6.5 Physical Applications

Calculus helps us understand the physical world by measuring changes over time or space. In this section, we'll see how integrals calculate mass, work, and fluid pressure in real-world situations.

We'll explore density functions to find object mass, variable forces to compute work done, and fluid pressure to determine forces on submerged surfaces. These applications show how calculus connects abstract math to practical problems in physics and engineering.

Mass and Density

Mass calculation with density functions

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• Linear density functions measure mass per unit length
  • Denoted as $\lambda(x)$, where $x$ represents position along the object (string, rod)
  • Calculate mass of an object with linear density $\lambda(x)$ from $a$ to $b$ using $\int_{a}^{b} \lambda(x) dx$
• Radial density functions measure mass per unit area
  • Denoted as $\rho(r)$, where $r$ represents distance from the object's center (disk, plate)
  • Calculate mass of a circular object with radial density $\rho(r)$ and radius $R$ using $2\pi \int_{0}^{R} r\rho(r) dr$
    • The $2\pi$ accounts for the circular symmetry
    • The $r$ inside the integral represents the radius at each point

Work and Pressure

Work computation for variable forces

• Variable force $[F(x)](https://www.fiveableKeyTerm:f(x))$ acting along a linear path from $a$ to $b$ does work calculated by $\int_{a}^{b} F(x) dx$
  • Constant force $F$ acting over a displacement $d$ does work calculated by $W = F \cdot d$
• Fluid pressure $P(x)$ acting on a vertical surface of width $w$ from depth $a$ to $b$ does work calculated by $w \int_{a}^{b} P(x) dx$
  • Fluid pressure at depth $x$ is given by $P(x) = \rho gx$ ($\rho$ is fluid density, $g$ is gravitational acceleration)
    • Pressure increases linearly with depth in a fluid

Hydrostatic force on submerged surfaces

• Hydrostatic force is the resultant force exerted by fluid pressure on a submerged surface
• For a rectangular surface of width $w$ and height $h$, submerged with its top edge at depth $d$, the hydrostatic force is $F = \rho gwh(d + \frac{h}{2})$
  • The term $(d + \frac{h}{2})$ represents the depth of the rectangular surface's centroid
    • The centroid is the point where the force acts as if concentrated

Integration for Physical Applications

Integration for physical applications

• Apply appropriate integration techniques to solve physical problems involving mass, work, and pressure
  • Substitution, integration by parts, trigonometric substitution, partial fractions
• Recognize the physical meaning of the integrals in the context of the problem
  • Mass integrals: $\int_{a}^{b} \lambda(x) dx$ or $2\pi \int_{0}^{R} r\rho(r) dr$
  • Work integrals: $\int_{a}^{b} F(x) dx$ or $w \int_{a}^{b} P(x) dx$
  • Hydrostatic force formula: $F = \rho gwh(d + \frac{h}{2})$

Interpretation of physical calculations

• Understand the units of the calculated quantities
  • Mass in kilograms (kg)
  • Work in joules (J)
  • Force in newtons (N)
• Relate the calculated values to the physical situation described in the problem
  • A large hydrostatic force value indicates a significant force exerted by the fluid on the submerged surface (dam, aquarium)
• Consider the implications of the results in real-world applications
  • Calculating work done by a variable force helps optimize machine or system designs to minimize energy consumption (car engines, hydraulic systems)

Mechanics and Motion

Newton's laws of motion

• First law: An object at rest stays at rest, and an object in motion stays in motion unless acted upon by an external force
• Second law: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (F = ma)
• Third law: For every action, there is an equal and opposite reaction

Energy and work

• Work is a form of energy transfer, measured as the product of force and displacement
• Kinetic energy is the energy of motion, related to an object's mass and velocity
• Potential energy is stored energy, often due to an object's position or configuration

Motion analysis

• Displacement is the change in position of an object, a vector quantity
• Velocity is the rate of change of displacement with respect to time
• Acceleration is the rate of change of velocity with respect to time