Mass and density functions are key tools for calculating mass in various objects. They help us understand how mass is distributed, whether it's along a thin rod or across a flat plate. These functions are crucial for engineering and physics applications.

Work and fluid systems calculations involve variable forces and fluid pressure. We use integrals to compute work done by changing forces or to find the force exerted by fluids on submerged surfaces. These concepts are vital in hydraulics and mechanical engineering.

Mass and Density Functions

Mass calculation with density functions

• Linear density functions
  • Express mass per unit length for thin rods or wires
  • Denoted by $\lambda(x) = \frac{dm}{dx}$, where $\lambda(x)$ represents linear density and $m$ represents mass
  • Compute total mass using $m = \int_a^b \lambda(x) dx$, integrating over the length from $a$ to $b$
• Radial density functions
  • Express mass per unit area for thin plates or sheets
  • Denoted by $\delta(r) = \frac{dm}{dA}$, where $\delta(r)$ represents radial density and $m$ represents mass
  • Compute total mass using $m = \int_a^b \delta(r) dA$, integrating over the area from $a$ to $b$, with $dA$ as the area element (circular plates, rectangular sheets)
• Center of mass
  • Represents the average position of mass in an object or system
  • Calculated using integrals involving density functions and position coordinates

Work and Fluid Systems

Work computation for variable forces

• Variable force in linear systems
  • Calculated using $W = \int_a^b F(x) dx$, where $W$ represents work, $F(x)$ represents the force function, and the integration is performed over the displacement from $a$ to $b$
  • Geometrically interpreted as the area under the force-displacement curve (springs, elastic materials)
• Fluid pressure work
  • Calculated using $W = \int_a^b pA(y) dy$, where $W$ represents work, $p$ represents constant fluid pressure, $A(y)$ represents the cross-sectional area function, and the integration is performed over the height from $a$ to $b$
  • Geometrically interpreted as the volume of fluid displaced (hydraulic lifts, pumps)
• Work-energy theorem
  • Relates the work done on an object to its change in kinetic energy

Hydrostatic force on submerged surfaces

• Hydrostatic force
  • Force exerted by a fluid at rest on a submerged surface
  • Calculated using $F = \rho g \int_a^b y w(y) dy$, where $F$ represents hydrostatic force, $\rho$ represents fluid density, $g$ represents acceleration due to gravity, $w(y)$ represents the width function of the surface, and the integration is performed over the depth from $a$ to $b$
• Hydrostatic pressure
  • Pressure exerted by a fluid at rest at a given depth
  • Calculated using $p = \rho g y$, where $p$ represents hydrostatic pressure, $\rho$ represents fluid density, $g$ represents acceleration due to gravity, and $y$ represents the depth below the surface (dams, tanks)

Rotational Dynamics

Angular motion and torque

• Newton's laws of motion applied to rotational systems
• Torque as the rotational equivalent of force
• Moment of inertia as a measure of an object's resistance to rotational acceleration
• Conservation of energy in rotational systems

Integration for Physical Applications

Integration techniques to solve physical problems

• Identify the appropriate density function, force function, or pressure formula based on the given problem (linear density, radial density, variable force, hydrostatic pressure)
• Set up the integral expression using the corresponding formula and given information
• Determine the limits of integration based on the physical dimensions or constraints
• Evaluate the integral using appropriate techniques such as substitution, integration by parts, or partial fractions (trigonometric substitution, tabular integration)

Analysis of real-world calculation results

• Relate the calculated values to their corresponding physical quantities
  • Mass expressed in units of kilograms (kg) or pounds (lbs)
  • Work expressed in units of joules (J) or foot-pounds (ft-lbs)
  • Hydrostatic force expressed in units of newtons (N) or pounds (lbs)
• Consider the practical implications and significance of the results
  1. Ensure safety by designing structures that can withstand calculated forces (bridges, buildings)
  2. Optimize energy efficiency by minimizing work required in various systems (engines, machines)
  3. Minimize material usage by accurately calculating mass and optimizing designs (aircraft, vehicles)