Name Physics 102



Physics 1405 Names______________________

RATIOS

Introduction

Many times in physics, the relationship between quantities can be described in terms of a ratio of one quantity to another. For example, the equation F = ma can be equally written as a = F/m.

If a ratio is constant, then there is a simple one-to-one relationship between the quantities. For a constant ratio, the relationship is linear. For a linear relationship between quantities in a ratio, if one quantity doubles, then the other quantity also doubles; if one quantity triples, then the other also triples. For our example of a = F/m, if a is constant, then when F triples, m must also triple, and so on.

If we plot a graph of the two quantities, for example F on the y-axis and m on the x-axis, then the slope of the graph is the ratio of these two quantities, a. If the slope is constant, then the ratio is constant and the graph is a straight line, or a linear graph.

Procedure, Data, and Results

Part A: Example of a Constant Ratio - Circles and (

1. Find four different-sized circular objects and trace each circle on a piece of paper.

2. Mark the diameter of each circle, measure each diameter in mm and record in Data Table 1.

3. Measure the circumference of each circle by winding a string around it and then measuring

the length of the string in mm. Record the measurements for each circle in Data Table 1.

4. Find the ratio of the circumference of each circle to its diameter. To do this, divide the

circumference C by the diameter D. Record the ratio for each trial in Data Table 1.

5. The ratio of the circumference of a circle to its diameter is known as pi (symbol (), which

has a value of about 3.14. Average the four values of ( in Data Table 1 to give your

experimental result. Calculate the experimental error using the following equation:

Experimental Error: [pic]

where A is the accepted value and E is the experimental value.

Data Table 1

| |Small Circle |Medium Circle |Large Circle |Largest |

| | | | |Circle |

| | | | | |

|Diameter, D | | | | |

|(mm) | | | | |

| | | | | |

|Circumference, C | | | | |

|(mm) | | | | |

| | | | | |

|Ratio of Circumference to | | | | |

|Diameter, C/D (() | | | | |

Average of C/D (Experimental Value, (): ____________

Accepted Value, A, of C/D: ( = 3.14

Experimental Error, [pic] _____________

6. Now, plot a graph of Circumference (y-axis) vs. Diameter (x axis) using Excel. Find the best fit line for your data and include the equation on the graph. It will be in the form y = mx + b where m is the slope of the line. The slope of this line is the ratio C/D, which theoretically is (.

7. Calculate the experimental error from your graphed line’s slope.

Slope from Graph (Experimental Value, (): ______________

Accepted Value, A, of C/D: ( = 3.14

Experimental Error, [pic] _____________

Part B: Example of a non-constant Ratio - Volume and Surface Area

1. Obtain four different-sized cubes or use the ones on the last pages. A cube has six faces each having the same area. The overall surface area of a cube can be found by finding the area of one face and then multiplying by six, the number of faces of the cube:

Area of a side = (length of the side)[pic]

Surface area = 6[pic]Area of a side

Measure one side of each cube in mm, then calculate the overall surface area and record your finding in Data Table 2.

2. The volume of a cube can be found by cubing the length of the side of one face. This is the same as multiplying length by length by length.

Volume of a cube = (length)[pic]

Use the length of each cube which you already measured and calculate the volume of the cube. Record your finding for the cube in Data Table 2.

3. For each cube, calculate the ratio of Volume, V, to Surface Area, A , or, mathematically, V/A. Record the result in Data Table 2.

4. Using Excel, plot a graph of Volume, (y-axis) to Surface Area (x-axis). Does the ratio of volume to surface area appear to be a straight line?

Optional: Try to find a “best fit” curve which will work for your graph.

Data Table 2

| |Small Cube |Medium Cube |Large Cube |Extra Large Cube |

| | | | | |

|Length | | | | |

|(mm) | | | | |

| | | | | |

|Volume | | | | |

|(mm)3 | | | | |

| | | | | |

|Surface Area | | | | |

|(mm)2 | | | | |

| | | | | |

|Volume /Area | | | | |

|(mm) | | | | |

Conclusions

1. When you plotted the Circumference vs. Diameter of the circles, did you get a straight line?

What does this tell you about the relationship between Circumference to Diameter of a

circle?

2. When you plotted the Volume vs. Surface Area of the cubes, did you get a straight line?

What does this tell you about the relationship between Volumes to Surface Area?

3. Using your own words, explain the meaning of the slope of a straight-line graph

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