Consistency of units
OBJECTIVE: To convert the given measurement values to different unit systems. (Note: Show the conversion calculation for the "g mm ms" unit system and arrange the rest of the values in the form of a table.)
Measurements: MASS =1kg, LENGTH =1m, TIME =1s, FORCE =1N, STRESS =1Pa, ENERGY =1Nm, DENSITY =1Kg/m3, YOUNG'S Modulus =210GPa, Velocity =56.33KMPH, GRAVITY =9.8m/s2
Unit systems: "kg m s", "kg cm s", "g mm ms", "kg cm ms", "g cm s", "g cm us", "g mm s", "ton mm s", "slug ft s", "kg mm s", "g cm ms", "kg mm ms"
CALCULATIONS:
The calculations for the conversion from "kg m s" to "g mm ms" is given below:
| Sl. No. | Parameter | Measurements in "kg m s" | Measurements in "g mm ms" |
| 1 | Mass | 1 kg | `10^3 g` |
| 2 | Length | 1 m | `10^3 mm` |
| 3 | Time | 1 sec | `10^3 ms` |
| 4 | Force | 1 N = `1 (kg)((m)/(s^2))` | `1000` x `1000/1000^2 (g)((mm)/(ms)^2) = 1 (g)((mm)/(ms)^2)` |
| 5 | Stress | 1 Pa = `1 ((kg)((m)/(s^2)))/m^2 = 1 N/m^2` | `1/ 1000^2(g((mm)/(ms)^2))/((mm)^2)=10^-6 g/((mm)(ms)^2)` |
| 6 | Energy | 1 J = `1 Nm` | 1 x 1000 `(g)((mm)/(ms)^2)mm=10^3(g)((mm)/(ms))^2` |
| 7 | Density | 1 `(kg)/(m^3)` | 1 x `1000/1000^3=10^-6 g/(mm)^3` |
| 8 | Young's Modulus | 210 GPa = 210 x `10^9` Pa = 210 x `10^9 N/m^2` |
210 x `10^9` x `1/1000^2 (g(mm)/(ms)^2)/(mm)^2=``210*10^9*10^-6 g/((ms)^2(mm))` `=210 * 10^3g/((ms)^2(mm))` |
| 9 | Velocity | 56.33 `(km)/(hr)`= `((56.33 )* 10^3)/(60*60) m/s = 15.64 m/s` | `15.64*1000/1000 (mm)/(ms) = 15.64 (mm)/(ms)` |
| 10 | Acc due to Gravity | 9.81 `m/s^2` | 9.81 x `1000/1000^2 (mm)/(ms)^2 = 9.81 * 10^-3 (mm)/(ms)^2` |
RESULTS:
The converted values for the rest of the unit systems are shown in the table below:
| Sl. No. | Measurements | Mass | Length | Time |
Force [Mass* accl] |
Stress [`F/(area)`] |
Energy [F*d] |
Density [`(Mass)/(vol)`] |
Young's Modulus [`(sigma)/(epsilon)`] |
Velocity (56.33 KmPH) [`(Dist)/(time)`] |
Acceler due to Gravity [`(Dist)/(time^2)`] |
| 1 | `kg` `m` `s` | `1 kg` | `1m` | `1s` | 1 N | 1 Pa | 1 J | 1 `(kg)/(m^3)` |
210 GPa = 210 x `10^9 ((kg)/(m(s^2)))` |
15.64 `m/s` | 9.81 `m/s^2` |
| 2 | `kg` cm `s` | `1 kg` | `10^2cm` | `1s` | `10^2` | `10^-2` | `10^4` | `10^-6(kg)/(cm)^3` | 210 x `10^7 (kg)/((cm)(s)^2)` | 15.64 x `10^2(cm)/(s)` | 9.81 x `10^2(cm)/(s)^2` |
| 3 | `g` `mm` `ms` | `10^3g` | `10^3mm` | `10^3ms` | 1 | `10^-6` | `10^3` | `10^-6(g)/(mm)^3` | 210 x `10^3g/((mm)(ms)^2)` | 15.64 `(mm)/(ms)` | 9.81 x `10^-3(mm)/(ms)^2` |
| 4 | `kg` `cm` `ms` | `1 kg` | `10^2cm` | `10^3ms` | `10^-4` | `10^-8` | `10^-2` | `10^-6(kg)/(cm)^3` | `2100 (kg)/((km)(ms)^2)` | 15.64 x `10^-1(cm)/(ms)` | 9.81 x `10^-4(cm)/(ms)^2` |
| 5 | `g` `cm` `s` | `10^3g` | `10^2cm` | `1s` | `10^5` | 10 | `10^7` | `10^-3(g)/(cm)^3` | 210 x `10^10g/((cm)(s)^2)` | 15.64 x `10^2(cm)/(s)` | 9.81 x `10^2(cm)/(s)^2` |
| 6 | `g` `cm` `mus` | `10^3g` | `10^2cm` | `10^6mus` | `10^-7` | `10^-11` | `10^-5` | `10^-3(g)/(cm)^3` | 210 x `10^-11g/((cm)(mus)^2)` | 15.64 x `10^-4(cm)/(mus)` | 9.81 x `10^-10(cm)/(mus)^2` |
| 7 | `g` `mm` `s` | `10^3g` | `10^3mm` | `1s` | `10^6` | 1 | `10^9` | `10^-6(g)/(mm)^3` | 210 x `10^9g/((mm)(s)^2)` | 15.64 x `10^3(mm)/(s)` | 9.81 x `10^3(mm)/(s)^2` |
| 8 | `Ton` `mm` `s` | `10^-3Ton` | `10^3mm` | `1s` | 1 | `10^-6` | `10^3` | `10^-12(Ton)/(mm)^3` | 210 x `10^3(Ton)/((mm)(s)^2)` | 15.64 x `10^3(mm)/(s)` | 9.81 x `10^3(mm)/(s)^2` |
| 9 | `slug` `ft` `s` | `0.0685 slug` | `3.28 ft` | `1s` | 0.224 | 0.021 | 0.735 | 0.002 `(slug)/(ft)^3` | 4.41 x `10^9(slug)/((ft)(s)^2)` | 51.29 `(ft)/s` | 32.2 `(ft)/s^2` |
| 10 | `kg` `mm` `s` | `1 kg` | `10^3mm` | `1s` | `10^3` | `10^-3` | `10^6` | `10^-9(kg)/(mm)^3` | 210 x `10^6(kg)/((mm)(s)^2)` | 15.64 x `10^3(mm)/(s)` | 9.81 x `10^3(mm)/(s)^2` |
| 11 | `g` `cm` `ms` | `10^3g` | `10^2cm` | `10^3ms` | 0.1 | `10^-5` | 10 | `10^-3(g)/(cm)^3` | 210 x `10^4g/((cm)(ms)^2)` | 15.64 x `10^-1(cm)/(ms)` | 9.81 x `10^-4(cm)/(ms)^2` |
| 12 | `kg` `mm` `ms` | `1 kg` | `10^3mm` | `10^3ms` | `10^-3` | `10^-9` | 1 | `10^-9(kg)/(mm)^3` | `210 (kg)/((mm)(ms)^2)` | 15.64 `(mm)/(ms)` | 9.81 x `10^-3(mm)/(ms)^2` |
The conversion factor for converting values in SI system to the respective unit systems are shown in the table below:
| Sl. No. | Unit systems | Multiplication factors for conversion | |||||||||
| Mass | Length | Time |
Force [Mass* accl] |
Stress [`F/(area)`] |
Energy [F*d] |
Density [`(Mass)/(vol)`] |
Young's Modulus [`(sigma)/(epsilon)`] |
Velocity [`(Dist)/(time)`] |
Acceleration [`(Dist)/(time^2)`] |
||
| 1 | `kg` `m` `s` | `1` | `1` | `1` | `1` | `1` | `1` | `1` |
`1` |
`1` | `1` |
| 2 | `kg` cm `s` | `1` | `10^2` | `1` | `10^2` | `10^-2` | `10^4` | `10^-6` | `10^-2` | `10^2` | `10^2` |
| 3 | `g` `mm` `ms` | `10^3` | `10^3` | `10^3` | 1 | `10^-6` | `10^3` | `10^-6` | `10^-6` | `1` | `10^-3` |
| 4 | `kg` `cm` `ms` | `1` | `10^2` | `10^3` | `10^-4` | `10^-8` | `10^-2` | `10^-6` | `10^-8` | `10^-1` | `10^-4` |
| 5 | `g` `cm` `s` | `10^3` | `10^2` | `1` | `10^5` | 10 | `10^7` | `10^-3` | `10` | `10^2` | `10^2` |
| 6 | `g` `cm` `mus` | `10^3` | `10^2` | `10^6` | `10^-7` | `10^-5` | `10^-11` | `10^-3` | `10^-11` | `10^-4` | `10^-10` |
| 7 | `g` `mm` `s` | `10^3` | `10^3` | `1` | `10^6` | `1` | `10^9` | `10^-6` | `1` | `10^3` | `10^3` |
| 8 | `Ton` `mm` `s` | `10^-3` | `10^3` | `1` | `1` | `10^-6` | `10^3` | `10^-12` | `10^-6` | `10^3` | `10^3` |
| 9 | `slug` `ft` `s` | `0.0685` | `3.28` | `1` | `0.224` | `0.021` | `0.735` | `0.002` | `0.21` | `51.29` | `3.282` |
| 10 | `kg` `mm` `s` | `1` | `10^3` | `1` | `10^3` | `10^-3` | `10^6` | `10^-9` | `10^-3` | `10^3` | `10^3` |
| 11 | `g` `cm` `ms` | `10^3` | `10^2` | `10^3` | 0.1 | `10^-5` | `10` | `10^-3` | `10^-5` | `10^-1` | `10^-4` |
| 12 | `kg` `mm` `ms` | `1` | `10^3` | `10^3` | `10^-3` | `10^-9` | `1` | `10^-9` | `10^-9` | `1` | `10^-3` |
CONCLUSION:
- The values were converted to respective units.
- The conversion factors were tabulated.