Dimensional Analysis

Dimensional analysis is a powerful tool for solving problems and making conversions. It can allow you to do calculations without memorizing specific formulas. Dimensional analysis will be used repeatedly, especially in the first four chapters.

Example

Provided with the following conversion factors, 5028 ft = 1 mi, 2.54 cm = 1 inch, determine the number of meters in 10.2 miles?

10.2 miles ´ [5028 ft/1 mi] ´ [12 inches/1 ft] ´ [2.54 cm/1 in] ´ [1 m/100 cm] = 15631 m = 1.56 ´ 104 m

To use dimensional analysis to solve problems we need to have at hand the following pieces of information:

(a) What data and units are given to start the problem?

(b) What quantity is needed as an answer?

(c) What conversion factors are available to go from the given data to the answer?

If you use appropriate conversion factors and cancel units you can check to see if you have done the problem correctly, by determining the units of your answer.

In the previous example we converted from one unit of length to another unit of length. Dimensional analysis can also be used when the starting point and ending point are different types of units, provided we use quantities with two types of units as conversions.

For Example

• Density ® Relates mass and volume
• Velocity ® Relates length and time

Example

If water has a density of 1.00 g/cm3, what would be the mass of water (in kg) inside a swimming pool 10 ft wide, 10 ft long and 8 ft deep?

Q) What quantity are we starting with?

A) The volume of a swimming pool 10 ft ´ 10 ft ´ 8 ft = 800 ft3

Q) What quantity do we want to determine?

A) The mass of water inside the pool.

[8 ´ 102 ft3] ´ [12 in/1 ft]3 ´ [2.54 cm/1 ft]3 ´ [1.00 g/cm3] ´ [1 kg/1000 g] = 22,653 kg = 2 ´ 104 kg

Example

(68 from BL&B). The distance from the earth to the moon is 240,000 miles (2.40 ´ 105 mi) and the Concorde flies at 2400 km/hr (2.4 ´ 103 km/hr). How many hours would it take to fly to the moon on the Concorde?

What quantity are we starting with?

2.40 ´ 105 miles to the moon.

What quantity do we want to determine?

How many hours will it take to get to the moon.

2.40 ´ 105 mi ´ [5280 ft/1 mi] ´ [12 in/1 ft] ´ [2.54 cm/1 in] ´ [1 m/100 cm] ´ [1 km/1000 m] ´ [2.4 ´ 103 km/1 hr]

= 160.9 hours = 1.6 ´ 102 hrs = 6.7 days