Density and Fluid Mechanics

The buoyant force on an object submerged in a fluid is calculated using Archimedes' principle, which states that the buoyant force equals the weight of the fluid displaced by the object. In this case, the displaced volume is 0.020 m³, and the density of water is 1000 kg/m³. The weight of the displaced water can be calculated using the formula: buoyant force = volume × density × gravitational acceleration. Thus, the buoyant force is 0.020 m³ × 1000 kg/m³ × 9.8 m/s² = 196 N.

Specific gravity is the ratio of the density of a substance to the density of water. To find the density of the substance, multiply the specific gravity (0.80) by the density of water (1000 kg/m³). This calculation yields 0.80 × 1000 kg/m³ = 800 kg/m³. Thus, the density of the substance is 800 kg/m³.

In a Venturi tube, fluid dynamics principles dictate that as fluid flows through a constricted section, its velocity increases due to the conservation of mass. According to Bernoulli's principle, an increase in fluid speed results in a decrease in pressure. Therefore, in the narrow section of the tube, the fluid exhibits higher speed and lower pressure compared to the wider sections. This relationship is fundamental in various applications, such as in carburetors and atomizers, where controlling fluid flow is essential.

A hydrometer measures the density of a liquid based on Archimedes' Principle, which states that an object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced. The hydrometer consists of a sealed glass tube with a weighted bottom that floats upright in the liquid. The level to which it sinks indicates the liquid's density; denser liquids will support the hydrometer more, causing it to float higher, while less dense liquids will allow it to sink lower. This relationship allows for precise density measurement.

According to the principle of conservation of mass, the flow rate must remain constant in a closed system. When a pipe narrows, the cross-sectional area decreases, leading to an increase in velocity. Using the equation of continuity, \(A_1V_1 = A_2V_2\), where \(A\) is the area and \(V\) is the velocity, we can calculate the new speed. Given the initial area (6.0 cm²) and speed (2.0 m/s), and the narrowed area (2.0 cm²), solving for the new speed yields 6.0 m/s, demonstrating the relationship between area and velocity in fluid dynamics.

Bernoulli's Principle states that in a fluid flowing through a pipe, an increase in fluid speed occurs simultaneously with a decrease in static pressure, provided the height remains constant. This relationship is due to the conservation of energy within the fluid. As the fluid accelerates, its kinetic energy increases, causing the potential energy in the form of static pressure to decrease. Thus, at the same height, higher fluid speed results in lower static pressure.

According to the continuity equation, which states that the mass flow rate must remain constant in a closed system, when a fluid flows through a narrower section of a pipe, the cross-sectional area decreases. To maintain the same mass flow rate, the fluid must speed up as it enters the narrower section. This increase in velocity compensates for the reduced area, ensuring that the same volume of fluid passes through per unit time, leading to an increase in the speed of fluid flow in the narrower section.

The continuity equation for fluid flow, expressed as A₁V₁ = A₂V₂, represents the principle of conservation of mass in a fluid system. Here, A₁ and A₂ are the cross-sectional areas of two different points in a flow, while V₁ and V₂ are the corresponding fluid velocities at those points. This equation indicates that the mass flow rate must remain constant throughout the flow, meaning that if the area decreases, the velocity must increase, and vice versa, ensuring that the fluid is incompressible and mass is conserved.

An object floats when the weight of the water displaced equals the weight of the object. If 75% of its volume is submerged, it displaces an amount of water equal to 75% of its own weight. The density of water is 1000 kg/m³, so the weight of the displaced water can be calculated as 0.75 times the volume of the object times the density of water. Setting this equal to the weight of the object (density times volume) leads to the conclusion that the density of the object must be 750 kg/m³.

A floating object at rest experiences a balance of forces, where the buoyant force acting upward is equal to the weight of the object acting downward. This equilibrium allows the object to float without sinking or rising. The buoyant force is determined by the volume of fluid displaced, which corresponds to the weight of the object when it is floating. Thus, for an object to remain at rest on the surface of the fluid, its buoyant force must equal its weight.

Density is defined as the mass of an object divided by its volume. The formula ρ = m/v represents this relationship, where ρ symbolizes density, m represents mass, and v denotes volume. This formula indicates how much mass is contained in a given volume, allowing for the comparison of different materials and their compactness. Understanding this concept is essential in fields like physics and engineering, where material properties are crucial.

Archimedes' Principle states that an object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. This means that the buoyant force acting on the object is directly related to the volume of fluid that is pushed out of the way (displaced) by the object. Therefore, the correct term to complete the statement is "displaced," as it accurately describes the fluid responsible for the buoyant force acting on the submerged object.

In an ideal hydraulic lift system, Pascal's principle states that pressure applied to a confined fluid is transmitted undiminished in all directions. The formula F₁/A₁ = F₂/A₂ illustrates this principle, where F₁ and F₂ are the forces applied on two different pistons with areas A₁ and A₂, respectively. This relationship shows that the ratio of force to area remains constant, allowing a small force applied over a smaller area to lift a larger force over a larger area, enabling efficient lifting mechanisms in hydraulic systems.

Pascal's Principle asserts that when pressure is applied to a confined fluid, the change in pressure is transmitted equally in all directions throughout the fluid. This means that any increase in pressure at one point in the fluid results in an equivalent increase at every other point, allowing for the effective transfer of force. This principle is foundational in hydraulics and explains how hydraulic systems operate, such as in car brakes and lifts. The term "transmitted" accurately reflects this uniform distribution of pressure changes within the fluid.

Absolute pressure is the total pressure exerted on a system, including the atmospheric pressure. Gauge pressure measures the pressure relative to atmospheric pressure, meaning it does not account for atmospheric pressure itself. To find absolute pressure, you add gauge pressure to atmospheric pressure. Therefore, the relationship can be expressed as: Absolute Pressure = Atmospheric Pressure + Gauge Pressure. This explains why the correct answer is "plus."

Gauge pressure measures the pressure of a fluid relative to atmospheric pressure. It indicates how much pressure is exerted by the fluid in excess of the surrounding atmospheric pressure. Therefore, when gauge pressure is described as "above" atmospheric pressure, it signifies that the measured pressure exceeds the baseline atmospheric pressure, reflecting the additional force exerted by the fluid. This distinction is crucial in various applications, such as in tire pressure measurements or industrial systems, where understanding the difference between gauge and absolute pressure is essential for safety and performance.

Gauge pressure can be calculated using the formula: \( P = \rho g h \), where \( P \) is the pressure, \( \rho \) is the density of the fluid, \( g \) is the acceleration due to gravity, and \( h \) is the depth. For fresh water at a depth of 5.0 m, using \( \rho = 1000 \, \text{kg/m}^3 \) and \( g = 9.8 \, \text{m/s}^2 \), the calculation yields \( P = 1000 \times 9.8 \times 5.0 = 49000 \, \text{Pa} \) or 49 kPa. This represents the gauge pressure at that depth.

Pressure in a fluid is defined as the force exerted per unit area. This relationship is captured by the formula P = f/a, where P represents pressure, f is the force applied, and a is the area over which the force is distributed. This principle is fundamental in understanding how fluids behave under different forces and is applicable in various fields, including physics and engineering. It highlights that increasing the force or decreasing the area will result in higher pressure.

Specific gravity (SG) is the ratio of the density of a substance to the density of water. To find the SG of the substance with a density of 800 kg/m³, divide its density by the density of water (1000 kg/m³). This calculation is 800 kg/m³ ÷ 1000 kg/m³ = 0.80. Therefore, the specific gravity of the substance is 0.80, indicating it is less dense than water.

Density is defined as mass divided by volume. If the mass of an object doubles while the volume remains constant, the formula for density (density = mass/volume) shows that the new density will be twice the original density. Therefore, when the mass increases to twice its original value, the density also doubles, as the volume does not change.