3.1 Dimensional Analysis

• What is the fundamental dimension for mass?

  M
• In mechanics, dimensions refer to the fundamental physical quantities that describe any other quantity
• What is the fundamental dimension for time?

  T
• What are fundamental dimensions?

  Independent physical quantities
• The dimension of acceleration is LT⁻².


  True
• Match the quantity with its dimension:

  Area ↔️ L²

  Volume ↔️ L³

  Speed ↔️ LT⁻¹

  Density ↔️ ML⁻³
• What are the dimensions of force in the equation $F =$$ma$?


  MLT⁻²
• The principle of homogeneity of dimensions ensures that only physical quantities of the same kind can be added, subtracted, or equated.


  True
• Match the mechanical quantity with its derived dimension:

  Area ↔️ L²

  Volume ↔️ L³

  Speed ↔️ LT⁻¹

  Density ↔️ ML⁻³
• The dimensions of mass are represented by M.


  True
• Arrange the following mechanical quantities from simplest to most complex derived dimensions:

  1️⃣ Area (L²)

  2️⃣ Volume (L³)

  3️⃣ Speed (LT⁻¹)

  4️⃣ Density (ML⁻³)

  5️⃣ Force (MLT⁻²)
• In the equation $F =$$ma$, both sides have dimensions of MLT⁻²
• What are the dimensions of force (F) in the equation $F =$$ma$?


  MLT^{-2}</latex>
• For an equation to be valid, all terms must have the same dimensions
• Match the quantity with its dimension:

  Volume ↔️ $L^{3}$

  Speed ↔️ $LT^{ - 1}$

  Density ↔️ $ML^{ - 3}$

  Force ↔️ $MLT^{ - 2}$
• Steps to solve a problem using dimensional analysis:

  1️⃣ Identify the quantities in the equation

  2️⃣ Determine the dimensions of each quantity

  3️⃣ Ensure all terms have the same dimensions

  4️⃣ Verify the equation's validity
• What are dimensions in mechanics used to describe?

  Physical quantities
• What is the fundamental dimension for length?

  L
• Match the quantity with its dimension:

  Area ↔️ L²

  Volume ↔️ L³

  Speed ↔️ LT⁻¹

  Density ↔️ ML⁻³
• Fundamental dimensions include length, mass, time, and temperature
• What are the three fundamental dimensions in mechanics?

  Length, mass, time
• The dimension of force is derived using the formula $F =$$ma$, resulting in MLT⁻²
• Dimensional analysis ensures that all terms in an equation have the same dimensions.


  True
• By applying dimensional analysis, you can verify the validity of an equation or even derive an equation by ensuring the dimensions on both sides
• The derived dimension of speed is LT⁻¹.


  True
• What are the three fundamental dimensions in mechanics?

  Length, Mass, Time
• The principle of homogeneity of dimensions states that all terms in a valid physical equation must have the same dimensions
• In the equation $F =$$ma$, both sides have the same dimensions.


  True
• The principle of homogeneity of dimensions states that all terms in a valid physical equation must have the same dimensions
• Dimensional analysis is used to check the validity of equations based on the principle of homogeneity of dimensions


  True
• What are the fundamental dimensions in mechanics?

  Length, mass, and time
• The dimension of force, MLT⁻², is derived from the fundamental dimensions of mass, length, and time


  True
• What condition must be satisfied for addition or subtraction in dimensional analysis?

  Terms must have same dimensions
• The dimension of force is MLT⁻².


  True
• What are derived dimensions?

  Combinations of fundamental dimensions
• Order the quantities from their simplest to most complex dimensions:

  1️⃣ Length

  2️⃣ Area

  3️⃣ Volume

  4️⃣ Speed
• The principle of homogeneity of dimensions states that every term in a valid physical equation must have the same dimensions.


  True
• What must all terms in a valid equation have according to the principle of homogeneity of dimensions?

  Same dimensions
• What are the three fundamental dimensions in mechanics?

  Length, Mass, Time
• The derived dimension of force is MLT⁻²