Physics of Fluids: Archimedes' Principle, Pascal's Law, and Hydraulic Systems

Learning Competencies and Objectives

• Primary Learning Competency: Research and explain the mechanisms by which simple hydraulic systems utilize fluid principles to enhance the performance of both simple and compound machines.
• Detailed Objectives: Discuss how simple hydraulic systems apply fluid principles to improve machine efficiency. This discussion is centered on the core concepts of Archimedes' Principle and Pascal's Principle.

Preliminary Drill and Foundational Questions

• Mechanical Advantage: How can one determine if a machine possesses a high mechanical advantage?
• Machine Efficiency: What specific factors must be taken into consideration to ensure that a machine operates with high efficiency?
• Buoyant Force: What is the fundamental definition of buoyant force?

Physics of Fluids: Buoyant Force and Upthrust

• Definition of Buoyant Force: Buoyant force is defined as the upward force exerted by a fluid upon any object immersed within it. This force acts in direct opposition to the object's weight. Its magnitude determines whether the object will float or sink, which is dependent on the object's density relative to the density of the fluid.
• Causes of Buoyancy:
  • Buoyant force is frequently referred to as "upthrust."
  • The force arises due to the fact that pressure within a fluid increases in proportion to depth.
  • When an object is submerged, the pressure exerted at the bottom of the object is significantly greater than the pressure at the top.
  • This pressure differential creates a net upward force.
  • This force opposes gravity and is the reason objects feel lighter when submerged in a fluid; if the force is greater than the weight of the object, the object floats.

Archimedes’ Principle

• The Principle Defined: Archimedes’ principle states that the buoyant force acting on an object is exactly equal to the weight of the fluid that the object displaces.
• Fully Submerged Objects: For an object that is completely underwater, the buoyant force is equal to the weight of a volume of fluid that matches the object's total volume.
• Floating Objects: For an object that is floating, the weight of the displaced fluid is equal to the total weight of the object.

Factors Affecting Buoyant Force

• Fluid Density ($\rho$): Denser fluids exert a larger buoyant force. This explains why it is easier for objects to float in saltwater (which is denser) than in freshwater.
• Submerged Volume ($V$): There is a direct correlation between the volume of the object submerged and the buoyant force; as more volume is submerged, more fluid is displaced, thereby increasing the force.
• Gravitational Acceleration ($g$): The buoyant force is directly proportional to gravity. Consequently, the force would change if the experiment were conducted on different planets with varying gravitational pulls.
• Object Density: This determines the ultimate behavior of the object in the fluid. Objects with a density greater than the fluid will sink, while those with a lower density will float.

Types of Buoyancy States

• Positive Buoyancy: This occurs when the buoyant force is greater than the object's weight ($F_B > \text{Weight}$). The result is that the object floats.
• Negative Buoyancy: This occurs when the buoyant force is less than the object's weight ($F_B < \text{Weight}$). The result is that the object sinks.
• Neutral Buoyancy: This occurs when the buoyant force is exactly equal to the object's weight ($F_B = \text{Weight}$). The result is that the object remains suspended at its current depth within the fluid.

Mathematical Calculation of Buoyant Force

• The Formula:$F_B = \rho \times V \times g$
• Variables Defined:
  • $\rho$ = density of the fluid
  • $V$ = volume of the fluid displaced
  • $g$ = acceleration due to gravity
• Implication: The formula demonstrates that buoyant force is a direct function of the fluid's density and the specific volume of the object that is submerged.
• Sample Problem 1:
  • Given: A block of wood with length = $2.5~m$, width = $0.5~m$, and height = $0.4~m$. The density of water is $1000~kg/m^3$. The acceleration due to gravity is $10~N/kg$.
  • Task: Determine the buoyant force acting on the block if it is placed in the water.

Pascal’s Law and Pressure in Fluids

• Pascal's Principle Defined: This principle states that pressure applied to a confined fluid is transmitted equally in all directions throughout the fluid. This ensures that any change in pressure at any specific point is felt equally at every other point within the fluid.
• Static Fluid Pressure: In a static fluid, the pressure at any given point is the same in all directions.
• Definition of Pressure: Pressure is defined as the force acting on a unit area. The magnitude of pressure depends on two factors: the size of the force and the area upon which that force acts.
• Pascal's Law Formula:$P = \frac{F}{A}$
• Variables Defined:
  • $F$ is the applied force
  • $P$ is the transmitted pressure
  • $A$ is the cross-sectional area

Sample Problems for Pressure and Hydraulics

• Sample Problem 2 (Pressure Calculation):

  • Question: What is the pressure exerted by a piston if a force of $10~N$ is applied over a cross-sectional area of $0.10~m^2$?
  • Formula: $P = \frac{F}{A}$

• Sample Problem 3 (Hydraulic System Force):

  • Principle: In a hydraulic system, pressure is constant throughout ($P_1 = P_2$), leading to the ratio: $\frac{F_1}{A_1} = \frac{F_2}{A_2}$.
  • Given:
    • Area of the input piston ($A_1$) = $0.001~m^2$
    • Area of the output piston ($A_2$) = $0.1~m^2$
    • External input force ($F_1$) = $100~N$
  • Task: Calculate the external output force ($F_2$).