Atmospheric Thermodynamics and Dynamics: The Earth’s Atmosphere

Dimensions and Units

Newton's Second Law and Physical Dimensions

Understanding the correct use of physical dimensions and units is essential in physics, especially when applying fundamental laws such as Newton's Second Law. The law states that the acceleration of an object is proportional to the force acting on it.

• Newton's Second Law: $\vec{F} = m \dfrac{d^2 \vec{x}}{dt^2}$

• Physical Dimensions: Ensuring equations use the correct units is crucial for clear scientific communication.

• International System of Units (SI): The SI system standardizes units globally for consistency.

Example: Both $\vec{F} = m \dfrac{d^2 \vec{x}}{dt^2}$ and $\vec{F} = 0.001 \times m \dfrac{d^2 \vec{x}}{dt^2}$ express Newton's Second Law, but the first uses standard SI units.

SI Base Units

Fundamental Physical Quantities

SI base units are the foundation for all physical measurements. Each base unit represents an independent physical property.

Independent Measurement	Property	Name	Symbol
Space	Length	Meter	m
Matter	Mass	Kilogram	kg
Time	Time	Second	s
Thermal State	Temperature	Kelvin	K

• Length (meter, m): Measures spatial extent.

• Mass (kilogram, kg): Measures amount of matter.

• Time (second, s): Measures duration.

• Temperature (kelvin, K): Measures thermal state.

SI Derived Units

Units Derived from Base Quantities

Derived units are combinations of base units used to express other physical properties.

Property	Name	Symbol	Physical Origin
Frequency	Hertz	Hz (s-1)	Number of events per second
Force	Newton	N (kg m s-2)	$F = m a$
Pressure	Pascal	Pa (N m-2)	$P = F / A$
Energy	Joule	J (N m)	$dE = F \cdot dx$
Power	Watt	W (J s-1)	$\text{Power} = dE/dt$

• Frequency (Hertz, Hz): Number of cycles per second.

• Force (Newton, N): Product of mass and acceleration.

• Pressure (Pascal, Pa): Force per unit area.

• Energy (Joule, J): Work done by a force over a distance.

• Power (Watt, W): Rate of energy transfer.

Standard vs Convenience in Units

Commonly Used Units in Atmospheric Science

While SI units are standard, other units are sometimes used for convenience in specific contexts.

• Time Units: Minute (min), hour (h), and day (d) may be used instead of seconds.

• Pressure Units: Hectopascal (hPa), also called millibar (mb), is preferred in atmospheric science.

• Temperature Units: Celsius (°C) is often used for observed temperature, but Kelvin (K) is used in physical laws.

Conversion between Celsius and Kelvin:

$T = T_0 + T_c = 273.15 + T_c$

where $T$ is temperature in Kelvin and $T_c$ is temperature in Celsius.

Note: Most physical laws use the Kelvin scale for temperature.

Decimal Multiples and Submultiples

Prefixes for SI Units

SI prefixes are used to express multiples and submultiples of units, making it easier to handle very large or small quantities.

Multiple	Prefix	Symbol
109	Giga	G
106	Mega	M
103	Kilo	k
102	Hecto	h
101	Deca	da
10-1	Deci	d
10-2	Centi	c
10-3	Milli	m
10-6	Micro	μ
10-9	Nano	n

• Example: 1 kilometer (km) = $10^3$ meters (m).

• Example: 1 millisecond (ms) = $10^{-3}$ seconds (s).

Additional info: These notes are part of a lecture series for a college-level course in atmospheric thermodynamics and dynamics, focusing on foundational physical concepts and their application in atmospheric science.