Questions and Answers
What is the purpose of dimensional analysis in physics?
To check the validity of formulas and equations by checking the units
What are the base units of force in the International System of Units (SI)?
kg m/s^2 or kg ms^(-2)
Why are radians ignored in dimensional analysis?
Because radians are a ratio and do not have a unit
What is the dimensional analysis of the area of a rectangle?
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What is the dimensional analysis of momentum?
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How do you validate an equation using dimensional analysis?
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What is the dimensional analysis of kinetic energy?
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Why are trigonometric ratios and logarithms ignored in dimensional analysis?
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What is the dimensional analysis of potential energy?
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What is the dimensional analysis of electric potential?
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Is the formula v=u+gt dimensionally valid?
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What is the dimensional analysis of the coefficient of friction?
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Can you add or subtract different units in dimensional analysis?
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What is the dimensional analysis of angular velocity?
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Is the formula F=ma dimensionally valid?
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What is the dimensional analysis of torque?
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Can you ignore constants in dimensional analysis?
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Is the formula E=mc^2 dimensionally valid?
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Study Notes
Dimensional Analysis
- Dimensional analysis is a method to check the validity of formulas and equations using units (dimensions).
- In dimensional analysis, measurable quantities are converted into their base units using SI base units.
Dimensionless Quantities
- A radian is a ratio and has no unit, so it is ignored in dimensional analysis.
- Coefficients of friction (μ) and restitution (e) are also ratios and can be ignored.
- Trigonometric ratios (like sin, cos, or tan) and logarithms (log) can be ignored in dimensional analysis.
How to Conduct 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.
Examples of Dimensional Analysis
- Area of a rectangle: A = (m)(m) = m^2
- Force: F = ma, where F = (kg)(ms^(-2)) = kg m/s^2
Exercises and Validating Equations
- Conduct dimensional analysis to find the dimensions of momentum, kinetic energy, potential energy, and electric potential.
- Use dimensional analysis to validate formulas, such as v = u - gt.
Worked Example: Validating Equations
- v = u - gt: LHS = v = m/s, RHS = m/s - m/s^2(s) = m/s - m/s = m/s, so RHS = LHS and the formula is dimensionally valid.
Finding Formulas using Dimensional Analysis
- Find the formula for maximum range R of a projectile on a horizontal plane, which involves the initial speed u and acceleration due to gravity g, with no dimensionless constants.
Dimensional Analysis
- Dimensional analysis is a method to check the validity of formulas and equations using units (dimensions).
- In dimensional analysis, measurable quantities are converted into their base units using SI base units.
Dimensionless Quantities
- A radian is a ratio and has no unit, so it is ignored in dimensional analysis.
- Coefficients of friction (μ) and restitution (e) are also ratios and can be ignored.
- Trigonometric ratios (like sin, cos, or tan) and logarithms (log) can be ignored in dimensional analysis.
How to Conduct 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.
Examples of Dimensional Analysis
- Area of a rectangle: A = (m)(m) = m^2
- Force: F = ma, where F = (kg)(ms^(-2)) = kg m/s^2
Exercises and Validating Equations
- Conduct dimensional analysis to find the dimensions of momentum, kinetic energy, potential energy, and electric potential.
- Use dimensional analysis to validate formulas, such as v = u - gt.
Worked Example: Validating Equations
- v = u - gt: LHS = v = m/s, RHS = m/s - m/s^2(s) = m/s - m/s = m/s, so RHS = LHS and the formula is dimensionally valid.
Finding Formulas using Dimensional Analysis
- Find the formula for maximum range R of a projectile on a horizontal plane, which involves the initial speed u and acceleration due to gravity g, with no dimensionless constants.
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Description
Test your understanding of dimensional analysis, a method to check formulae and equations using units and SI base units.
