Density and Fluid Mechanics Fundamentals

Density is defined as the mass of an object divided by its volume. This relationship indicates how much mass is contained in a given volume of a substance. The formula ρ = m/v succinctly captures this concept, where ρ represents density, m is mass, and v is volume. Understanding this formula is fundamental in various scientific fields, as it helps in determining how substances interact and behave under different conditions.

Specific gravity is the ratio of the density of a substance to the density of water. A specific gravity of 0.80 indicates that the substance's density is 80% that of water. Given that the density of water is 1000 kg/m³, the density of the substance can be calculated by multiplying the specific gravity by the density of water: 0.80 × 1000 kg/m³ = 800 kg/m³. Thus, the density of the substance is 800 kg/m³.

Density is defined as mass per unit volume. If the mass of an object doubles while its volume remains constant, the new density can be calculated by dividing the new mass (which is twice the original) by the unchanged volume. This results in a density that is also twice the original density. Therefore, when mass increases while volume stays the same, density increases proportionally, leading to a doubling of the density.

Gauge pressure in a fluid is defined as the pressure relative to atmospheric pressure. It can be calculated using the formula Pg = ρgh, where Pg is the gauge pressure, ρ is the fluid's density, g is the acceleration due to gravity, and h is the height of the fluid column above the measurement point. This relationship arises from the hydrostatic pressure principle, which states that pressure increases with depth in a fluid due to the weight of the fluid above.

To calculate the gauge pressure of fresh water at a depth of 5.0 m, we use the formula \( P = \rho g h \), where \( \rho \) is the density of water (1000 kg/m³), \( g \) is the acceleration due to gravity (9.8 m/s²), and \( h \) is the depth (5.0 m). Substituting the values, we get \( P = 1000 \times 9.8 \times 5.0 = 49000 \) Pa, which is equivalent to 49 kPa. Gauge pressure measures the pressure relative to atmospheric pressure, hence the answer is 49 kPa.

Absolute pressure is the total pressure exerted on a system, including atmospheric pressure. Gauge pressure measures pressure relative to atmospheric pressure and does not include it. Therefore, to find absolute pressure, you need to add gauge pressure to atmospheric pressure. This relationship shows that absolute pressure is the sum of atmospheric pressure and gauge pressure, making the correct operation addition.

Pascal's Principle asserts that when pressure is applied to an enclosed fluid, that pressure change is transmitted undiminished throughout the fluid in all directions. This means that any increase in pressure at one point in the fluid will be felt equally at every other point, allowing for efficient force transfer. This principle is fundamental in hydraulic systems, where it enables the multiplication of force and the functioning of various machinery. Thus, the pressure is not absorbed, reduced, or lost, but rather transmitted uniformly.

In an ideal hydraulic lift system, the principle of Pascal's law applies, stating that pressure applied to a confined fluid is transmitted undiminished in all directions. The formula F₁/A₁ = F₂/A₂ illustrates that the ratio of the force applied (F₁) to the area (A₁) of the smaller piston is equal to the ratio of the force (F₂) exerted by the larger piston to its area (A₂). This relationship allows for the amplification of force, enabling a small input force to lift a larger load, making hydraulic systems efficient and effective for lifting heavy objects.

Archimedes' Principle states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid that the object displaces. When an object is placed in a fluid, it pushes aside or displaces a volume of that fluid. The weight of this displaced fluid generates an upward force, which is the buoyant force. Therefore, the correct term to complete the statement is "displaced," as it directly relates to the fluid that is moved out of the way by the submerged object.

To calculate the buoyant force on an object, we use Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object. The buoyant force can be calculated using the formula: \( F_b = \rho \cdot V \cdot g \), where \( \rho \) is the density of the fluid (1000 kg/m³ for water), \( V \) is the displaced volume (0.020 m³), and \( g \) is the acceleration due to gravity (9.8 m/s²). Substituting these values gives \( F_b = 1000 \cdot 0.020 \cdot 9.8 = 196 \, \text{N} \).

An object floats when the weight of the water displaced equals the weight of the object. If 75% of the object's volume is submerged, it displaces an amount of water equal to 75% of its own weight. Given that the density of water is 1000 kg/m³, the submerged volume corresponds to a weight of 750 kg/m³. Therefore, the density of the object must be 750 kg/m³ for it to float with 75% of its volume submerged.

The continuity equation A₁V₁ = A₂V₂ illustrates the principle of conservation of mass in fluid dynamics. When a fluid flows from a wider section of a pipe (A₁) to a narrower section (A₂), the cross-sectional area decreases. To maintain a constant flow rate, the velocity of the fluid must increase in the narrower section. This relationship shows that as the area decreases, the speed of the fluid increases, ensuring that the mass flow rate remains constant throughout the pipe.

Bernoulli's Principle states that in a flowing fluid, an increase in fluid speed occurs simultaneously with a decrease in static pressure, provided the height remains constant. This relationship arises because the total energy in a streamline flow must remain constant. As the kinetic energy of the fluid increases due to higher velocity, the potential energy in the form of static pressure must decrease to maintain this balance. Thus, at the same height, a higher fluid speed results in lower static pressure.

According to the principle of conservation of mass, the flow rate must remain constant in a pipe. The flow rate is the product of the cross-sectional area and the velocity of the fluid. As the pipe narrows, the cross-sectional area decreases, which causes the velocity to increase to maintain the same flow rate. Using the equation \(A_1 v_1 = A_2 v_2\), where \(A\) is the area and \(v\) is the velocity, we can calculate the new velocity in the narrower section. Given the initial conditions, the velocity in the narrow section is found to be 6.0 m/s.

A Venturi tube operates based on Bernoulli's principle, which states that as the speed of a fluid increases, its pressure decreases. In the narrow section of the tube, the fluid's velocity increases due to the reduced cross-sectional area, resulting in higher speed. Consequently, this increase in speed leads to a decrease in pressure in that section. This relationship between fluid speed and pressure is fundamental to the functioning of the Venturi effect, which is used in various applications, such as flow measurement and fluid mixing.