Putting Scientific Notation to Work

Physics deals with some very large and very small numbers. To work with such numbers, you use scientific notation. Scientific notation is expressed as a number multiplied by a power of 10.

For example, suppose you're measuring the mass of an electron in the MKS system. You put an electron on a scale (in practice, electrons are too small to measure on a scale — you have to see how they react to the pull of magnetic or electrostatic forces to measure their mass), and you measure the following:

0.00000000000000000000000000000091 kg

What the heck is that? That's a lot of zeros, and it makes this number very unwieldy to work with. Fortunately, you know all about scientific notation, so you can convert the number into the following:

9.1 × 10-31 kg

That is, 9.1 multiplied by a power of 10, 10-31. Scientific notation works by extracting the power of 10 and putting it on the side, where it's handy. You convert a number to scientific notation by counting the number of places you have to move the decimal point to get the first digit in front of that decimal point. For example, 0.050 is 5.0 × 10-2 because you move the decimal point two places to the right to get 5.0. Similarly, 500 is 5.0 × 102 because you move the decimal point two places to the left to get 5.0.

Check out this practice question about scientific notation:

EXAMPLE

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.

5. What is 0.0043 in scientific notation?

Solve It

6. What is 430,000.0 in scientific notation?

Solve It

7. What is 0.00000056 in scientific notation?

Solve It

8. What is 6,700.0 in scientific notation?

Solve It

Converting between Units

Physics problems frequently ask you to convert between different units of measurement. For example, you may measure the number of feet your toy car goes in three minutes and thus be able to calculate the speed of the car in feet per minute, but that's not a standard unit of measure, so you need to convert feet per minute to miles per hour, or meters per second, or whatever the physics problem asks for.

For another example, suppose you have 180 seconds — how much is that in minutes? You know that there are 60 seconds in a minute, so 180 seconds equals three minutes. Here are some common conversions between units:

[check] 1 m = 100 cm = 1,000 mm (millimeters)

[check] 1 km (kilometer) = 1,000 m

[check] 1 kg (kilogram) = 1,000 g (grams)

[check] 1 N (newton) = 105 dynes

[check] 1 J (joule) = 107 ergs

[check] 1 P (pascal) = 10 Ba

[check] 1 A (amp) = 0.1 Bi

[check] 1 T (tesla) = 104 G (gauss)

[check] 1 C (coulomb) = 2.9979 × 109 Fr

The conversion between CGS and MKS almost always involves factors of 10 only, so converting between the two is simple. But what about converting to and from the FPI and other systems of measurement? Here are some handy conversions that you can come back to as needed:

[check] Length:

• 1 m = 100 cm

• 1 km = 1,000 m

• 1 in (inch) = 2.54 cm

• 1 m = 39.37 in

• 1 mile = 5,280 ft = 1.609 km

• 1 Å (angstrom) = 10-10 m

[check] Mass:

• 1 kg = 1,000 g

• 1 slug = 14.59 kg

• 1 u (atomic mass unit) = 1.6605 × 10-27 kg

[check] Force:

• 1 lb (pound) = 4.448 N

• 1 N = 105 dynes

• 1 N = 0.2248 lb

[check] Energy:

• 1 J = 107 ergs

• 1 J = 0.7376 ft-lb

• 1 BTU (British thermal unit) = 1,055 J

• 1 kWh (kilowatt hour) = 3.600 × 106 J

• 1 eV (electron volt) = 1.602 × 10-19 J

[check] Power:

• 1 hp (horsepower) = 550 ft-lb/s

• 1 W (watt) = 0.7376 ft-lb/s

Because conversions are such an important part of physics problems, and because you have to keep track of them so carefully, there's a systematic way of handling conversions: You multiply by a conversion constant that equals 1, such that the units you don't want cancel out.

EXAMPLE

Q. A ball drops 5 meters. How many centimeters did it drop?

A. The correct answer is 500 centimeters. To perform the conversion, you do the following calculation:

5.0 meters × [100 centimeters/1 meter] = 500 centimeters

Note that 100 centimeters divided by 1 meter equals 1 because there are 100 centimeters in a meter. In the calculation, the units...