Everything we do in physics is about describing the physical world; every number and letter that you write represents some real, physical quantity. There are two main ramifications of this:
- There are restrictions to the calculations that we can do.
- We need a common way of making and reporting measurements so we can effectively and consistently communicate with each other.
There are many kinds of measurements that we can make on a physical system. The three that we will be working with the most in this book are length, mass, and time. We will work with other quantities (such as energy and force), but they can ultimately all be described in terms of length, mass, and time. Length and time are intuitive enough that I will not define them here. Mass is a bit more nuanced.
A decent high school level description of mass is the amount of “stuff” in an object. Note that this is different from the space an object takes up (it’s volume); you can take something soft and compress it down to a smaller size, and you have not changed how much “stuff” is there. Mass is also different that how much an object weighs; weight is a measure of how hard gravity pulls on an object, and the strength of gravity changes from place to place. You can weigh the same object at sea level and on top of a mountain, and find different results, even though the amount of “stuff” in your object has not changed.
However, referring to “the amount of stuff” is not precise enough for good scientific discussion. Mass is best thought of as a measurement of how much an object resists changes in motion. We’ll develop this description of mass more in chapter 6.
3.1 Physically meaningful calculations
Remember, in physics class every number and symbol represents some measurable physical quantity. We need to follow these rules for everything we do mathematically, including calculations of numbers and algebraic manipulations of equations.
3.2 Significant figures
Measurements are not infinitely precise. For example, if you measure something with a ruler that has markings every eighth of an inch, and the measurement lies between two markings, you need to estimate the length. If you perform a calculation involving this measurement, your final answer is only going to be as precise as the ruler you used to make the measurement.
Now, remember that every number that goes into a physics calculation represents a real physical quantity that was measured with some device or another. So, the result of any calculation is inherently limited by the precision of the tools used to get the measurements. In practical terms, this is to say that the string of decimals your calculator gives you as a final result is mostly meaningless.
The term significant figures refers to the rules we use to determine how to handle this limitation. A figure (or digit) is significant if it has real meaning in regards to measurement. Different scientific disciplines—and even different groups of researchers within the same discipline—have different ways of defining what “significant” means in practice. A general rule of thumb is the result of a calculation is rounded to the number of significant figures of the least precise measurement that went into the calculation. You will certainly come across situations where there are more strict rules than this. The important thing is that everyone involved uses the same set of rules for determining significant figures. (In fact, you will probably also run across situations where the rules are less strict that this; it is not uncommon in introductory physics to just round all answers to three significant figures, no matter what went into the calculation.)
If you have never encountered significant figures previously, it can be difficult to wrap your head around—numbers are numbers, right? The thing to keep in mind is that everything we do in physics needs to relate back to some physical quantity that must be measured. Here are some examples to help you figure it out:
Example
Add two following lengths together: 2.5 cm and 4.509 cm. You would get 2.5 + 4.509 = 7.009, but result is not correct. The first number has two significant figures, and the second has four. Since the least precise measurement only had 2 significant figures, we need to round our result to two significant figures as well, giving us 7.0 cm as the correct result.
In general, a zero after a decimal place is significant. So, the zero in the number 4.509 cm and the zero in 7.0 cm would both be significant. However, leading zeros are not significant. For example, if we write 4.509 cm as 0.04509 m, the two zeros at the very beginning are not significant, but the zero between the 5 and 9 is significant.
Example
Add the two lengths: 0.0025 cm and 0.045 09 cm. Your calculator would give you 0.0025 + 0.04509 = 0.04759, but your calculator knows nothing about significant figures. The first value has two significant figures, and the second has four significant figures. Again, the least precise value is still only known to two significant figures. So our result needs to be rounded to just two significant figures, and we have 0.048 cm.
In this example, the zero between the 5 and 9 in 0.04509cm was significant, but none of the other zeros were.
Example
Multiply 12.4571 m/s and 13.28 s. The calculation is: (12.4571)(13.28) = 165.430288
The first value has six significant figures and the second one has four. Our answer is 164.4 m, which has four significant figures.
Example
Divide 195.0 m by 42 s. The calculation is 195.0/42 = 4.642857143
The fist number has four significant figures, and the second only has two; we need to round our result to two significant figures, giving us 4.6 m/s.