Dimensional Analysis and Formulae - UNSOLVED PRACTICE SET
Chapter: Physical World and Measurement | Topic: Dimensional Analysis and Formulae
DIMENSIONAL ANALYSIS AND FORMULAE - UNSOLVED PRACTICE SET
Topic: Dimensional Analysis and Formulae
Multiple Choice Questions
Q1. The dimensional formula for force is:
- [M L T⁻²]
- [M L² T⁻²]
- [M L T⁻¹]
- [M² L T⁻²]
Q2. Which of the following is a dimensionless quantity?
- Velocity
- Acceleration
- Strain
- Force
Q3. The dimensional formula for gravitational constant G is:
- [M⁻¹ L³ T⁻²]
- [M L³ T⁻²]
- [M⁻¹ L² T⁻²]
- [M L² T⁻²]
Q4. If the dimensions of a physical quantity are [M L² T⁻²], the quantity could be:
- Force
- Power
- Work or Energy
- Momentum
Q5. The principle of homogeneity of dimensions states that:
- Only quantities with the same dimensions can be added or subtracted
- All physical quantities must have the same dimensions
- Dimensions can be changed by changing units
- Derived quantities cannot have dimensions
Q6. Dimensional analysis can be used to:
- Derive the exact numerical constants in a formula
- Check the dimensional consistency of a physical equation
- Determine the exact value of a physical quantity
- Replace the need for experiments
Short Answer Questions
Q7. State the principle of homogeneity of dimensions. Explain its significance in checking the correctness of physical equations.
Q8. Write the dimensional formulae for:
(a) Pressure
(b) Power
(c) Frequency
Q9. Derive the dimensional formula for the coefficient of viscosity (η), given that the viscous force F = ηA(dv/dx), where A is area, v is velocity, and x is distance.
Q10. A student writes a formula for the time period of a simple pendulum as T = k√(m/g), where m is mass and g is acceleration due to gravity. Use dimensional analysis to check if this formula is dimensionally correct. If not, correct it.
Q11. What are the limitations of dimensional analysis? List any two limitations.
Q12. The velocity of sound in a gas is given by v = √(γP/ρ), where P is pressure and ρ is density. Check the dimensional consistency of this formula, given that γ is dimensionless.
Long Answer Questions
Q13. Explain the method of dimensions. Discuss how dimensional analysis can be used to:
(i) Check the dimensional correctness of a formula
(ii) Derive relationships between physical quantities
(iii) Convert units from one system to another
Illustrate each with a suitable example.
Q14. Using dimensional analysis, derive the expression for the time period of a simple pendulum. Assume that the time period T depends on the length l, mass m, and acceleration due to gravity g. Explain why the mass does not appear in the final formula.
Q15. A student proposes that the velocity v of a wave on a stretched string depends on the tension T in the string, the mass per unit length μ, and the wavelength λ. Using dimensional analysis, derive the relationship between these quantities. Discuss what the dimensional method cannot tell you about this relationship.
Application-Based Problems
Q16. Check the dimensional correctness of the following equations:
(a) s = ut + ½at² (equation of motion)
(b) F = ma (Newton's second law)
(c) E = mc² (Einstein's mass-energy relation)
(d) v² = u² + 2as
(e) T = 2π√(l/g) (time period of simple pendulum)
For each equation, show the dimensional analysis step-by-step.
Q17. The escape velocity from a planet is given by v = √(2GM/R), where G is the gravitational constant, M is the mass of the planet, and R is its radius.
(a) Check the dimensional consistency of this formula.
(b) If the radius of Earth is 6.4 × 10⁶ m and its mass is 6 × 10²⁴ kg, calculate the escape velocity from Earth. (G = 6.67 × 10⁻¹¹ Nm²/kg²)
(c) The student wants to express this velocity in km/h. Convert your answer.
Q18. In a school project, a student is trying to find the relationship between the frequency f of a stretched string and the tension T, length l, and mass per unit length μ. The student hypothesises that f = kTᵃlᵇμᶜ, where k is a dimensionless constant.
(a) Use dimensional analysis to find the values of a, b, and c.
(b) Write the final formula for frequency.
(c) What does this tell you about how frequency changes when:
(i) Tension is doubled?
(ii) Length is halved?
(iii) Mass per unit length is tripled?