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DIMENSIONAL ANALYSIS PROBLEMS WITH ANSWERS

Dimensional analysis is a powerful tool used in physics, engineering, and other scientific disciplines to check the consistency of equations and solve problems involving physical quantities. By analyzing the dimensions of the quantities involved in a problem, one can derive relationships between them and simplify complex equations. In this article, we will explore some common dimensional analysis problems and provide answers to help you understand the concept better.

Problem 1: Velocity Calculation

Suppose you are given the equation for velocity as:

Velocity (v) = Distance (d) / Time (t)

Using dimensional analysis, determine the dimensions of velocity in terms of distance and time.

Solution:

Velocity (v) = Distance (d) / Time (t)
Dimensions of Distance (d) = [L]
Dimensions of Time (t) = [T]
Therefore, dimensions of Velocity (v) = [L] / [T] = [LT^-1]

So, the dimensions of velocity are length per time, represented as [LT^-1].

Problem 2: Force Calculation

Consider the equation for force as:

Force (F) = Mass (m) x Acceleration (a)

Using dimensional analysis, find the dimensions of force in terms of mass and acceleration.

Solution:

Force (F) = Mass (m) x Acceleration (a)
Dimensions of Mass (m) = [M]
Dimensions of Acceleration (a) = [LT^-2]
Therefore, dimensions of Force (F) = [M] x [LT^-2] = [MLT^-2]

Thus, the dimensions of force are mass times length per time squared, denoted as [MLT^-2].

Problem 3: Work Calculation

Given the equation for work as:

Work (W) = Force (F) x Distance (d)

Using dimensional analysis, determine the dimensions of work in terms of force and distance.

Solution:

Work (W) = Force (F) x Distance (d)
Dimensions of Force (F) = [MLT^-2]
Dimensions of Distance (d) = [L]
Therefore, dimensions of Work (W) = [MLT^-2] x [L] = [ML²T^-2]

Hence, the dimensions of work are mass times length squared per time squared, represented as [ML²T^-2].

Summary

Dimensional analysis is a valuable technique for verifying equations and solving problems in physics and engineering. By understanding the dimensions of physical quantities and using them to derive relationships, one can simplify complex calculations and ensure the consistency of equations. The examples provided in this article demonstrate how dimensional analysis can be applied to various scenarios, helping to enhance problem-solving skills and deepen understanding of fundamental concepts in science and engineering.

For more information on dimensional analysis and its applications, you can visit Khan Academy’s Dimensional Analysis Guide.