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What is the purpose of Dimensional Analysis in Physics?
To check the validity of formulae and equations by converting measurable quantities into their base units using SI Base units.
Why can ratios like radian, coefficient of friction, and coefficient of restitution be ignored in Dimensional Analysis?
Because they do not have units, as they are ratios.
What is the base unit of Area?
$m^2$
What is the base unit of Force?
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What is the dimension of Momentum?
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What is the dimension of Kinetic Energy?
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How do you validate an equation using Dimensional Analysis?
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What is the importance of leaving the left- and right-hand sides of an equation unchanged during Dimensional Analysis?
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What is Dimensional Analysis?
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The unit of time is the ______.
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The unit of length is the ______.
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The unit of mass is the ______.
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A radian has a unit.
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What cannot be included during dimensional analysis?
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What is the first step to conduct dimensional analysis?
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What must be true for addition or subtraction of units?
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What is the formula for Speed?
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What is the formula for Force?
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What is the formula for Energy/Work?
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What must be true for an equation to be valid in dimensional analysis?
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What does the formula t = 2π√(ml/F) represent?
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The unit of energy/work is the ______.
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The unit of power is the ______.
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Match the physical quantity with its unit:
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What is the formula for calculating Speed?
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Which unit is used to measure Force?
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The coefficients of friction and restitution are always included in dimensional analysis.
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What is the formula for Momentum?
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What does the acronym SI stand for in SI Base Units?
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The formula for Kinetic Energy (K.E) is K.E = ______ Mass × (velocity)². What is the value of the coefficient?
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What must be true for an equation to be dimensionally valid?
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Match the following concepts with their corresponding formulas:
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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.
Overview of Dimensional Analysis
- Dimensional analysis checks the consistency of equations and formulae using units (dimensions).
- Important in various fields for validating calculations and relationships.
SI Base Units
- Second (s) - Time
- Meter (m) - Length
- Kilogram (kg) - Mass
- Radian is a unitless ratio; unitless terms can be omitted in analysis.
Conducting Dimensional Analysis
- Replace variables with their respective units.
- Unit addition/subtraction must be from the same type, e.g., 4m + 5m = 9m.
- Different units can be multiplied/divided for simplification.
Common Physical Quantities and their Units
- Speed/Velocity: Distance / Time; unit: m/s
- Acceleration: Change in velocity / Time; unit: m/s²
- Force: Mass x Acceleration; unit: Newton (N) = kg·m/s²
- Momentum: Mass x Velocity; unit: kg·m/s
- Energy/Work: Potential Energy (P.E = Mass x Gravity x Height), Kinetic Energy (K.E = 0.5 x Mass x Velocity²); unit: Joule (J) = kg·m²/s²
- Power: Work / Time; unit: Watt (W) = kg·m²/s³
- Angular Velocity: Radians / Second; unit: rad/s
Validating Dimensional Equations
- Units on the Left Hand Side (LHS) must equal those on the Right Hand Side (RHS).
- Constants are often ignored in dimensional checks.
Worked Examples
- Various equations involving velocities (u, v), acceleration (g), distance (s), and time (t) can be validated dimensionally.
- Period of vibration for a piano wire relates to mass (m), length (l), and tension (F).
- Viscosity in fluids is analyzed through relationships involving force (F), area (A), speed (s), and distance (y).
Key Questions for Practice
- Validate dimensional correctness for formulas regarding velocity, distance, and time.
- Derive the correct period formula for a vibrating piano wire using dimensional analysis.
- Determine the unit of viscosity based on the relationship involving force, area, speed, and distance.
These notes encapsulate the essential concepts of dimensional analysis, key units, formula validation methods, and examples for understanding and application.
Overview of Dimensional Analysis
- Dimensional analysis checks the consistency of equations and formulae using units (dimensions).
- Important in various fields for validating calculations and relationships.
SI Base Units
- Second (s) - Time
- Meter (m) - Length
- Kilogram (kg) - Mass
- Radian is a unitless ratio; unitless terms can be omitted in analysis.
Conducting Dimensional Analysis
- Replace variables with their respective units.
- Unit addition/subtraction must be from the same type, e.g., 4m + 5m = 9m.
- Different units can be multiplied/divided for simplification.
Common Physical Quantities and their Units
- Speed/Velocity: Distance / Time; unit: m/s
- Acceleration: Change in velocity / Time; unit: m/s²
- Force: Mass x Acceleration; unit: Newton (N) = kg·m/s²
- Momentum: Mass x Velocity; unit: kg·m/s
- Energy/Work: Potential Energy (P.E = Mass x Gravity x Height), Kinetic Energy (K.E = 0.5 x Mass x Velocity²); unit: Joule (J) = kg·m²/s²
- Power: Work / Time; unit: Watt (W) = kg·m²/s³
- Angular Velocity: Radians / Second; unit: rad/s
Validating Dimensional Equations
- Units on the Left Hand Side (LHS) must equal those on the Right Hand Side (RHS).
- Constants are often ignored in dimensional checks.
Worked Examples
- Various equations involving velocities (u, v), acceleration (g), distance (s), and time (t) can be validated dimensionally.
- Period of vibration for a piano wire relates to mass (m), length (l), and tension (F).
- Viscosity in fluids is analyzed through relationships involving force (F), area (A), speed (s), and distance (y).
Key Questions for Practice
- Validate dimensional correctness for formulas regarding velocity, distance, and time.
- Derive the correct period formula for a vibrating piano wire using dimensional analysis.
- Determine the unit of viscosity based on the relationship involving force, area, speed, and distance.
These notes encapsulate the essential concepts of dimensional analysis, key units, formula validation methods, and examples for understanding and application.
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Test your understanding of dimensional analysis, a method to check formulae and equations using units and base units in physics.
