This article will talk about what is Dimensional Analysis with examples. It has worksheet for Dimensional Analysis with answers. It also covers 5 steps for performing Dimensional Analysis.
Dimensional Analysis Introduction:
In order to understand what is dimensional analysis we first need to understand what dimension is?
What is dimension?
Every physical quantity can be expressed in terms of fundamental dimensions, such as length (L), mass (M), and time (T). Other derived dimensions, like velocity (LT⁻¹) or force (MLT⁻²), can be constructed from these fundamental dimensions.
What is dimensional analysis?
So, dimensional analysis is technique to analyze and understand the relationships between physical quantities and their units of measurement. It involves examining the dimensions (units) of various physical quantities and using these dimensions to derive relationships, make predictions, or check the consistency of equations.
The fundamental idea behind dimensional analysis is that physical equations must be dimensionally consistent, which means that the dimensions (e.g., length, mass, time) on both sides of an equation must be the same. If the dimensions do not match, it indicates an error or inconsistency in the equation.
Key principles of dimensional analysis:
- Identifying the dimensions of the physical quanntity.
- Using dimensional homogeneity: In a valid equation, the dimensions of each term on one side of the equation must be the same as the dimensions of the terms on the other side. This ensures that the equation makes sense from a dimensional perspective.
- Checking equations: Dimensional analysis can be used to check the validity of equations, predict the form of unknown equations, or derive dimensionless parameters that are important in many areas of science and engineering.
For example, consider the equation for the period (T) of a pendulum:
T=2π√(L/g)
Where:
T is the period of the pendulum.
L is the length of the pendulum.
g is the acceleration due to gravity.
Using dimensional analysis, you can check the equation for dimensional consistency:
Dimension of T (time).
Dimension of L (length).
Dimension of g (length/time²).
In this case, the dimensions of both sides of the equation are consistent, as they are all expressed in terms of time, length, and time squared. This consistency confirms that the equation is dimensionally valid.
Dimensional analysis worksheet with answers:
Below is dimensional analysis worksheet for your practice:
- Convert 15 meters per second (m/s) to kilometers per hour (km/h).
- Determine the dimensional consistency of the equation F = ma, where F is force (N), m is mass (kg), and a is acceleration (m/s²).
- Calculate the volume of a cube with sides measuring 2 feet each in cubic inches (in³).
- The formula for the area of a circle is A = πr², where A is area (m²) and r is the radius (m). Convert this formula to find the area in square centimeters (cm²).
- You have a recipe that calls for 250 milliliters (mL) of water, but you only have a measuring cup in fluid ounces (fl oz). Convert 250 mL to fluid ounces, rounding to the nearest tenth.
- The speed of sound in air is approximately 343 meters per second (m/s). How many kilometers per hour is this?
- A car’s fuel efficiency is measured in miles per gallon (mpg). Convert a fuel efficiency of 30 mpg to liters per 100 kilometers (L/100 km).
- In a chemistry experiment, you need 0.025 moles of a substance. Calculate the mass of this substance in grams.
- The period (T) of a pendulum is given by the formula T = 2π√(L/g), where L is the length of the pendulum (m) and g is the acceleration due to gravity (m/s²). Determine the units of T.
- An object has a weight of 150 pounds. Calculate its mass in kilograms.
Answers for your reference:
- 54 km/h
- Dimensionally consistent (N = kg⋅m/s²)
- 64 in³
- A = πr², so A = π(r² * 10,000), where 1 m² = 10,000 cm²
- 8.5 fl oz
- Approximately 1235 km/h
- Approximately 7.8 L/100 km
- 1.5 grams
- The units of T are seconds (s).
- Approximately 68.18 kg
You can also download Dimensional Analysis Worksheet pdf from Boston University from below link:
Ch0-DimensionalAnalysis (bu.edu)
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