Question

Someone claims that the absolute pressure in a liquid of constant density doubles when the depth is doubled. Do you agree? Explain.

Short answer

Answer: Yes, the absolute pressure in a liquid of constant density doubles when the depth is doubled.

Step-by-step solution

Understanding the Hydrostatic Pressure Formula

The hydrostatic pressure formula is used to calculate the pressure at a certain depth within a liquid. The formula is given by: P = P0 + ρgh Where: P is the pressure at the depth h, P0 is the pressure at the surface (i.e., atmospheric pressure), ρ is the density of the liquid (in kg/m³), g is the acceleration due to gravity (9.81 m/s²), and h is the depth (in meters). Since the density of the liquid remains constant in this problem, we will analyze the relationship between pressure and depth.

Find the Pressure at Initial Depth h

Let's first find the pressure at the initial depth h using the hydrostatic pressure formula: P1 = P0 + ρgh Note that we are only considering the absolute pressure, so P0 is 0. P1 = 0 + ρgh P1 = ρgh

Find the Pressure at the New Depth 2h (Double the Initial Depth)

Now let's find the pressure at the new depth, which is double the initial depth, 2h: P2 = P0 + ρg(2h) Again, we are only considering the absolute pressure, so P0 is 0. P2 = 0 + ρg(2h) P2 = 2ρgh

Analyzing the Relationship Between the Pressures at Depth h and 2h

Now that we have expressions for the pressure at both depths, let's analyze the relationship between the two. Let's divide the pressure at the new depth (P2) by the pressure at the initial depth (P1): (P2)/(P1) = (2ρgh)/(ρgh) Simplifying the equation, we find: (P2)/(P1) = 2 This relationship shows that when the depth is doubled, the pressure in the liquid also doubles.

Conclusion

Based on our analysis of the hydrostatic pressure formula, the claim that the absolute pressure in a liquid of constant density doubles when the depth is doubled is indeed valid.