• mass (M)
• length or distance (L)
• time (T)
• current (I)

• If is a variable representing the length of a soccer field, then .
• If is a variable representing the mass of a car, then

• is the ratio of a circle's circumference to its diameter, so .
• The Mach Number (M) is the ratio of the speed of an object moving through a fluid to the speed of sound in that fluid, so [M] = 1.
• An angle is measured as the ratio of an arc-length of a circle to its radius. If is the measure of some angle, then .

• How many people here have a bank account? What is stored in the bank account?
• What about computer or phone memory? RAM?

• Fourier's Principle of Dimensional Homogeneity: The universe operates under strict physical logic; you can never add or subtract variables that have different dimensions. When you add or subtract variables with the same dimensions, the result has that same dimension. So if you ever find yourself trying to add a Length to a Time, or a Mass to a Length, then you're doing something wrong.
• Dimensional Balance: Every valid equation must balance perfectly on both sides. If the left side reduces to Length, the right side must reduce to Length. If the left side is a Force (remember that ), then the right side of the equation must also be a Force.
• The Transcendental Rule: Certain functions need to have a dimensionless argument. We won't go over this rule in this class, but just including it here for future reference. It basically means there are certain functions, called "transcendental functions," including sine, cosine, exponential, and logarithms, that can only take dimensionless numbers as input.

• Check whether an equation is valid
• Derive formulas
• Understand how the physics of a system will change as you scale up or down in size (For example, if you build a working 1/10-scale prototype of a drone, will it work the same way if you build it at full size?)

• Displacement ( ): Measured in meters (m). .
• Initial Velocity Term ( ): Units are meters per second multiplied by seconds: . The seconds cancel out in the numerator and denominator, leaving .
• Acceleration Term ( ): Units are meters per second squared multiplied by seconds squared: . The scalar is dimensionless. The time units cancel out, leaving .
• Therefore, right-hand side of the equation, , has dimensions . Since each term has the dimension of Length, it is ok to add them together, according to Fourier's Principle of Dimensional Homogeneity. A Length plus a Length gives you a Length.
• Since the left-hand side of the equation is a Length ( ) and the right-hand side is also a Length, then we can see the equation is dimensionally consistent by the Dimensional Balance rule.

Reconstructing Centripetal Force ( )

• Target Benchmark: Remember Newton's 2nd Law: . The dimensional form of this is . In S.I. units, force is measured in Newtons, defined as: .
• Variables: Mass ( ), Velocity ( ), and Radius ( ).
• Step-by-Step Logic:
  • To get mass ( ) on top, keep in the numerator.
  • To get seconds squared ( ) on the bottom, we can square velocity ( ). This gives us units of . So now we have
  • To eliminate the extra meter remaining on top, divide the expression by radius ( ).
  • Our final expression, , has units .
• The Final Derived Formula:
  The dimensions balance and you didn't need to rely on memorization.

The Period of a Pendulum

• [PLACEHOLDER]

• The Battery Check: Knowing that a typical alkaline AA battery provides around 1.5V and safely delivers 100–500 mA under a medium load gives an engineer an instant, rough sense of how many cells are required to drive a specific circuit before they ever touch a calculator.

  

• The Machinist's Feel: A machinist knows that a heavy steel component shouldn't deflect by more than a few micrometers under tool pressure. If a formula spits out an answer in millimeters, their intuition flags a catastrophic design flaw immediately.
• The Thermal Reality: A thermal engineer knows a small cooling fan moves roughly 10–50 CFM (cubic feet per minute) of air. If their math says they need 5,000 CFM to cool a laptop, they know they used the wrong equation.

• 1 mA: Threshold of human perception; produces a faint tingling sensation.
• 6-10 mA: The "Let-Go" threshold. Involuntary muscle contractions. You may be unable to release a live wire.
• 50-100 mA: Triggers ventricular fibrillation and respiratory paralysis; typically fatal within seconds.
• 50 V: Universal regulatory safety cutoff where voltage overcomes dry skin resistance to drive lethal current.

• 1,500 W: Standard power ceiling for high-draw appliances to avoid tripping 15A household circuit breakers. Because standard US outlets top out around 15 Amps at 120 Volts ( ), manufacturers design high-draw appliances right at 1,500 W to avoid tripping standard circuit breakers.
  • toaster: ~1,500 W
  • microwave oven: ~1000 W
  • computer: ~130 W
  • 52-in LED TV: ~100 W
• 4.18 Joules: Specific heat capacity required to raise the temperature of 1 gram of water by 1°C. This means a 1,000W electric kettle will take roughly 4.2 seconds to heat a tiny shot of espresso water by 1°C, allowing you to estimate heating time.
• 37°C (98.6°F): Normal human core body temperature. Knowing that room temperature is typically 20°C to 22°C (68°F - 72°F) reminds you that your body is constantly radiating heat into its environment to dump metabolic waste heat.
• 21°C (70°F): Average comfortable room temperature used as a standard ambient baseline in thermal equations.

• 100 kPa (14.7 psi): Ambient atmospheric air pressure exerted at sea level under standard conditions. This is approximately the air pressure around us right now.
• 2,000 kg: Average mass of a modern consumer SUV.
• 9.81 m/s²: Approximate acceleration due to gravity on the earth's surface, used to quickly convert mass into Newtons of force.
• 1,000 kg/m³: Volumetric density of pure water, equating exactly to one metric ton per cubic meter.

• 100 ms: Maximum human perception latency threshold for an interaction to feel completely instantaneous.
• 25 Mbps: Minimum sustained internet download bandwidth required to stream uncompressed 4K video.
• 12 Wh: Typical energy capacity of a standard smartphone battery.
• 75,000 Wh: Average energy capacity of an electric vehicle battery pack.

• Find the variables: List everything affecting your problem (e.g., speed, size, density).
• Count the units: Count the fundamental units involved (like Mass, Length, and Time).
• Subtract to simplify: Subtract the number of units from the number of variables.