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Questions and Answers
What are the dimensions and SI units of pressure?
M L-1 T-2
What are the dimensions and SI units of work?
M L^2 T-2
What are the dimensions and SI units of kinetic energy?
M L^2 T-2
Is the equation [P] = M L-1 T-2 correct?
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Derive the equation for the speed of sound in a gas.
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What is the formula for potential energy?
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Is the equation P = ρgh correct?
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How do you convert units of energy from Joules to ergs?
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Convert the unit of force from Newton to Dyne.
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What is the formula for the force of viscosity?
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What are the dimensions of the coefficient of viscosity (η)?
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What are the dimensions of Young's modulus?
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Identify the dimensions of area.
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Identify the dimensions of volume.
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Identify the dimensions of velocity.
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Identify the dimensions of density.
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Identify the dimensions of acceleration.
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Identify the dimensions of force.
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Identify the dimensions of frequency.
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Identify the dimensions of momentum.
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What is the relationship for the period of a simple pendulum?
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What is the formula for centrifugal force?
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What are the dimensions of energy from the equation K.E = mc^2?
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Convert the unit of pressure from Pascal to Bar.
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Find the dimensions of Planck's constant from the equation E = hν.
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What are the units and dimensions of bulk modulus?
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Study Notes
Dimensions and Units
-
Pressure
- Dimensions: ML⁻¹T⁻²
- SI Unit: Pascal (Pa)
-
Work
- Dimensions: ML²T⁻²
- SI Unit: Joule (J)
-
Kinetic Energy
- Dimensions: ML²T⁻²
- SI Unit: Joule (J)
Checking Equations
-
Equation 1: P = ρgh (Pressure, density, gravity, height)
-
Dimensions:
- Left-hand side (L.H.S.): [P] = ML⁻¹T⁻²
- Right-hand side (R.H.S.): [ρ][g][h] = (ML⁻³) (LT⁻²) (L) = ML⁻¹T⁻²
- Conclusion: The equation is dimensionally correct
-
Dimensions:
Deriving Equations
-
Speed of Sound in a Gas:
- Derivation: v = √(K * P/ρ), where K is a constant
-
Dimensions:
- [v] = LT⁻¹
- [K] = dimensionless
- [P] = ML⁻¹T⁻²
- [ρ] = ML⁻³
- Equation: v = √(K * P/ρ)
- SI Unit: meters per second (m/s)
Finding Dimensions and Units
-
Retarding Force on a Sphere:
- Equation: F = Kvr²ρ
-
Dimensions:
- [F] = ML⁻¹T⁻²
- [K] = dimensionless
- [v] = LT⁻¹
- [r] = L
- [ρ] = ML⁻³
- Value of exponents: x = 2, y = 1, z = 2
-
Potential Energy:
- Dimensions: [P.E] = [m][g][h] = (M)(LT⁻²)(L) = ML²T⁻²
- SI Unit: Joule (J)
-
Force of Viscosity:
- Equation: F = 6πηvr
-
Dimensions of η:
- [F] = ML⁻¹T⁻²
- [v] = LT⁻¹
- [r] = L
- [η] = ML⁻¹T⁻¹ (coefficient of viscosity)
-
Young's Modulus:
- Equation: Y = (Stress/Strain) = (F/A)/(ΔL/L)
- Dimensions: [Y] = ML⁻¹T⁻²
- SI unit: Pascal (Pa)
Converting Units
-
Energy:
- Conversion: 1 Joule = 1 kg⋅m²⋅s⁻² = 10⁷ g⋅cm²⋅s⁻² = 10⁷ ergs
-
Force:
- Conversion: 1 Newton = 1 kg⋅m⋅s⁻² = 10⁵ g⋅cm⋅s⁻² = 10⁵ dynes
Other Relevant Concepts
-
Area:
- Dimensions: L²
- SI Unit: square meter (m²)
-
Volume:
- Dimensions: L³
- SI Unit: cubic meter (m³)
-
Velocity:
- Dimensions: LT⁻¹
- SI Unit: meters per second (m/s)
-
Density:
- Dimensions: ML⁻³
- SI Unit: kilograms per cubic meter (kg/m³)
-
Acceleration:
- Dimensions: LT⁻²
- SI Unit: meters per second squared (m/s²)
-
Frequency
- Dimensions: T⁻¹
- SI Unit: Hertz (Hz)
-
Momentum
- Dimensions: MLT⁻¹
- SI Unit: kilogram meter per second (kg⋅m/s)
Dimensional Analysis
-
Checking Equation Correctness:
- Determine dimensions of each term in the equation.
- Check if the dimensions on the left-hand side (L.H.S) are equal to the dimensions on the right-hand side (R.H.S).
- If they are equal, the equation is dimensionally correct.
-
Deriving Equations:
- Determine the dimensions of the quantities involved.
- Use the principle of dimensional homogeneity – the dimensions on both sides of an equation must be the same.
- This allows you to find the relationships between the quantities.
Additional Information
-
Planck's Constant (h):
- Equation: E = hν (Energy, frequency)
- Dimensions: ML²T⁻¹
- SI Unit: Joule-second (J⋅s)
-
Bulk Modulus (B):
- Equation: B = (ΔP)/(ΔV/V) (Pressure change, volume change, original volume)
- Dimensions: ML⁻¹T⁻²
- SI Unit: Pascal (Pa)
-
Simple Pendulum:
- Time Period (T): T = 2π√(L/g) (length, acceleration due to gravity)
- Dimensions: [T] = T
-
Centrifugal Force (F):
- Equation: F = mv²/r (mass, velocity, radius)
- Dimensions: [F] = ML⁻¹T⁻²
-
Converting Units:
- Use conversion factors between different systems of units.
-
Key Notes:
- Dimensional analysis helps in understanding the relationships between physical quantities.
- It is a powerful tool for checking the consistency of equations and for deriving new equations.
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Description
Test your understanding of dimensions and units in physics, focusing on pressure, work, and kinetic energy. The quiz includes derivations of equations and evaluations of their dimensional correctness. Challenge yourself with questions on the dimensional analysis of various physical quantities.
