Calculus helps us understand the physical world by measuring changes over time or space. In this section, we'll see how integrals calculate mass, , 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 λ(x), where x represents position along the object (string, rod)
Calculate mass of an object with linear density λ(x) from a to b using ∫abλ(x)dx
functions measure mass per unit area
Denoted as ρ(r), where r represents distance from the object's center (disk, plate)
Calculate mass of a circular object with radial density ρ(r) and radius R using 2π∫0Rrρ(r)dr
The 2π 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 ∫abF(x)dx
Constant force F acting over a displacement d does work calculated by W=F⋅d
Fluid pressure P(x) acting on a vertical surface of width w from depth a to b does work calculated by w∫abP(x)dx
Fluid pressure at depth x is given by P(x)=ρgx (ρ 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=ρgwh(d+2h)
The term (d+2h) 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: ∫abλ(x)dx or 2π∫0Rrρ(r)dr
Work integrals: ∫abF(x)dx or w∫abP(x)dx
Hydrostatic force formula: F=ρgwh(d+2h)
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
Key Terms to Review (12)
Area density: Area density is a measure of mass per unit area. It is commonly used in physical applications to quantify how mass is distributed over a surface.
Density function: A density function in calculus represents the distribution of mass or probability over a given interval. It is often used to calculate properties like total mass, center of mass, and moments through integration.
F(x): f(x) is a mathematical function that represents a relationship between an independent variable x and a dependent variable y. It is a fundamental concept in calculus that describes how a quantity varies with respect to changes in another quantity.
Hooke’s law: Hooke's law states that the force needed to extend or compress a spring by some distance is proportional to that distance. Mathematically, it is expressed as $F = -kx$ where $F$ is the force applied, $k$ is the spring constant, and $x$ is the displacement.
Hoover Dam: Hoover Dam is a concrete arch-gravity dam in the Black Canyon of the Colorado River, on the border between Nevada and Arizona. It was constructed during the Great Depression and provides water storage, hydroelectric power, and flood control.
Hydrostatic pressure: Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It increases linearly with depth in a fluid.
Joule: A joule is a unit of energy in the International System of Units (SI). It is defined as the amount of work done when a force of one newton displaces an object by one meter in the direction of the force.
Pascal’s principle: Pascal's principle states that a change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container. This principle is fundamental in understanding hydraulic systems and their applications.
Pascals: Pascals (Pa) are the SI unit of pressure, defined as one newton per square meter. They are used to quantify internal pressure, stress, Young's modulus, and tensile strength.
Radial density: Radial density is a measure of how mass or other quantities vary with distance from a central point. It is commonly used in physics and engineering to analyze symmetrical objects.
Spring constant: The spring constant, denoted by $k$, is a measure of the stiffness of a spring. It quantifies the amount of force required to compress or extend the spring by a unit length.
Work: Work is the integral of force over a distance. In calculus, it is often represented as $W = \int_a^b F(x) \, dx$.