Basic Physical Concepts in Atmospheric Thermodynamics and Dynamics

Newton's Second Law

Definition and Application

Newton's Second Law is a fundamental principle in classical mechanics, describing the relationship between force, mass, and acceleration. It is essential for understanding motion in atmospheric dynamics.

• Force (F): A vector quantity containing both magnitude and direction, representing the push or pull on an object.

• Mass (m): A scalar quantity representing the amount of matter in an object.

• Position (\vec{x}): A vector indicating the location of an object in a coordinate system.

The law is mathematically expressed as:

$\vec{F} = m \frac{d^2 \vec{x}}{dt^2}$

This equation states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.

Example: Calculating the force required to accelerate a mass in a given direction.

Vectors in Physics

Expression of a Vector

Vectors are quantities that have both magnitude and direction, crucial for describing physical phenomena such as velocity, force, and acceleration.

• General form: $\vec{V}_1 = x_1 \vec{i} + y_1 \vec{j} + z_1 \vec{k}$

• Magnitude: $|\vec{V}_1| = \sqrt{x_1^2 + y_1^2 + z_1^2}$

• Unit vectors: i, j, k represent the x, y, z directions, respectively.

Example: A velocity vector in three dimensions.

Basic Vector Operations

Vectors can be manipulated through various operations, which are foundational in physics calculations.

• Addition/Subtraction: $\vec{A}_1 + \vec{A}_2 = (x_1 + x_2)\vec{i} + (y_1 + y_2)\vec{j} + (z_1 + z_2)\vec{k}$

• Multiplication by Scalar: $k\vec{a}$ scales the vector by a constant k.

• Dot Product: $\vec{A}_1 \cdot \vec{A}_2 = x_1x_2 + y_1y_2 + z_1z_2$ (results in a scalar)

• Cross Product: $\vec{A}_1 \times \vec{A}_2$ (results in a vector perpendicular to both)

Example: Calculating the resultant force from two vectors.

Vector Operation: Example

Consider two vectors: $\vec{a} = (3, -1, 2)$ and $\vec{b} = (1, -2, 3)$.

• Dot Product: $\vec{a} \cdot \vec{b} = 3 \times 1 + (-1) \times (-2) + 2 \times 3 = 3 + 2 + 6 = 11$

• Cross Product: $\vec{a} \times \vec{b}$ (calculated using determinant method)

Example: Finding the angle between two velocity vectors.

Vector Operation: Example 2

Calculating average velocity from displacement and time.

• A particle moves 80 cm South in 0.12 s and then 60 cm West in 0.08 s.

• Displacement vector: $\Delta \vec{x} = -80\vec{i} - 60\vec{j}$

• Total time: $\Delta t = 0.12 + 0.08 = 0.2$ s

• Magnitude of displacement: $||\Delta \vec{x}|| = \sqrt{(-80)^2 + (-60)^2} = 100$ cm

• Average velocity: $\frac{||\Delta \vec{x}||}{\Delta t} = \frac{100}{0.2} = 500$ cm/s

Example: Determining the speed of a moving particle.

Pressure

Definition and Formula

Pressure is a measure of the force applied perpendicular to the surface of an object per unit area.

• Formula: $p = \frac{F}{A}$

• Units: Pascal (Pa), where 1 Pa = 1 N/m2

Example: Calculating the pressure exerted by a force on a surface.

Pressure Inside Fluid Under Gravity

Within a fluid, the pressure at any location is exerted equally in all directions on neighboring fluid parcels.

• Pressure is isotropic at a point inside a fluid.

• Important for understanding atmospheric and oceanic pressure distributions.

Example: Pressure at a given depth in water.

Buoyancy Law

Archimedes' Principle

Archimedes' principle describes the buoyant force experienced by objects immersed in a fluid.

• The upward buoyant force is equal to the weight of the fluid displaced by the object.

• Applies to both fully and partially submerged objects.

Formula: $F_{\text{buoyant}} = \rho_{\text{fluid}} V_{\text{displaced}} g$ Example: Floating and sinking of objects in water.

Work and Energy

Work

Work is the energy transferred to or from an object via the application of force along a displacement.

• Formula: $w = \int_{x_1}^{x_2} F \cdot dx$

• Only the component of force in the direction of displacement contributes to work.

Example: Lifting an object against gravity.

Momentum

Momentum is the product of an object's mass and velocity, representing the quantity of motion.

• Formula: $\vec{p} = m\vec{v}$

• Units: kg·m/s

• Change in momentum: $dp = Fdt$

Example: Calculating the momentum of a moving car.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion.

• Formula: $E_k = \frac{1}{2}mv^2$

• Change in kinetic energy: $dE = F \cdot d\vec{x}$

Example: Energy of a moving ball.

Potential Energy

Potential energy is the energy held by an object due to its position relative to other objects or forces.

• Formula (gravitational): $U = mgh$

• Relationship with force: $F = -\frac{dU}{dx}$

Example: Energy stored in a stretched spring.

Dimensions and Units

SI Base and Derived Units

Physical quantities are measured using standardized units in the International System of Units (SI).

Quantity	SI Base Unit
Length	meter (m)
Mass	kilogram (kg)
Time	second (s)

Derived Quantity	SI Derived Unit
Frequency	Hertz (Hz)
Pressure	Pascal (Pa)

• Other units such as minute (min), hour (h), and day (d) may be used for convenience.

• Hectopascal (hPa), also called millibar (mb), is commonly used for atmospheric pressure.

• Temperature is often expressed in Celsius (°C), but Kelvin (K) is used for most physical laws: $T_{\text{Kelvin}} = T_{\text{Celsius}} + 273.15$

Example: Reporting atmospheric pressure in hPa.

Decimal Multiples and Submultiples

SI units use prefixes to indicate multiples or submultiples for convenience in expressing large or small quantities.

• Kilo- (k): $10^3$

• Milli- (m): $10^{-3}$

• Micro- (μ): $10^{-6}$

Example: 1 kilometer (km) = 1000 meters (m).

Additional info: Some context and examples have been expanded for clarity and completeness.