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DePaul HON 225 - Measurements Graphs

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HON 225 Graded Lab Activity (50 points) Measurements and Graphs INTRODUCTION A graph is a pictorial representation of ordered pairs of numbers from which you may quickly determine relationships between quantities. A change in one quantity may cause another to change, e.g. the area of a circle increases as its radius increases. The radius is the independent variable and the area is the dependent variable, because the area depends on the size of the radius. After the ordered pairs of numbers are plotted on graph paper, the plotted points are connected by a smooth line or curve, depending on the trend these points follow on the graph. This curve can be described by a mathematical equation. The two quantities plotted are called variables. PURPOSE Because some information presented in lecture is done so graphically, it is important to be able to understand and interpret various types of graphs or plots. The easiest way to learn this is to construct some yourself. By the end of this lab, you should be able to plot graphs displaying both linear and curved relationships. You should be able to distinguish between dependent and independent variables, determine the equation of a straight line that passes through the origin (0, 0) and determine the slope of the line. The purpose of the measurement portion of this lab is to make a series of measurements and interpret those measurements and resultant calculations in terms of accuracy, precision, and significant figures. You will determine the density of different substances by determining their mass and volume. You will learn to use balances and graduated cylinders. Formulas The volume V of a regularly shaped object can be calculated from the measurements of the characteristic dimensions of the object. The following formulas will be needed in the experiment: For a rectangular solid, V = lwh , where l is length, w is width, and h is height. For a circular cylinder, V = πr2h, where r = radius of circular base and h = height. The volume V of an irregularly shaped object, such as a piece of rock, is determined by immersing it in a liquid (water) in a graduated vessel. Since the object will displace a volume of water equal to its own volume, the difference in the level readings before and after immersion gives the volume of the object. Graduated cylinders typically have scale divisions of 5 or 10 milliliters (mL). A measurement can be made (estimated) to 1 mL. [Note: 1 mL = 1 cm3 = 1 cubic centimeter.] The density ρ of a substance is defined as its mass per unit volume, ρ = M/V. Density is commonly expressed in g/mL or g/cm3. Density is a characteristic property of a substance and can often be used to identify unknown materials. Also, if the density and volume of an object are known, its mass may be obtained.EQUIPMENT Rulers are provided. Bring a sharp pencil and a calculator. You will use graduated cylinders, balances, wooden blocks, and unidentified minerals/rocks. PROCEDURE 1: Plotting a line In this exercise, calculate the ordered pairs you will need to plot by using the mathematical equation for the circumference of a circle as a function of the diameter (Eq. 1), then record the values in Data Table 1 (on your report sheet). Values should be rounded off to two decimal places. C = πd (1) where C = the circumference in centimeters d = the diameter in centimeters π = 3.14. Plot a graph (Graph number 1) of the circumference of a circle as a function of the diameter using the information in Data Table 1. The following guidelines should be used when plotting graphs: 1. Make the proper choice of graph paper 2. Use a sharp pencil (not a pen!) and write legibly 3. Choose scales so that the major portion of the graph is used 4. Plot the independent variable on the x axis and the dependent variable on the y axis. 5. Choose scales for the x and y axes that are easy to read and plot 6. Plot each ordered pair of numbers as a single point (d1, C1), (d2, C2), etc. 7. Plot each point clearly, using a dot surrounded by a small circle. 8. Label the axes with the quantity plotted, including units, e.g. distance (cm) 9. Draw a smooth line connecting the plotted points 10. Give the graph a title (usually the upper center of the graph). The name includes the axes labels plus the object to which the data refer, e.g. "volume vs radius for a sphere" 11. Write your name, date, and section on the graph (usually upper right corner). 13. Determine the slope of the curve. Slope is the change in the y values divided by the change in the x values. In symbol notation this is written Δy/Δx, read as "delta y over delta x" (delta means "change in"). Using m as the symbol for slope: Slope = m = Δy/Δx = (y2 - y1)/(x2 - x1). If the straight line passes through the origin (0, 0) then you can write: Slope = m = (y2 - y1)/(x2 - x1) = (y2 - 0)/(x2 - 0) = y2 /x2 m = y/x or y = mx. This is the general equation for a straight line that passes through the origin. Write the equation of the line on the graph close to the line. Slope (m) of a straight line has a constant value. 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 you will have curves that do not pass the origin, and y intercept will have a numerical value.PROCEDURE 2: Plotting a parabola The relationship between the area of a circle and the radius of the circle is written as follows: Area = π(radius)2 A = πr2 (2) where A = the area in square meters r = the radius in meters π = 3.14. Using equation 2, calculate the ordered pairs of numbers for seven values of the radius and record them in Data Table 2 (on your report sheet). Plot a graph (Graph number 2) of the area (y axis) as a function of the radius (x axis). Label the graph completely. The equation y = kx2 is the general equation for a curve called a parabola. In this equation y is a function of x squared (x2) unlike the equation for a straight line (y = mx), where y is a function of the linear value of x. PROCEDURE 3: Measurement of regularly shaped objects 1. Using your ruler, measure each dimension of a small block of wood. Record all data in the table provided in the Report Sheet. Calculate the volume of the block of wood. All data and calculations


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