Dimensional Analysis: Videos & Practice Problems

Dimensional analysis is a mathematical method used to convert between units by employing conversion factors that relate the units being converted. To perform dimensional analysis, start with the given amount and use conversion factors to cancel out unwanted units, thus isolating the desired end unit. For example, to convert inches to centimeters, use the conversion factor that 1 inch equals 2.54 centimeters. When multiplying, the number of significant figures (sig figs) in the final answer should match the least number of sig figs in the given data. In more complex scenarios involving multiple conversion factors, such as converting minutes to years, each step must cancel out the previous unit until the final unit is reached. The accuracy of the final value is determined by the significant figures of the initial given value, not the conversion factors. Remember, the goal is to systematically cancel out units to arrive at the correct end amount with the appropriate number of significant figures.

Dimensional Analysis

Dimensional analysis is a systematic method used to convert units from one measurement system to another, ensuring accuracy in calculations. This process begins with a known quantity, referred to as the given amount, and aims to arrive at the desired unit, known as the end amount. The key to successful dimensional analysis lies in the use of conversion factors, which are ratios that express how one unit relates to another.

For instance, when converting inches to centimeters, the conversion factor is that 1 inch equals 2.54 centimeters. To perform the conversion, you set up the equation so that the units cancel appropriately. Starting with 32 inches, you would multiply by the conversion factor:

\[32 \, \text{inches} \times \frac{2.54 \, \text{cm}}{1 \, \text{inch}} = 81.28 \, \text{cm}\]

However, it’s crucial to consider significant figures in your final answer. Since 32 has two significant figures, the result should be rounded to 81 cm.

In more complex conversions, such as converting minutes to years, multiple conversion factors are employed. For example, to convert 115 minutes to years, you would use the following conversion factors:

• 1 hour = 60 minutes
• 1 day = 24 hours
• 1 year = 365 days

Setting up the dimensional analysis, you would have:

\[115 \, \text{minutes} \times \frac{1 \, \text{hour}}{60 \, \text{minutes}} \times \frac{1 \, \text{day}}{24 \, \text{hours}} \times \frac{1 \, \text{year}}{365 \, \text{days}}\]

Calculating the total number of minutes in a year gives:

\[60 \times 24 \times 365 = 525,600\]

Thus, the calculation becomes:

\[\frac{115}{525,600} \approx 2.19 \times 10^{-4} \, \text{years}\]

In this case, the significant figures are determined by the given amount, which is 115, having three significant figures. Therefore, the final answer is expressed as 2.19 x 10^-4 years.

Overall, dimensional analysis is a powerful tool that relies on conversion factors to facilitate unit conversions while maintaining the integrity of significant figures. By canceling out units and isolating the desired measurement, this method ensures accurate and reliable results in various scientific calculations.

Dimensional Analysis Example

To determine how many questions a teaching assistant (TA) can grade in 130 minutes, we can utilize dimensional analysis, a method that allows us to convert units and solve problems systematically.

First, we identify the given amount, which is 130 minutes. Our goal is to find the total number of questions graded. We know that:

• The TA can grade 4 assignments per hour.
• Each assignment contains 12 questions.

Next, we need to convert the given time from minutes to hours since our conversion factors involve hours. The conversion factor for this is:

1 hour = 60 minutes

Now, we can set up our conversion factors:

• Convert minutes to hours: 130 minutes × (1 hour / 60 minutes)
• Convert hours to assignments: result in hours × (4 assignments / 1 hour)
• Convert assignments to questions: result in assignments × (12 questions / 1 assignment)

Now, we can combine these conversions into a single equation:

Questions = 130 minutes × (1 hour / 60 minutes) × (4 assignments / 1 hour) × (12 questions / 1 assignment)

When we perform the calculations, we first convert 130 minutes to hours:

130 minutes × (1 hour / 60 minutes) = 2.1667 hours

Next, we calculate the number of assignments graded in that time:

2.1667 hours × 4 assignments/hour = 8.6667 assignments

Finally, we convert assignments to questions:

8.6667 assignments × 12 questions/assignment = 104 questions

However, we must consider significant figures in our final answer. The values we used have the following significant figures: 130 (2 sig figs), 4 (1 sig fig), 12 (2 sig figs), and 60 (1 sig fig). The limiting factor here is the conversion factor with 1 significant figure, which means our final answer should be rounded to 1 significant figure.

Thus, the TA can grade approximately 100 questions in 130 minutes, demonstrating the effectiveness of dimensional analysis in solving unit conversion problems.

If the distance between Washington, D.C. and New York City is 224.9 miles, determine the distance in centimeters.

• A. 8.110 x 10⁹ cm
• B. 3.619 x 10⁷ cm
• C. 4.207 x 10^-3 cm
• D. 1.913 x 10^-5 cm
• E. 5.000 x 10¹¹ cm

Answer: (B)

A large scale ranching farm possesses 22.5 million acres with 5.0 x 10^-5 % reserved for housing. Determine how many square kilometers this area for dwelling represents. (1 km = 0.6214 miles) (1 mile = 5280 ft) (1 acre = 43,560 ft²)

• A. 9.2 km²
• B. 0.115 km²
• C. 1.8 km²
• D. 0.046 km²

Answer: (D)

Here’s what students ask on this topic:

What is dimensional analysis?

Dimensional analysis is a mathematical technique used to convert one set of measurements to another, ensuring that the equations used for the conversion are dimensionally consistent. This means that you can only equate or compare quantities that have the same fundamental dimensions (such as length, mass, time, electric current, etc.).

In practice, dimensional analysis involves using conversion factors, which are expressions that relate different units of measurement. By multiplying the quantity by the appropriate conversion factor, you can change the units without altering the actual value of the quantity. This is particularly useful in physics, chemistry, and engineering to check the plausibility of physical equations, to convert units from one system to another, and to compute numerical solutions even without knowing the precise values of the quantities involved.

For example, if you have a speed given in miles per hour and you want to express it in meters per second, dimensional analysis would guide you in setting up the conversion factors to correctly make this change in units.

How to do dimensional analysis?

Dimensional analysis is a method used to convert one unit of measurement to another. To perform dimensional analysis, follow these steps:

1. Identify the starting unit and the desired unit: Determine what unit you're starting with and what unit you want to end up with.
2. Find conversion factors: Look for relationships between the units that express how one unit can be converted into another. A conversion factor is a fraction that represents the equivalence between units.
3. Set up the problem: Write down the quantity you're converting as a fraction, and multiply it by the conversion factor(s). Arrange the conversion factor so that the unit you want to eliminate is opposite of the same unit in your original quantity. This way, the units you want to cancel out will be diagonal from each other.
4. Cancel units: Cross out any units that appear in both the numerator and denominator.
5. Perform the math: Multiply all the top numbers (numerators), then multiply all the bottom numbers (denominators), and divide the top by the bottom to get your final answer.
6. Check your work: Ensure that the units you're left with are the ones you wanted, and the answer makes sense in the context of the problem.

Remember, each step in the multiplication should bring you closer to the desired unit. Keep track of your units throughout the process.

What is the purpose of dimensional analysis?

Dimensional analysis is a mathematical technique used to convert one set of measurements to another, ensuring that equations and calculations are dimensionally consistent. This means that all terms in an equation must have the same dimensional units for the equation to be physically meaningful. It is also used to check the plausibility of derived equations, to plan experiments, and to deduce relationships between physical quantities that are not immediately obvious.

The technique involves using conversion factors that describe the relationship between different units of measurement. By multiplying the quantity by these conversion factors, you can systematically cancel out the unwanted units and end up with the desired units. This is particularly useful in science and engineering where calculations often involve multiple units, and ensuring the correct application of units can be the difference between success and potentially costly or dangerous errors.