A horizontal pipe is fitted with a pressure gauge. When a downstream valve is completely closed, the gauge reads 6.0 10^4 Pa . When the valve is fully opened, an unknown liquid flows steadily through the pipe at a velocity of 10 m s ⁻¹ and the gauge reading drops to 2.0 10^4 Pa . The density of the unknown liquid is _________ kg m ⁻³ .

Step-by-step solution

According to Bernoulli's principle for a horizontal pipe, the total pressure head remains constant. When the valve is closed, the velocity of the liquid is zero. Thus, the initial static pressure is P₁ = 6.0 10⁴ Pa . When the valve is opened, the liquid flows with a velocity v = 10 m s ⁻¹ and the pressure drops to P₂ = 2.0 10⁴ Pa . Applying Bernoulli's equation: P₁ = P₂ + 1 2 v² P₁ - P₂ = 1 2 v² Substitute the given values: 6.0 10⁴ - 2.0 10⁴ = 1 2 (10)² 4.0 10⁴ = 50 = 40000 50 = 800 kg m ⁻³ Answer: 800