Week - 1 - Consistency of Units
AIM:
To convert the SI system of units to given system of units (g-mm-ms) and showing the conversion factor detailed calculation for one system of units.
Based on this determining the conversion factor matrix for a set of units given in the problem statement.
DATA GIVEN:
Mass=1 kg
Length= 1 m
Time= 1 s
Force= 1 N
Stress= 1 Pa
Energy=1 Nm = 1 J
Density = 1 `(kg)/m^3`
Young's Modulus = 210 GPa
Velocity = 56.33 MPH = 15.65 `m/s`
Acceleration due to gravity = 9.8 `m/s^2`
SOLUTION:
In the below mentioned table, Conversion factors were mentioned for g-mm-ms system of units.
Detailed calculation procedure for calculating the converison factor is mentioned below and accordingly the values in the table were filled.
| System | Mass | Length | Time | Force | Stress | Energy | Density | Young's Modulus | Velocity | Acceleration due to Gravity |
| kg-m-s (SI) | 1 kg | 1 m | 1 s | 1 N | 1 Pa | 1 Nm = 1 J | 1 `(kg)/m^3` | `2.1e+11`Pa | 15.65`m/s` | 9.8`m/s^2` |
| g-mm-ms | `1e+3` | `1e+3` | `1e+3` | 1 | `1e-6` | `1e+3` | `1e-6` | `2.1e+5` | 15.65 | `9.8e-3` |
Calculation Procedure for determining the Conversion factor:
SI (kg/m/s) units to gm/mm/ms (1)
1 kg = 1000 g
1 m = 1000 mm
1 s = 1000 ms
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Force :
In SI units,
Force = 1N = 1 `kg*m/s^2`
In (1) units,
Force (N) = `(1000)*(1000/1000^2)``g*(mm)/(ms^2)`
Force (N) = 1 `g*(mm)/(ms^2)`
Conversion Factor = 1
_______________________________________________________
Stress :
In SI units,
Stress = 1Pa = 1`N/m^2`= 1 `(kg)/(m*s^2)`
In (1) units,
Stress (Pa) = `(1000)/(1000*1000^2)``g/((mm)*(ms^2))`
Stress (Pa) = `10^-6 g(mm)/(ms^2)`
Conversion Factor = `10^-6` = `1e-6`
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Energy :
In SI units,
Energy = 1 Nm = 1 `kg*m^2/s^2`
In (1) units,
Energy (Nm) = `1000*1000^2/1000^2` `g*(mm^2)/(ms^2)`
Energy (Nm) = `10^3` `g*(mm^2)/(ms^2)`
Conversion Factor = `10^3` = `1e+3`
__________________________________________________________
Density:
In SI units,
Density = 1 `(kg)/m^3`
In (1) units,
Density = `1000/1000^3` `(g)/(mm^3)`
Density = `10^-6` `(g)/(mm^3)`
Conversion Factor = `10^-6` = `1e-6`
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Young's Modulus :
E = 210 GPa = `210*10^9` Pa = `2.1e+11` Pa
since the units of E and Stress are same, the same conversion factor is used for E.
Hence, E = `210*10^9*10^-6`= `2.1e+5` Pa
____________________________________________________________
Velocity:
In SI units,
Velocity = 15.65 `m/s`
In (1) units,
Velocity = `15.65*1000/1000` `(mm)/(ms)`
Velocity = 15.65 `(mm)/(ms)`
Coversion Factor = 1
______________________________________________________________
Acceleration due to gravity:
In SI units,
Acceleration = 9.8 `m/s^2`
In (1) units,
Acceleration = `9.8*1000/1000^2` `(mm)/(ms)^2`
Acceleration = `9.8e-3` `(mm)/(ms)^2`
Conversion Factor = `10^-3` = `1e-3`
RESULT:
Conversion factor Matrix:
Following the above mentioned conversion factor procedure, the remaining unit system conversion factors were calculated and tabulated.
Note:
- 1 kg = `6.85e-2`slug
- 1 m = 3.28 ft
| System | Mass | Length | Time | Force | Stress | Energy | Density | Young's Modulus | Velocity | Acceleration due to Gravity |
| kg-m-s (SI) | 1 kg | 1 m | 1 s | 1 N | 1 Pa | 1 Nm = 1 J | 1 `(kg)/m^3` | `2.1e+11`Pa | 15.65 m/s | 9.8 `m/s^2` |
| g-mm-ms | `1e+3` | `1e+3` | `1e+3` | 1 | `1e-6` | `1e+3` | `1e-6` | `2.1e+5` | 15.65 | `9.8e-3` |
| kg-cm-ms | 1 | `1e+2` | `1e+3` | `1e-4` | `1e-8` | `1e-2` | `1e-6` | `2.1e+3` | 1.565 | `9.8e-4` |
| kg-cm-s | 1 | `1e+2` | 1 | `1e+2` | `1e-2` | `1e+4` | `1e-6` | `2.1e+9` | 1565 | `9.8e+2` |
| g-cm-s (CGS) | `1e+3` | `1e+2` | 1 | `1e+5` | 10 | `1e+7` | `1e-3` | `2.1e+12` | 1565 | `9.8e+2` |
| g-cm-`mu s` | `1e+3` | `1e+2` | `1e+6` | `1e-7` | `1e-11` | `1e-5` | `1e-3` | 2.1 | `1.565e-3` | `9.8e-10` |
| g-mm-s | `1e+3` | `1e+3` | 1 | `1e+6` | 1 | `1e+9` | `1e-6` | `2.1e+11` | 15650 | `9.8e+3` |
| ton-mm-s | `1e-3` | `1e+3` | 1 | 1 | `1e-6` | `1e+3` | `1e-12` | `2.1e+5` | 15650 | `9.8e+3` |
| slug-ft-s | `6.85e-2` | 3.28 | 1 | 0.225 | 0.021 | 0.738 | `2e-3` | `4.4e+9` | 51.33 | 32.14 |
| kg-mm-s | 1 | `1e+3` | 1 | `1e+3` | `1e-3` | `1e+6` | `1e-9` | `2.1e+8` | 15650 | `9.8e+3` |
| g-cm-ms | `1e+3` | `1e+2` | `1e+3` | 0.1 | `1e-5` | 10 | `1e-3` | `2.1e+6` | 1.565 | `9.8e-4` |
| kg-mm-ms | 1 | `1e+3` | `1e+3` | `1e-3` | `1e-9` | 1 | `1e-9` | `2.1e+2` | 15.65 | `9.8e-3` |