Static Equilibrium, Elasticity, and Fluid Dynamics

Flashcards covering the fundamental vocabulary of static equilibrium, elasticity, and fluid dynamics as presented in the Chapter 2 and 3 lecture notes.

Rigid body

A solid body in which deformation is neglected and the distance between any two given points remains constant in time regardless of external forces.

Free-body diagram

A diagram showing the magnitude and direction of all the forces acting on a chosen body, where the body is shown on its own, free of its surroundings.

Equilibrium

The state of balance in a system where there is no linear motion and no rotational motion; linear and angular velocities must be constant.

Torque ($\tau$)

The turning or twisting effectiveness of a force, defined mathematically as $\tau = \mathbf{r} \times \mathbf{F}$ with a magnitude given by $|\tau| = rF \sin(\theta)$.

Lever arm ($r$)

The distance from the axis of rotation to the point of application of force.

Stress

The applied force per unit cross sectional area of a material that describes the magnitude of forces causing deformation.

Pascal (Pa)

The SI unit of stress, equivalent to one newton of force pressing on a unit surface area of one meter squared ($1\,N/m^2$).

Tensile stress

Stress that occurs when forces pull on an object and cause its elongation, acting at right angles to a surface.

Compressive stress ($\sigma$)

Stress that occurs when forces cause the compression of an object, calculated as $\sigma = F/A$.

Bulk stress

Also known as volume stress, it is the pressure ($p$) resulting from a force $F$ normal to a surface that presses on the unit surface area $A$ of a submerged object.

Shear stress ($\tau$)

Stress caused by forces acting tangentially or in parallel to a surface, calculated as $F/A$.

Strain

A dimensionless quantity that defines the change in the shape of a body as the ratio of deformation to the initial size.

Longitudinal strain ($\varepsilon$)

Also called tensile strain, it is the ratio of the change in length ($\Delta L$) to the original length ($L$) of the body.

Volumetric strain

The strain produced by compressive stress, defined as the ratio of change in volume ($\Delta V$) to the original volume ($V$).

Bulk modulus ($B$)

The proportionality constant relating bulk stress to bulk strain, calculated as $B = - \frac{\Delta p}{\Delta V/V_0}$.

Compressibility ($k$)

The reciprocal of the bulk modulus ($k = 1/B$), used in relation to fluids to describe volume change per unit increase in pressure.

Shearing strain ($\gamma$)

A change in shape expressed as $\gamma = \tan(\theta)$ or $\Delta x/L$, defined as the angle through which a plane perpendicular to a fixed surface gets turned.

Shear modulus ($G$)

The proportionality constant defined by the ratio of shear stress to shear strain.

Elastic modulus

A measure of a material's rigidity or stiffness, defined as the ratio of stress to strain ($\Delta \sigma / \Delta \varepsilon$).

Young's modulus ($Y$)

A property of a material measuring its ability to stretch, defined as the ratio of tensile stress ($\sigma$) to tensile strain ($\varepsilon$).

Elasticity

The property of a solid material that allows it to restore its shape after an external load is removed.

Elastic limit

The maximum stress a material can withstand before permanent deformation occurs; the highest limit before plastic deformation begins.

Plasticity

The property of a solid substance that allows it to keep its deformed shape even when the external load is removed.

Fluid

A collection of molecules randomly arranged and held together by weak cohesive forces; both liquids and gases are classified as fluids.

Laminar flow

Characterized by fluid particles following smooth paths in layers with little or no mixing, where each layer has constant velocity relative to neighboring layers.

Turbulent flow

Fluid motion characterized by irregular, chaotic changes in pressure and velocity, resulting in lateral mixing, eddies, or swirls.

Reynolds number

A dimensionless parameter that determines the threshold velocity at which laminar flow becomes turbulent.

Equation of Continuity

An expression for incompressible fluids ($A_1 v_1 = A_2 v_2$) stating that the product of the cross-sectional area and fluid speed is constant.

Flow rate

Also called volume flux, it is the product $Av$ which represents the volume of fluid passing through a cross-section per unit time.

Bernoulli's Equation

A consequence of energy conservation stating that $p + \frac{1}{2} \rho v^2 + \rho g y = \text{constant}$ at all points along a streamline.

Surface tension ($\gamma$)

The force per unit length ($\gamma = F/l$) acting in the plane of the interface of a liquid, causing it to shrink to the minimum possible surface area.

Cohesive forces

Attractive forces acting between molecules of the same type.

Adhesive forces

Forces of attraction acting between molecules of different types.

Contact angle ($\theta$)

The angle between a solid surface and a line drawn tangent to the liquid at the surface; it indicates whether a liquid will "wet" a surface.

Capillary action

The process of a liquid flowing in a narrow space without external forces like gravity, caused by surface tension and adhesive forces.

Viscosity ($\eta$)

A physical property describing a fluid's internal friction or resistance to flow and deformation, measured in Pascal-seconds ($Pa\,s$).

Poiseuille's law

A principle expressing the rate of steady flow ($Q$) through a horizontal pipe as $Q = \frac{\Delta p \pi r^4}{8 \eta l}$.