Week - 1 - Consistency of Units

The detailed conversion calculations including the conversion factors for each quantity from SI units to the gm/mm/ms unit system are explained below:

1. Mass:

- SI Unit: 1 kg

- Conversion:

- 1 kg = 1000 grams

- Conversion Factor: 1000 (1 kg = 1000 grams)

Result: 1000 grams

2. Length:

- SI Unit: 1 meter

- Conversion:

- 1 meter = 1000 millimeters

- Conversion Factor: 1000 (1 meter = 1000 millimeters)

Result: 1000 millimeters

3. Time:

- SI Unit: 1 second

- Conversion:

- 1 second = 1000 milliseconds

- Conversion Factor: 1000 (1 second = 1000 milliseconds)

Result: 1000 milliseconds

4. Force:

- SI Unit: 1 Newton (N)

- Formula: 1 Newton = 1 kg·m/s²

- Conversion:

- 1 kg = 1000 grams

- 1 meter = 1000 millimeters

- 1 second = 1000 milliseconds

- 1 Newton = 1000 grams × (1000 millimeters / 1000 milliseconds²)

- 1 Newton = 1 grams·millimeters / milliseconds²

- Conversion Factor: 1 (1 Newton = 1 grams·millimeters / milliseconds²)

Result: 1 grams·millimeters / milliseconds²

5. Stress:

- SI Unit: 1 Pascal (Pa)

- Formula: 1 Pascal = 1 Newton / meter²

- Conversion:

- 1 Pascal = 1 Newton / (1000 millimeters)²

- 1 Newton = 1000 grams·millimeters / milliseconds²

- Therefore, 1 Pascal = 1000 grams·millimeters / (1000 millimeters)²·milliseconds²

- 1 Pascal = 1 × 10⁻³ grams / (millimeters·milliseconds²)

- Conversion Factor: 10⁻³ (1 Pascal = 10⁻³ grams / millimeters·milliseconds²)

Result: 1 × 10⁻³ grams / millimeters·milliseconds²

6. Energy:

- SI Unit: 1 Newton-meter (Nm)

- Formula: 1 Newton-meter = 1 Newton × meter

- Conversion:

- 1 Newton = 1000 grams·millimeters / milliseconds²

- 1 meter = 1000 millimeters

- Therefore, 1 Newton-meter = 1000 grams·millimeters / milliseconds² × 1000 millimeters

- 1 Newton-meter = 1 × 10⁶ grams·millimeters² / milliseconds²

- Conversion Factor: 10⁶ (1 Newton-meter = 10⁶ grams·millimeters² / milliseconds²)

Result: 1 × 10⁶ grams·millimeters² / milliseconds²

7. Density:

- SI Unit: 1 kilogram per cubic meter (kg/m³)

- Conversion:

- 1 kilogram = 1000 grams

- 1 cubic meter = (1000 millimeters)³

- Therefore, 1 kg/m³ = 1000 grams / (1000 millimeters)³

- 1 kg/m³ = 1 × 10⁻⁶ grams / millimeters³

- Conversion Factor: 10⁻⁶ (1 kg/m³ = 10⁻⁶ grams / millimeters³)

Result: 1 × 10⁻⁶ grams / millimeters³

8. Young's Modulus:

- SI Unit: 210 Gigapascals (GPa)

- Conversion:

- 1 Gigapascal = 1 × 10⁹ Pascals

- 1 Pascal = 1 gram·millimeter / (millimeters·milliseconds²)

- Therefore, 210 GPa = 210 × 10⁹ × (1 × 10⁻³ grams / millimeters·milliseconds²)

- 210 GPa = 210 grams / millimeters·milliseconds²

- Conversion Factor: 1 (210 GPa = 210 grams / millimeters·milliseconds²)

Result: 210 grams / millimeters·milliseconds²

9. Velocity:

- SI Unit: 56.33 kilometers per hour (km/h)

- Conversion:

- 1 kilometer = 1000 meters

- 1 hour = 3600 seconds

- Therefore, 56.33 km/h = (56.33 × 1000 meters) / 3600 seconds

- 56.33 km/h = 15.647 meters / second

- 15.647 meters / second = 15.647 × 1000 millimeters / 1000 milliseconds

- 56.33 km/h = 15.647 millimeters / milliseconds

- Conversion Factor: 1 (56.33 km/h = 15.647 millimeters / milliseconds)

Result: 15.647 millimeters / milliseconds

10. Gravity:

- SI Unit: 9.8 meters per second squared (m/s²)

- Conversion:

- 1 meter = 1000 millimeters

- 1 second = 1000 milliseconds

- Therefore, 9.8 m/s² = 9.8 × (1000 millimeters / (1000 milliseconds)²)

- 9.8 m/s² = 9.8 × 10⁻³ millimeters / milliseconds²

- Conversion Factor: 10⁻³ (9.8 m/s² = 9.8 × 10⁻³ millimeters / milliseconds²)

Result: 0.0098 millimeters / milliseconds²

Now we carry out the conversions for the different unit systems. The results for them are tabulated in the table below.

Appendix:

This table was created in LATEX, and then the a snapshot of the output table was made. I have also attached the pdf file for the same.

One can Regenerate the pdf file for these table of conversion units by using the following code in an latex editor (if the file isnt accessible):

Code:

\documentclass{article}

\usepackage{amsmath}

\usepackage{array}

\usepackage{adjustbox}

\begin{document}

\begin{table}[h!]

\centering

\caption{Physical Quantities and Units}

\begin{adjustbox}{max width=\textwidth}

\begin{tabular}{|c|c|c|c|c|c|c|}

\hline

\textbf{Units} & \textbf{Mass} & \textbf{Length} & \textbf{Time} & \textbf{Force} & \textbf{Stress} & \textbf{Energy} \\

\hline

Kg cm ms & $1\,\text{kg}$ & $10^2\,\text{cm}$ & $10^3\,\text{ms}$ & $10^{-4}\,\text{kg}\cdot\text{cm}/\text{ms}^2$ & $10^{-8}\,\text{kg}/(\text{cm}\cdot\text{ms}^2)$ & $10^{-2}\,\text{kg}\cdot\text{cm}^2/\text{ms}^2$ \\

\hline

Kg cm s & $1\,\text{kg}$ & $10^2\,\text{cm}$ & $1\,\text{s}$ & $10^2\,\text{kg}\cdot\text{cm}/\text{s}^2$ & $10^{-2}\,\text{kg}/(\text{cm}\cdot\text{s}^2)$ & $10^4\,\text{kg}\cdot\text{cm}^2/\text{s}^2$ \\

\hline

gm mm ms & $10^3\,\text{gm}$ & $10^3\,\text{mm}$ & $10^3\,\text{ms}$ & $1\,\text{gm}\cdot\text{mm}/\text{ms}^2$ & $10^{-6}\,\text{gm}/(\text{mm}\cdot\text{ms}^2)$ & $10^3\,\text{gm}\cdot\text{mm}^2/\text{ms}^2$ \\

\hline

Kg m s & $1\,\text{kg}$ & $1\,\text{m}$ & $1\,\text{s}$ & $1\,\text{kg}\cdot\text{m}/\text{s}^2$ & $1\,\text{kg}/(\text{m}\cdot\text{s}^2)$ & $1\,\text{kg}\cdot\text{m}^2/\text{s}^2$ \\

\hline

gm cm s & $10^3\,\text{gm}$ & $10^2\,\text{cm}$ & $1\,\text{s}$ & $10^5\,\text{gm}\cdot\text{cm}/\text{s}^2$ & $10\,\text{gm}/(\text{cm}\cdot\text{ms}^2)$ & $10^7\,\text{gm}\cdot\text{cm}^2/\text{s}^2$ \\

\hline

gm cm µs & $10^3\,\text{gm}$ & $10^2\,\text{cm}$ & $10^6\,\text{µs}$ & $10^{-7}\,\text{gm}\cdot\text{cm}/\text{µs}^2$ & $10^{-11}\,\text{gm}/(\text{cm}\cdot\text{µs}^2)$ & $10^{-5}\,\text{gm}\cdot\text{cm}^2/\text{µs}^2$ \\

\hline

gm mm s & $10^3\,\text{gm}$ & $10^3\,\text{mm}$ & $1\,\text{s}$ & $10^6\,\text{gm}\cdot\text{mm}/\text{s}^2$ & $1\,\text{gm}/(\text{mm}\cdot\text{s}^2)$ & $10^9\,\text{gm}\cdot\text{mm}^2/\text{s}^2$ \\

\hline

Ton mm s & $10^{-3}\,\text{gm}$ & $10^3\,\text{mm}$ & $1\,\text{s}$ & $1\,\text{ton}\cdot\text{mm}/\text{s}^2$ & $10^{-6}\,\text{ton}/(\text{mm}\cdot\text{s}^2)$ & $10^3\,\text{ton}\cdot\text{mm}^2/\text{s}^2$ \\

\hline

Slug ft s & $0.068\,\text{slug}$ & $3.28\,\text{ft}$ & $1\,\text{s}$ & $0.224\,\text{slug}\cdot\text{ft}/\text{s}^2$ & $0.0208\,\text{slug}/(\text{ft}\cdot\text{s}^2)$ & $0.7366\,\text{slug}\cdot\text{ft}^2/\text{s}^2$ \\

\hline

Kg mm s & $1\,\text{kg}$ & $10^3\,\text{mm}$ & $1\,\text{s}$ & $10^3\,\text{kg}\cdot\text{mm}/\text{s}^2$ & $10^{-3}\,\text{kg}/(\text{mm}\cdot\text{s}^2)$ & $10^6\,\text{kg}\cdot\text{mm}^2/\text{s}^2$ \\

\hline

gm cm ms & $10^3\,\text{gm}$ & $10^2\,\text{cm}$ & $10^3\,\text{ms}$ & $10^{-1}\,\text{gm}\cdot\text{cm}/\text{ms}^2$ & $10^{-5}\,\text{gm}/(\text{cm}\cdot\text{ms}^2)$ & $10\,\text{gm}\cdot\text{cm}^2/\text{ms}^2$ \\

\hline

Kg mm ms & $1\,\text{kg}$ & $10^3\,\text{mm}$ & $10^3\,\text{ms}$ & $10^{-3}\,\text{kg}\cdot\text{mm}/\text{ms}^2$ & $10^{-9}\,\text{kg}/(\text{mm}\cdot\text{ms}^2)$ & $1\,\text{kg}\cdot\text{mm}^2/\text{ms}^2$ \\

\hline

\end{tabular}

\end{adjustbox}

\end{table}

\begin{table}[h!]

\centering

\caption{Additional Physical Quantities and Units}

\begin{adjustbox}{max width=\textwidth}

\begin{tabular}{|c|c|c|c|c|}

\hline

\textbf{Units} & \textbf{Density} & \textbf{Young’s Modulus} & \textbf{Velocity} & \textbf{Gravity} \\

\hline

Kg cm ms & $10^{-6}\,\text{kg}/\text{cm}^3$ & $2.1 \times 10^3\,\text{kg}/(\text{cm}\cdot\text{ms}^2)$ & $1.56472\,\text{cm/ms}$ & $9.8 \times 10^{-4}\,\text{cm/ms}^2$ \\

\hline

Kg cm s & $10^{-6}\,\text{kg}/\text{cm}^3$ & $2.1 \times 10^7\,\text{kg}/(\text{cm}\cdot\text{s}^2)$ & $1564.72\,\text{cm/s}$ & $9.8 \times 10^2\,\text{cm/s}^2$ \\

\hline

gm mm ms & $10^{-6}\,\text{gm}/\text{mm}^3$ & $2.1 \times 10^3\,\text{gm}/(\text{mm}\cdot\text{ms}^2)$ & $15.6472\,\text{mm/ms}$ & $9.8 \times 10^{-3}\,\text{mm/ms}^2$ \\

\hline

Kg m s & $1\,\text{kg}/\text{m}^3$ & $2.1 \times 10^9\,\text{N}/\text{m}^2$ & $15.6472\,\text{m/s}$ & $9.8\,\text{m/s}^2$ \\

\hline

gm cm s & $10^{-3}\,\text{gm}/\text{cm}^3$ & $2.1 \times 10^{10}\,\text{gm}/(\text{cm}\cdot\text{s}^2)$ & $1564.72\,\text{cm/s}$ & $9.8 \times 10^2\,\text{cm/s}^2$ \\

\hline

gm cm µs & $10^{-3}\,\text{gm}/\text{cm}^3$ & $2.1\,\text{gm}/(\text{cm}\cdot\text{µs}^2)$ & $15.64 \times 10^{-4}\,\text{cm/µs}$ & $9.8 \times 10^{-10}\,\text{cm/µs}^2$ \\

\hline

gm mm s & $10^{-6}\,\text{gm}/\text{mm}^3$ & $2.1 \times 10^{11}\,\text{gm}/(\text{mm}\cdot\text{s}^2)$ & $156472\,\text{mm/s}$ & $9.8 \times 10^3\,\text{mm/s}^2$ \\

\hline

Ton mm s & $10^{-12}\,\text{ton}/\text{mm}^3$ & $2.1 \times 10^3\,\text{ton}/(\text{mm}\cdot\text{s}^2)$ & $156472\,\text{mm/s}$ & $9.8 \times 10^3\,\text{mm/s}^2$ \\

\hline

Slug ft s & $1.93 \times 10^{-3}\,\text{slug}/\text{ft}^3$ & $4.368\,\text{slug}/(\text{ft}\cdot\text{s}^2)$ & $51.322\,\text{ft/s}$ & $32.144\,\text{ft/s}^2$ \\

\hline

Kg mm s & $10^{-9}\,\text{kg}/\text{mm}^3$ & $2.1 \times 10^6\,\text{kg}/(\text{mm}\cdot\text{s}^2)$ & $156472\,\text{mm/s}$ & $9.8 \times 10^3\,\text{mm/s}^2$ \\

\hline

gm cm ms & $10^{-3}\,\text{gm}/\text{cm}^3$ & $2.1 \times 10^4\,\text{gm}/(\text{cm}\cdot\text{ms}^2)$ & $1.56472\,\text{cm/ms}$ & $9.8 \times 10^{-4}\,\text{cm/ms}^2$ \\

\hline

Kg mm ms & $10^{-9}\,\text{kg}/\text{mm}^3$ & $2.1\,\text{kg}/(\text{mm}\cdot\text{ms}^2)$ & $15.6472\,\text{mm/ms}$ & $9.8 \times 10^{-3}\,\text{mm/s}^2$ \\

\hline

\end{tabular}

\end{adjustbox}

\end{table}

\end{document}

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