Question

A large cylinder is filled with equal volumes of two immiscible fluids. A balloon is submerged in the first fluid; the gauge pressure in the balloon at the deepest point in the first fluid is found to be 3 atm. Next, the balloon is lowered all the way to the bottom of the second fluid, where the hydrostatic pressure in the balloon reads 8 atm. What is the ratio of the gauge pressure accounted for by the first fluid to the gauge pressure accounted for by the second fluid? A. 1: 3 B. 3: 4 C. 3: 5 D. 3: 8

Short answer

C. 3:5

Step-by-step solution

Understand the problem

We need to find the ratio of the gauge pressure attributed to the first fluid to the gauge pressure attributed to the second fluid. We know the gauge pressures at two different points in the fluids.

Gauge pressure at the deepest point in the first fluid

The gauge pressure in the balloon at the deepest point in the first fluid is given as 3 atm. This is solely due to the first fluid, as it is stated that it is submerged only in the first fluid at this point.

Total pressure at the bottom of the second fluid

When the balloon is lowered to the bottom of the second fluid, the total gauge pressure is 8 atm. This pressure includes the contribution from both the first and second fluids.

Calculate gauge pressure contribution from the second fluid

The gauge pressure due to the second fluid can be computed by subtracting the gauge pressure of the first fluid from the total gauge pressure at the bottom of the second fluid:\[\text{Gauge pressure from second fluid} = 8 \text{ atm} - 3 \text{ atm} = 5 \text{ atm}\]

Find the ratio of gauge pressures

The gauge pressure from the first fluid is 3 atm and from the second fluid is 5 atm. The ratio is given by:\[\frac{\text{Gauge pressure from first fluid}}{\text{Gauge pressure from second fluid}} = \frac{3 \text{ atm}}{5 \text{ atm}} = \frac{3}{5}\]

Final answer

The ratio of the gauge pressure from the first fluid to the gauge pressure from the second fluid is 3:5 (option C).

Key concepts

gauge pressure

Gauge pressure is the pressure of a fluid measured relative to the ambient atmospheric pressure. It is different from absolute pressure, which includes the atmospheric pressure.
For example, if you inflate a balloon, the pressure inside is higher than the atmospheric pressure outside. The difference between these pressures is the gauge pressure.
In the given exercise, the gauge pressure is influenced by the depth and density of the fluid in which the balloon is submerged. At different depths within immiscible fluids, the gauge pressures vary due to the varying contributions from each fluid layer.

hydrostatic pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium at any given point due to the force of gravity. This concept helps in understanding how pressure varies with depth.
The formula for hydrostatic pressure is given by: \[ P = \rho gh \] where:
\( P \) = hydrostatic pressure
\( \rho \) = density of the fluid
\( g \) = acceleration due to gravity
\( h \) = height (depth) of the fluid column
In this exercise, the balloon submerged at various depths in two immiscible fluids experiences different hydrostatic pressures. This difference helps determine how gauge pressures are attributed to each fluid.

ratio calculation

Ratio calculation is crucial for comparing different quantities. It shows the relative size of two numbers and is often expressed as 'a to b' or \( a : b \).
In the exercise, we calculated the ratio of the gauge pressure from the first fluid to that from the second fluid. This was done by first identifying the individual contributions of each fluid to the total gauge pressure.
Initially, the balloon was submerged, giving a gauge pressure of 3 atm due to the first fluid. Then, it was lowered to the bottom of the second fluid with a total gauge pressure of 8 atm. Subtracting 3 atm (contributed by the first fluid) from 8 atm gave us 5 atm, which is the pressure due to the second fluid.
Therefore, the ratio is: \[ \frac{3 \text{ atm}}{5 \text{ atm}} = 3:5 \].

fluids in physics

Fluids in physics are substances that can flow and take the shape of their container. They include both liquids and gases. Several key principles are used to describe fluid behavior:

• Density: Mass per unit volume, often denoted by \( \rho \). Different fluids have different densities.
• Pressure: Force applied perpendicular to the surface of an object per unit area.
• Buoyancy: The upward force exerted by a fluid, opposing the weight of an object submerged in it.

In the exercise provided, we deal with immiscible fluids, meaning they do not mix and have distinct layers. Each fluid exerts pressure based on its density and the depth of submersion, contributing differently to the gauge pressure read by the balloon. Understanding these concepts helps in solving fluid-related problems in physics effectively.