Measure density of glasses by buoyancy



Dimensional Analysis Lab

General Chemistry Laboratory

Syracuse University Chemistry 107

Introduction

In mathematics and science, dimensional analysis is a tool to understand the properties of physical quantities independent of the units used to measure them. Every physical quantity is some combination of mass, length, time, electric charge, and temperature, (denoted M, L, T, Q, and Θ, respectively). For example, velocity, which may be measured in meters per second (m/s), miles per hour (mi/h), or some other units, has dimension L/T.

Dimensional analysis is routinely used to check the plausibility of derived equations and computations. It is also used to form reasonable hypotheses about complex physical situations that can be tested by experiment or by more developed theories of the phenomena, and to categorize types of physical quantities and units based on their relations to or dependence on other units, or their "dimensions", or their lack thereof.

Dimensional analysis was developed by the 19th century French mathematician Joseph Fourier,[1] based on the idea that the physical laws like F = ma should be independent of the units employed to measure the physical variables. This led to the conclusion that meaningful laws must be homogeneous equations in their various units of measurement, a result which was eventually formalized in the Buckingham π theorem. This theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n − m dimensionless parameters, where m is the number of fundamental dimensions used. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables.

A dimensional equation can have the dimensions reduced or eliminated through nondimensionalization, which begins with dimensional analysis, and involves scaling quantities by characteristic units of a system or natural units of nature.

A young student hasn’t been feeling well and decides to take 2 Tylenol. The student collapses during the middle of study hall, and you are trying to help the nurse understand what happened. The only information you have is that she mentioned she had taken Tylenol. The nurse wants to know what strength, how many tablets, how many times in the last 24 hours… You feel that information cannot be that important, “I mean, after all, it’s just Tylenol…”

In this lab we will explore some of the methods that may be employed in determining mathematical answers using dimensional analysis.

Objectives

The objective of this laboratory is to use dimensional analysis to solve somewhat silly problems.

Background

A Guide to Dimensional Analysis

For examples and more see:

1. Determine what you want to know. Read the problem and identify what you're being asked to figure out, e.g. "how many milligrams are in a liter of solution."

a. Rephrase if necessary using "per." Example: You want to know "milligrams per liter."

b. Translate into "math terms" using appropriate abbreviations to end up with "mg/L" as your answer unit (AU). Write this down, e.g. "AU= mg/L"

2. Determine what you already know.

a. What are you given by the problem, if anything? Example: "In one minute, you counted 45 drops."

▪ Rephrase if necessary. Think: "Drip rate is 45 drops per minute."

▪ Translate into math terms using abbreviations, e.g. "45 gtt/min"

• If a given is in the form mg/kg/day, rewrite as mg/kg x day (see example 4)

• If a percentage is given, e.g. 25%, rewrite as 25/100 with appropriate labels (see example 5)

b. Determine conversion factors that may be needed and write them in a form you can use, such as "60 min/1 hour." You will need enough to form a "bridge" to your answer unit(s). See example 1.

▪ Factors known from memory: You may know that 1 kg = 2.2 lb, so write down "1 kg/2.2 lb" and/or "2.2 lb/1 kg" as conversion factors you may need.

▪ Factors from a conversion table: If the table says "to convert from lb to kg multiply by 2.2," then write down "2.2 lb/1 kg"

3. Setup the problem using only what you need to know.

a. Pick a starting factor.

▪ If possible, pick from what you know a factor having one of the units that's also in your answer unit and that's in the right place. See example 1.

▪ Or pick a factor that is given, such as what the physician ordered.

▪ Note that the starting factor will always have at least one unit not in the desired answer unit(s) that will need to be changed by canceling it out.

b. Pick from what you know a conversion factor that cancels out a unit in the starting factor that you don't want. See example 1.

c. Keep picking from what you know factors that cancel out units you don't want until you end up with only the units (answer units) you do want.

d. If you can't get to what you want, try picking a different starting factor, or checking for a needed conversion factor.

e. If an intermediate result must be rounded to a whole number, such as drops/dose which can only be administered in whole drops, setup as a separate sub-problem, solve, then use the rounded off answer as a new starting factor. See example 9.

4. Solve: Make sure all the units other than the answer units cancel out, then do the math.

a. Simplify the numbers by cancellation. If the same number is on the top and bottom, cancel them out.

b. Multiply all the top numbers together, then divide into that number all the bottom numbers.

c. Double check to make sure you didn't press a wrong calculator key by dividing the first top number by the first bottom number, alternating until finished, then comparing the answer to the first one. Miskeying is a significant source of error, so always double check.

d. Round off the calculated answer to the correct amount of significant figures. Remember, most of the time with conversion factors the ‘1’ is an exact number with infinite sig figs.

e. Add labels (the answer unit) to the appropriately rounded number to get your answer. Compare units in answer to answer units recorded from first step.

Take a few seconds and ask yourself if the answer you came up with makes sense. If it doesn't, start over.

Calculations:

(Adapted from Dimensional Analysis Olympics – Stumpff and Dieckmann, Columbus North High School)

#1 Broad Jump:

Measure your standing broad jump in cm. Average three jumps. How many broad jumps would it take for you to “jump” a mile?

|1st jump | |

|2nd jump | |

|3rd jump | |

|average distance | |

#2 Candy Around the World:

Go to the front and select a candy bar. Measure the distance of the candy bar wrapped. How many candy bars would you have to put end-to-end to wrap around the equator of the earth? (the earth’s circumference at the equator is 24,902.32 miles).

|length of candy bar | |

#3 Mass of a Kiss:

Go to the front and grab a Hershey’s kiss. Measure the mass of one single kiss. One 5 lb. Hershey’s Kiss is equivalent to how many little Hershey’s Kisses?

|mass of one Hershey’s Kiss | |

#4 Hoppin’ Good Times:

Go out into the hallway. Mark a distance of five meters by placing a strip of tape as the start, and a second strip of tape as the finish line five meters away. Time how long it takes you to hop those five meters. Calculate your hopping speed in meters/second. How many hours would it take you to hop from Middletown to New York City nonstop? (71 miles).

|time to hop five meters | |

#5 Wall to Wall Cars:

Using one of the hot wheel cars provided, and a large graduated cylinder, find the volume of the car in cm3 (remember 1mL = 1 cm3). Calculate how many cars it would take to fill the volume of your locker.

|volume of hot wheel | |

|length of locker | |

|height of locker | |

|depth of locker | |

#6 Tootsie Roll Chew:

Go to the front and grab a Tootsie Roll. Measure the mass of the Tootsie Roll with wrapper. Measure the time it takes you to chew and swallow a single Tootsie Roll. How many hours would it take you to eat 10 pounds of Tootsie Rolls? (one piece at a time).

|mass of Tootsie Roll w/ wrapper | |

|mass of Tootsie Roll wrapper alone | |

|mass of Tootsie Roll | |

|time to chew and swallow | |

#7 Speed Walking:

Go out into the hallway. Using the previously marked distance of five meters time how long it takes you to speed walk those five meters. Calculate your walking speed in mph. Calculate how many days it would take you to walk from Watertown to New York City if you only walked for 5 hours each day. (Middletown to New York is 72 miles).

|time to speed walk five meters | |

#8 Heart Smart:

Cardiac output is the total volume of blood pumped by the ventricle per minute, or simply the product of heart rate (HR) and stroke volume (SV). The stroke volume at rest in the standing position averages between 60 and 80 mL of blood in most adults. (April 1998, Montana State University – Bozeman) If your stroke volume is 70 mL, how many liters of blood would your body pump in one minute?

|resting heart rate for 1 minute | |

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