# Measurements and Uncertainty in Physics

Measurements and dealing with their uncertainties are some first skills you learn in a lab. In this article, I will go over what you need to know when dealing with data.

**Introduction**

The beginning of physics started with measuring quantities. It soon became apparent that not all data meant the same. It was important to have reliable measurements. However, that was often a problem. Measuring things was often a random affair. The devices used were not all made the same either. Many measuring devices were poorly made and inaccurate at first. So measurements will never be perfect. They will not because humans are error prone and the devices we use are as well.

**Uncertainty**

Every device, like a ruler, has a built in amount of uncertainty. It is vital to state this when publishing measurements. It also gives the uncertainty as a percentage. This can help make it clear. It is the ratio of the uncertainty to the measured value, multiplied by 100.

If we measure a piece of metal to be 6.5 cm long, we can assume it has an uncertainty of .1 cm. To get the percentage of uncertainty we divide:

\[ \frac{0.1}{6.5} = .015 = 1% \]

**Significant Figures**

This is the number of reliably known digits in a number. Non-zero digits are usually significant. Zeroes vary according to the situation. In basic lab work, you never want to state more precision than you have. This is just inaccuracy. If your measurements have 3 significant digits, then your calculations should have only 3 significant digits. Many people do this, don’t be one of them, please. If you are dealing with numbers that contain different amounts of significant digits, then always use the least amount.Consider the number 12 and 23.4. I use these two numbers in a calculation. The result should have 2 significant figures in it.

**Scientific Notation**

When you are dealing with numbers that contain many digits it is convenient to use scientific notation. A number like 100,00,000 can be written as \( 1.0 * 10^7 \). This is just a standard way to display numerical values. Another example is the number .00000383. I would write it as \( 3.83 * 10^-6 \).

**Derived Quantities**

Derived quantities come from the known base quantities. Base quantities are length, time, mass, electric current, temperature, substances, and luminosity. So a derived quantity uses these to form a different unit. Speed uses time and length units, for example.

**Conversion of Units**

Anything that we can measure comprises, a number, and the associated unit. Without the unit, the number means nothing. Sometimes there are multiple units involved. When this happens, we must convert to something meaningful to us. Lets convert 15 inches to centimeters. 1 in = 2.54 cm.

\[ 15 in * \frac{2.54 cm}{in} = 38.1 = 38 cm \]

You see that the inches cancelled out. This is the trick to knowing if you set it upright. The correct units will cancel out and it will leave you with the one unit that you want.

**Example**

We consider the age of the universe to be 14 billion years old. Let’s write this in scientific notation.

\[ 14,000,000,000 = 1.4 * 10^10 \]

**Example**

How many significant figures do the following numbers have? 214, 81.6, and .0086?

214 = 3 significant digits, 81.6 = 3 significant digits, and .0086 has 2 significant digits

**Example**

Write out the following numbers in scientific notation. 21.8, .0068

21.8 = 2.18 * 10^1 and .0068 = 6.8 8 10^-3

Write out the following number. \( 8.69 * 10^4 \)

\[ 8.6900 \]

**Conclusion**

This section was about the basics of measurements and uncertainty. When measuring, you must always take into account the uncertainty of your tool. As you state your calculation, give the correct number of significant digits.