the MKS system. What will your measurement’s units be?
2. You need to measure the force a tire exerts on the road as it’s moving using the MKS system. What are the units of your answer?
3. You’re asked to measure the amount of energy released by a firecracker when it explodes using the CGS system. What are the units of your answer?
Practice Answers
1. seconds. The unit of time in the MKS system is the second.
2. newtons. The unit of force in the MKS system is the newton.
3. ergs. The unit of energy in the CGS system is the erg.
Eliminating Some Zeros: Using Scientific Notation
Physicists have a way of getting their minds into the darndest places, and those places often involve really big or really small numbers. Physics has a way of dealing with very large and very small numbers; to help reduce clutter and make them easier to digest, it uses scientific notation.
Remember: In scientific notation, you write a number as a decimal (with only one digit before the decimal point) multiplied by a power of ten. The power of ten (10 with an exponent) expresses the number of zeroes. To get the right power of ten for a vary large number, count all the places in front of the decimal point, from right to left, up to the place just to the right of the first digit (you don’t include the first digit because you leave it in front of the decimal point in the result).
For example, say you’re dealing with the average distance between the sun and Pluto, which is about 5,890,000,000,000 meters. You have a lot of meters on your hands, accompanied by a lot of zeroes. You can write the distance between the sun and Pluto as follows:
The exponent is 12 because you count 12 places between the end of 5,890,000,000,000 (where a decimal would appear in the whole number) and the decimal’s new place after the 5.
Scientific notation also works for very small numbers, such as the one that follows, where the power of ten is negative. You count the number of places, moving left to right, from the decimal point to just after the first nonzero digit (again leaving the result with just one digit in front of the decimal):
Remember: If the number you’re working with is larger than ten, you have a positive exponent in scientific notation; if it’s smaller than one, you have a negative exponent. As you can see, handling super large or super small numbers with scientific notation is easier than writing them all out, which is why calculators come with this kind of functionality already built in.
Scientists have come up with a handy notation that helps take care of variables that have very large or very small values in their standard units. Say you’re measuring the thickness of a human hair and find it to be 0.00002 meters thick. You could use scientific notation to write this as
A more familiar unit prefix is k, as in kilo, which represents 103 times the unit. For example the kilometer, km, is 103 meters, which equals 1,000 meters. The following table shows other common unit prefixes that you may see.
Examples
Q. How does the number 1,000 look in scientific notation?
A. The correct answer is 1.0 × 103. You have to move the decimal point three times to the left to get 1.0.
Q. What is 0.000037 in scientific notation?
A. The correct answer is 3.7 × 10– 5. You have to move the decimal point five times to the right to get 3.7.
Practice Questions
1. What is 0.0043 in scientific notation?
2. What is 430,000 in scientific notation?
3. What is 0.00000056 in scientific notation?
4. What is 6,700 in scientific notation?
Practice Answers
1. 4.3 × 10– 3. You have to move the decimal point three places to the right.
2. 4.3 × 105. You have to move the decimal point five places to the left.
3. 5.6 × 10– 7. You have to move the decimal point seven places to the right.
4. 6.7 × 103. You have to move the decimal point three places to the left.
From Meters to Inches and Back Again: Converting Between Units
Physicists use various measurement systems to record numbers from their observations. But what happens when you have to convert between those systems? Physics problems sometimes try to trip you up here, giving you the data you need in mixed units: centimeters for this measurement but meters for that measurement – and maybe even mixing in inches as well. Don’t be fooled. You have to convert everything to the same measurement system before you can proceed. How do you convert in the easiest possible way? You use conversion factors, which we explain in this section.
Tip: To convert between measurements in different measuring systems, you can multiply by a conversion factor. A conversion factor is a ratio that, when you multiply it by the item you’re converting, cancels out the units you don’t want and leaves those that you do. The conversion factor must equal 1.
Here’s how it works: For every relation between units – for example, 24 hours = 1 day – you can make a fraction that has the value of 1. If, for example, you divide both sides of the equation 24 hours = 1 day by 1 day, you get
Suppose you want to convert 3 days to hours. You can just multiply your time by the preceding fraction. Doing so doesn’t change the value of the time because you’re multiplying by 1. You can see that the unit of days cancels out, leaving you with a number of hours:
Remember: Words such as days, seconds, and meters act like the variables x and y in that if they’re present in both the numerator and the denominator, they cancel each other out.
To convert the other way – hours into days, in this example – you simply use the same original relation, 24 hours = 1 day, but this time divide both sides by 24 hours to get
Then multiply by this fraction to cancel the units from the bottom, which leaves you with the units on the top.
Consider the following problem. Passing the state line, you note that you’ve gone 4,680 miles in exactly three days. Very impressive. If you went at a constant speed, how fast were you going? Speed is just as you may expect – distance divided by time. So you calculate your speed as follows: