according to the above instructions. 13. Determine the slope of the curve. The slope of a curve (for this graph the curve is a straight line) is defined as the change in the y or vertical values divided by the change in the x or horizontal values. This can be stated in symbol notation as Ay/Ar. Read this as "delta y over delta .x" (delta means "change in"). Using m as the symbol for the slope we can write Ay y2-yi Slope Ar x2-x₂ If the straight line passes through the origin (ordered pair, 0, 0) then we can write 32-0 y2 X2-0 X2 Slope or = M = y2-y₁ X2-X1 y 771=- x 2 M y = mx This is the general equation for a straight line that passes through the origin. Remember, the slope (m) of a straight line has a constant value. Therefore, the values (V2. x2) can be any ordered pair of points on the plotted graph. The point where the curve crosses the y axis is known as the y intercept. In the equation above, when x is zero, y is zero. Thus the curve crosses the y axis at the origin. Later we shall have curves that do not pass through the origin, and they intercept will have a numerical value. PROCEDURE 2 The relationship between the area of a circle and the radius of the circle can be stated as follows: The area of a circle equals the constant pi (r) times the radius squared, or written in symbol notation, or where A= the arca in square meters, R the radius in meters, T -3.14. Using Eq: 1.2, calculate ordered pairs of numbers for seven values of the radius. Record the cal- culated values in Data Table 1.2. Plot a graph [Graph number two] of the area (y axis) as a function of the radius (r axis). Label the graph completely. The equation y - k is the general equation for a curve called a parabola. In this equation y is a function of x squared (²) where, as in the equation for a straight line (v mux), y is a function of the linear value of x. When the relationship between pressure (p) and volume (V) for an ideal gas (Boyle's law) is studied, you will be asked to plot the equation pl k. This is the equation for a hyperbola. Stated in terms of x and y, the equation is y= k/x. See Part 2, Section 8.4 in the Study Guide for more information on the parabola and the hyperbola. Data Table 1.2 Radius (R) meters 1 PROCEDURE 3 Many times the data plotted from an experiment in the laboratory yield a curve whose true nature is difficult to observe because of limited data. The true nature (the equation of the curve) can be deter- mined when the data are extended and replotted in order to obtain a straight line. The third column in Data Table 1.2 is labeled (R2). Calculate the seven values for the radius squared and record them in Data Table 1.2. Plot a graph (Graph number threel with the area as a function of the radius squared. Label the graph completely and calculate the slope. 2 2 3 Area = x(radius)² Á TRỊ 4 5 6 7 Area (4) (meters) 3.14 13.0 28.2 50.2 22=4 3²=9 4²=16 78.552=25 113. 154 314x R 1st graph R² (meters)² (1.2) 6²=36 72:49 2nd graph A(R)² y (x) QUESTIONS 1. Describe the method you used to determine the magnitude of the divisions for the x and y axes in order to plot a graph that uses the major portion of the graph paper. 2. Distinguish between dependent and independent variables. 3. Name the dependent and independent variables in Procedure 2. 4. How is the name for the graph selected? 5. How are the units for the slope determined? 6. What is the relationship between the units for the x and y axes and the units for the slope? 7. Is the relationship between the circumference and the diameter of a circle a linear relationship? Explain your answer. 8. What is the relationship between the slope of the curve in Procedure 1 and the ratio of C/d? 9. When the value for x increases faster than the values for y, which way (toward or away from) will the graph curve with respect to the y axis? 10. State in words the relationship between the area of a circle and the radius².