Podcast
Questions and Answers
What is the purpose of dimensional analysis?
What is the purpose of dimensional analysis?
What is the dimensions of a radian?
What is the dimensions of a radian?
What should be done with constants and unitless terms in dimensional analysis?
What should be done with constants and unitless terms in dimensional analysis?
What is the result of conducting dimensional analysis on the formula for the area of a rectangle?
What is the result of conducting dimensional analysis on the formula for the area of a rectangle?
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What is the base unit of force?
What is the base unit of force?
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What is the purpose of leaving the left- and right-hand side of an equation unchanged in dimensional analysis?
What is the purpose of leaving the left- and right-hand side of an equation unchanged in dimensional analysis?
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What should be done with trigonometric ratios and logarithms in dimensional analysis?
What should be done with trigonometric ratios and logarithms in dimensional analysis?
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What is the result of conducting dimensional analysis on the formula for momentum?
What is the result of conducting dimensional analysis on the formula for momentum?
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Study Notes
Dimensional Analysis
- Dimensional analysis is a method to check formulae and equations using units (dimensions).
- It involves converting measurable quantities into their base units using SI Base units.
Ignoring Units
- A radian is a ratio, so it doesn't have a unit and can be ignored in dimensional analysis.
- Ratios such as coefficients of friction (μ) and restitution (e) can also be ignored.
- Trigonometric ratios (like sin, cos, or tan) and logarithms (log) can be ignored in dimensional analysis.
Conducting Dimensional Analysis
- Replace variables with their units, ignoring constants and unitless terms.
- Only add or subtract the same units.
- Multiply and divide different units to simplify the expression or equation.
Example: Area of a Rectangle
- The formula for area of a rectangle is length × width.
- The base unit of length and width is m, so the unit of area is m².
Base Units of Force
- The formula for force is F = ma, where mass has a base unit of kg and acceleration has a base unit of ms⁻².
- The base unit of force is kg × ms⁻² = kgm/s².
Exercises
- Find the dimensions of:
- Momentum = Mass × Velocity
- Kinetic Energy = ½ × mass × velocity²
- Potential Energy = mass × gravity × height
- Electric Potential = Current × resistance
Validating Equations
- Dimensional analysis can be used to validate equations.
- A formula is dimensionally valid if the units of the left-hand side (LHS) are equal to the units of the right-hand side (RHS).
Worked Example: v = u - gt
- The formula v = u - gt is dimensionally valid because:
- LHS = v = ms⁻¹
- RHS = ms⁻¹ - ms⁻² × s = ms⁻¹ - ms⁻¹ = ms⁻¹
- RHS = LHS
Finding the Formula for Maximum Range
- The formula for maximum range R of a projectile on a horizontal plane involves initial speed u and acceleration due to gravity g.
- The formula can be found using dimensional analysis, and it does not include dimensionless constants.
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Description
This quiz covers dimensional analysis, a method for checking formulae and equations using units and SI base units. It also explains how to handle ratios and coefficients in dimensional analysis.
