Comprehensive Study Notes on Mechanics: 0.2 Fundamentals, Statics, Kinematics, Dynamics, and Fluid Mechanics

Fundamentals of Mass, Density, and Weight

Mechanics begins with the conceptualization of density, which is defined as the mass per unit volume of a substance. In the study of physical quantities, a fundamental distinction is made between scalar and vector types. A scalar quantity is characterized by having magnitude only, whereas a vector quantity is defined by having both magnitude and direction. Mass is defined as the amount of matter contained within a body or, alternatively, it is described as the measure of a body's inertia. This is contrasted with weight, which is the force exerted on a body when it is placed within a gravitational field, a relationship quantified by the equation $W = m \times g$.

Principles of Statics, Moments, and Equilibrium

The moment of a force about a specific point $O$ is calculated as the magnitude of the force multiplied by the perpendicular distance from the point $O$, expressed by the formula $T = F \times d$. This lead to the principle of moments, which states that for any body in a state of equilibrium, the sum of the clockwise moments calculated about a certain point is equal to the sum of the anticlockwise moments about that same point. In practical application, there are two definitive conditions required for a body to be considered in equilibrium: first, the resultant force acting on the body must be zero; and second, the sum of the clockwise moments about a point must be equal to the sum of the anti-clockwise moments about that same point.

Elasticity and Linear Kinematics

Hooke's law governs the behavior of materials under stress, stating that the extension experienced by a material is proportional to the applied force, provided that the limit of proportionality is not exceeded. This relationship is mathematically represented by the formula $F = k \times e$. Linear kinematics describes the motion of bodies through concepts of displacement, speed, and velocity. Displacement is defined as the distance moved in a particular direction. Speed is the rate of change of distance, given by $s = \frac{d}{t}$. Velocity is defined as the rate of change of displacement. For a moving body, average speed can be calculated using the formula $\text{Average Speed} = \frac{u + v}{t}$, although it is also defined more generally as the total distance divided by the time. Acceleration is understood as the rate of change of velocity, represented by the equation $a = \frac{v - u}{t}$.

Momentum and Newton's Laws of Motion

Linear momentum is the product of a body's mass and its velocity, designated as $p = m \times v$. According to the principle of conservation of momentum, the total momentum before a collision is equal to the total momentum after the collision, given that no external forces act on the system. These dynamics are governed by Newton's three laws of motion. Newton's first law states that a body stays at rest or, if it is in motion, continues to move with uniform velocity unless it is acted upon by an external force. Newton's second law states that the rate of change of momentum is proportional to the applied force and takes place in the direction in which the force acts, expressed through the formulas $F = \frac{mv - mu}{t}$ and $F = m \times a$. Newton's third law states that if a body $A$ exerts a force on body $B$, then body $B$ exerts an equal and opposite for on body $A$; this is also interpreted to mean that to every action there is an equal and opposite reaction.

Work, Energy, and Power

Energy is broadly defined as the capacity to do work. Kinetic energy is the specific form of energy associated with a body by virtue of its motion, calculated as $E_k = \frac{1}{2}mv^2$. Gravitational potential energy is the energy associated with a body by virtue of its position, given by $E_p = mgh$. Work is defined as the product of a force and the distance moved in the direction of that force, $W = F \times d$. The standard unit of work, the Joule, is defined such that $1\,\text{Joule}$ is the work done when a force of $1\,N$ moves through a distance of $1\,m$. The principle of conservation of energy states that energy can neither be created nor destroyed, but can be converted from one form to another. Power is defined as the rate at which work is done, $P = \frac{E}{t}$, where $1\,\text{Watt}$ is the power when $1\,\text{Joule}$ of work is done in $1\,\text{second}$.

Pressure and Fluid Mechanics

Pressure is defined as the force acting normally per unit area, using the formula $P = \frac{F}{A}$. When considering pressure in a fluid, the value is calculated using the formula $P = pgh$. This leads to Archimedes' principle, which states that when a body is wholly or partially submerged in a fluid, it experiences an upthrust. This upthrust is defined as being equal to the weight of the fluid displaced by the body.