Solid Mechanics

Fluid mechanics is the branch of applied mechanics concerned with the statics (fluids at rest) and dynamics (fluids in motion) of liquids and gases. It applies fundamental principles of mass, momentum, and energy conservation to analyze and predict fluid behavior. Its applications are vast, ranging from aerodynamics and hydraulics to meteorology and oceanography.

In continuum mechanics, the principle of mass conservation states that the mass of a closed system must remain constant over time. For a fluid, this is expressed by the continuity equation. In its Eulerian differential form, it is written as \(\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0\), where \(\rho\) is the density and \(\mathbf{u}\) is the velocity field.

These are two ways to describe motion in continuum mechanics:

• the Lagrangian specification follows individual material particles, tracking their properties over time, like watching a specific car in traffic
• the Eulerian specification focuses on fixed points in space, observing the properties (velocity, density) of whatever particles pass through those points, like a traffic camera observing a fixed intersection.

Electromotive force (\(\mathcal{E}\)) is the work done per unit of electric charge by a non-electrical source, such as a battery or generator. Despite its name, it is not a mechanical force but an electric potential, measured in volts. It represents the energy converted from another form (chemical, mechanical) into electrical energy as a charge traverses the source.

The electric current in a Voltaic pile is produced by a redox reaction. At the zinc anode, zinc metal is oxidized, releasing two electrons per atom (\(Zn \rightarrow Zn^{2+} + 2e^{-}\)). These electrons travel through the external circuit to the copper cathode. There, hydrogen ions from the aqueous electrolyte are reduced, forming hydrogen gas (\(2H^{+} + 2e^{-} \rightarrow H_2\)).

Absolute humidity (\(d_v\)) is the total mass of water vapor (\(m_{H_2O}\)) present in a given volume of air (\(V_{air}\)). It is expressed as grams of water vapor per cubic meter of air (\(g/m^3\)). The formula is \(d_v = \frac{m_{H_2O}}{V_{air}}\). Unlike relative humidity, it is not dependent on the air's temperature, but it is affected by changes in air pressure or volume.

In a reference frame rotating with an angular velocity \(\boldsymbol{\omega}\), the centrifugal force \(\mathbf{F}_{\mathrm{cf}}\) acting on an object of mass \(m\) at a position vector \(\mathbf{r}\) from the origin is given by the vector formula: \(\mathbf{F}_{\mathrm{cf}} = -m \boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf{r})\). This formula shows the force is directed perpendicular to the axis of rotation and outwards.

Coulomb's Law quantifies the electrostatic force between two stationary, electrically charged particles. The force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. The formula is \(F = k_e \frac{q_1 q_2}{r^2}\). This force can be either attractive or repulsive.

A reformulation of classical mechanics based on the principle of stationary action. It uses a scalar quantity called the Lagrangian, defined as kinetic energy minus potential energy (\(L = T - V\)). The equations of motion are derived from the Euler-Lagrange equation, \(\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0\), using generalized coordinates, which simplifies analysis of complex systems with constraints.

Galvanic corrosion is an electrochemical process where one metal corrodes preferentially when in electrical contact with another in the presence of an electrolyte. This occurs because the dissimilar metals create a bimetallic couple, with the less noble (more active) metal acting as the anode and corroding, while the more noble metal acts as the cathode.

The first electric battery, it produces a direct current by stacking pairs of dissimilar metal discs (e.g., zinc and copper) separated by brine-soaked cloth. Each pair forms a galvanic cell, and stacking them in series increases the overall voltage. This arrangement demonstrated the first continuous conversion of chemical energy into electrical energy, paving the way for modern electrochemistry.

The Voltaic pile established the principle of adding voltages by connecting electrochemical cells in series. Each zinc-copper pair, or cell, generates a characteristic electromotive force (EMF) of approximately 0.76 volts. By physically stacking these cells, Volta demonstrated that the total voltage across the pile is the sum of the individual EMFs of each cell, as described by \(V_{total} = n \times V_{cell}\).

A Fresnel zone is one of a series of confocal prolate ellipsoidal regions in space between a transmitter and a receiver. The primary zone's boundary is the ellipsoid where the path length from transmitter to a point on the surface to the receiver is half a wavelength longer than the direct path, causing a 180-degree phase shift.