Equation of Continuity - UNSOLVED PRACTICE SET

Q7. State the equation of continuity. Derive it for an incompressible fluid flowing through a pipe of varying cross-section.

Q8. Water flows through a pipe of radius 2 cm with velocity 3 m/s. Calculate the volume flow rate.

Q9. In a pipe, the velocity of water is 2 m/s where the diameter is 4 cm. What is the velocity where the diameter is 2 cm?

Q10. In your school, a student covers half the opening of a garden hose with her thumb. The water sprays out much faster. Explain using the equation of continuity.

Q11. Why does a river flow faster in a narrow gorge than in a wide plain?

Q12. Blood flows through an artery of radius 3 mm at 30 cm/s. If the artery narrows to a radius of 2 mm, calculate the new velocity of blood flow.

Q13. Derive the equation of continuity for fluid flow. Explain its physical basis (conservation of mass). Discuss its applications in:

(i) Pipe flow

(ii) Blood circulation

(iii) River flow

(iv) Jet engines

Show that for compressible fluids, the equation becomes ρ₁A₁v₁ = ρ₂A₂v₂.

Q14. Water flows through a horizontal pipe system as follows:

Section 1: Diameter 6 cm, velocity 2 m/s

Section 2: Diameter 4 cm

Section 3: Diameter 3 cm

(a) Calculate the velocity in sections 2 and 3.

(b) Calculate the volume flow rate through the pipe.

(c) Calculate the mass flow rate. (Density of water = 1000 kg/m³)

(d) If section 3 is vertical and water rises to a height of 5 m, calculate the velocity at the top.

(e) Verify that the equation of continuity holds throughout the system.

Q15. Analyse the equation of continuity in the human circulatory system:

(a) The aorta has a radius of about 1 cm and blood flows at 30 cm/s. Calculate the volume flow rate.

(b) The total cross-sectional area of all capillaries is about 2000 cm². Calculate the velocity of blood in capillaries.

(c) Explain why blood flows so slowly in capillaries despite the heart pumping vigorously.

(d) Discuss the significance of slow capillary flow for nutrient and gas exchange.

(e) What happens if an artery becomes narrowed (stenosis)?

Q16. A horizontal pipe has three sections with diameters 8 cm, 4 cm, and 2 cm. Water enters at 1 m/s in the widest section.

(a) Calculate the velocity in each section.

(b) Calculate the volume flow rate.

(c) If the pressure in the widest section is 2 × 10⁵ Pa, use Bernoulli's equation to find the pressure in the narrowest section. (Assume the pipe is horizontal)

(d) Calculate the pressure difference between the widest and narrowest sections.

(e) Discuss what would happen if the narrowest section were made even narrower.

Q17. In a school project, students build a venturi tube (a pipe with a constricted middle section) to demonstrate fluid flow.

(a) If the inlet diameter is 4 cm, the throat diameter is 2 cm, and the inlet velocity is 0.5 m/s, calculate the velocity at the throat.

(b) If water is used, calculate the pressure difference between inlet and throat. (Inlet pressure = 1.5 × 10⁵ Pa)

(c) Explain why the pressure drops in the constricted section.

(d) How could this device be used to measure flow rate?

(e) Discuss why this principle is important in carburettors and atomisers.

Q18. A fire hose has a diameter of 6 cm and delivers water at 4 m/s. The nozzle at the end has a diameter of 2 cm.

(a) Calculate the velocity of water emerging from the nozzle.

(b) Calculate the volume flow rate.

(c) Calculate the maximum horizontal distance the water jet can reach if fired from 1 m height.

(d) If the nozzle is pointed vertically upward, calculate the maximum height reached by the water.

(e) Discuss why fire fighters hold the hose firmly when water is flowing.