Consistency of Units
OBJECTIVE
To convert the values of given parameters from one unit system to another, and find out the conversion factor.
The parameters with their values is as shown below:
Mass = 1 kg
Length = 1 m
Time =1 s
Force = 1 N
Stress = 1 Pa
Energy = 1 N.m
Density = 1 kg/m^3
Young's Modulus = 210 GPa
Velocity = 56.33 kmph
Gravity = 9.8 m/s^2
Conversions are as shown below.
- kg cm ms
Mass `=1.kg =1.kg`
Conversion Factor = 1
Length `=1.m = 100.cm`
Conversion Factor = 100
Time `=1.s = 1000.ms`
Conversion Factor = 1000
Force `=1.N=1.(kg.m)/s^2 =1.(kg.100.cm)/(1000.ms)^2 =1.(kg.10^2 cm)/(10^6 ms^2 )=10^(-4).(kg.cm)/(ms^2)`
Conversion Factor = `10^-4`
Stress `=1.Pa=1.N/m^2 =1.(kg.m)/(m^2.s^2 )=1.(kg)/(m.s^2 )=1.(kg)/(10^2.cm.10^6.ms^2 )=10^(-8).(kg)/(cm.ms^2 )`
Conversion Factor = `10^-8`
Energy `=1.N.m=1.(kg.m)/s^2 .m=1.(kg.m^2)/s^2 =1.(kg.(100.cm)^2)/(1000.ms)^2 =1.(kg.10^4.cm^2)/(10^6.ms^2 )=10^(-2) (kg.cm^2)/(ms^2)`
Conversion Factor = `10^-2`
Density `=1.(kg)/(m^3) =1.(kg)/(10^2.cm)^3 =1.(kg)/(10^6.cm^3 )=10^(-6)(kg)/(cm^3)`
Conversion Factor = `10^-6`
Young's Modulus `=210.GPa=210×10^9.N/m^2 =210×10^9.(kg.m)/(m^2.s^2 )=210×10^9.(kg)/(m.s^2 )=210×10^9.(kg)/(10^2.cm.10^6.ms^2 )=2.1×10^3.(kg)/(cm.ms^2 )`
Conversion Factor = `10^-8`
Velocity `=56.33.(km)/(hr)=(56.33×(10^5.cm))/(3600.s)=(56.33×10^(5).cm)/(3600×(10^3.ms))=1.56.(cm)/(ms)`
Conversion Factor = 0.0277
Gravity `=9.81.m/s^2 =9.81.((10^2.cm))/(10^3.ms)^2 =9.81.(10^2.cm)/(10^6.ms^2)=9.81×10^(-4).(cm)/(ms^2)`
Conversion Factor = `10^-4`
- kg cm s
Mass `=1.kg =1.kg`
Conversion Factor = 1
Length `=1.m = 100.cm`
Conversion Factor = 100
Time `=1.s = 1.s`
Conversion Factor = 1
Force `=1.N=1.(kg.m)/s^2 =1.(kg.(100.cm))/(s^2) =1.(kg.10^2.cm)/(s^2)=10^(2).(kg.cm)/(s^2)`
Conversion Factor = 100
Stress `=1.Pa=1.N/m^2 =1.(kg.m)/(m^2.s^2 )=1.(kg)/(m.s^2 )=1.(kg)/((10^2.cm).s^2 )=10^(-2).(kg)/(cm.ms^2 )`
Conversion Factor = `10^-2`
Energy`=1.N.m=1.(kg.m)/s^2 .m=1.(kg.m^2)/s^2 =1.(kg.(100.cm)^2)/(s^2) =1.(kg.10^4.cm^2)/(s^2 )=10^(4) (kg.cm^2)/(s^2)`
Conversion Factor = `10^4`
Density `=1.(kg)/(m^3) =1.(kg)/(10^2.cm)^3 =1.(kg)/(10^6.cm^3 )=10^(-6)(kg)/(cm^3)`
Conversion Factor = `10^-6`
Young's Modulus`=210.GPa=210×10^9.N/m^2 =210×10^9.(kg.m)/(m^2.s^2 )=210×10^9.(kg)/(m.s^2 )=210×10^9.(kg)/(10^2.cm.s^2 )=210×10^7.(kg)/(cm.s^2 )`
Conversion Factor = `10^-2`
Velocity`=56.33.(km)/(hr)=(56.33×10^5.cm)/(3600.s)=(56.33×10^5.cm)/(3600.s)=1564.72.(cm)/(s)`
Conversion Factor = 27.77
Gravity`=9.81.m/s^2 =9.81.(10^2.cm)/(s^2) =981 (cm)/(s^2)`
Conversion Factor = 100
- gm mm ms
Mass `=1.kg =1000.gm`
Conversion Factor = 1000
Length `=1.m = 1000.mm`
Conversion Factor = 1000
Time `=1.s = 1000.ms`
Conversion Factor = 1000
Force `=1.N=1.(kg.m)/(s^2) =1.((10^3.gm).(10^3.mm))/(10^3.ms)^2 =1.(10^6.gm.mm)/(10^6.ms^2 )=1.(gm.mm)/(ms)`
Conversion Factor = 1
Stress `=1.Pa=1.N/m^2 =1.(kg.m)/(m^2.s^2 )=1.(kg)/(m.s^2 )=1.((10^3.gm))/(10^3.mm.10^6.ms^2 )=10^(-6).(gm)/(mm.ms^2)`
Conversion Factor = `10^-6`
Energy`=1.N.m=1.(kg.m)/s^2 .m=1.(kg.m^2)/s^2 =1.((10^3.gm).(10^3.mm)^2)/(10^3.ms)^2 =1.(10^3.gm.mm^2)/(ms^2)=10^(3). (gm.mm^2)/(ms^2)`
Conversion Factor = `10^3`
Density`=1.(kg)/(m^3) =1.((10^3.gm))/(10^3.mm)^3 =1.((10^3.gm))/(10^9.mm^3 )=10^(-6)(gm)/(mm^3)`
Conversion Factor = `10^-6`
Young's Modulus`=210.GPa=210×10^9.N/m^2 =210×10^9.(kg.m)/(m^2.s^2 )=210×10^9.(kg)/(m.s^2 )=210×10^9.((10^3.gm))/((10^3.mm).(10^6.ms^2))=210×10^3.(gm)/(mm.ms^2 )`
Conversion Factor = `10^-6`
Velocity`=56.33.(km)/(hr)=(56.33×(10^6.mm))/(3600.s)=(56.33×10^(6).mm)/(3600×(10^3.ms))=15.64.(gm)/(ms)`
Conversion Factor = 0.277
Gravity`=9.81.m/s^2 =9.81.((10^3.mm))/(10^3.ms)^2 =9.81.(10^3.mm)/(10^6.ms^2)=9.81×10^(-3).(mm)/(ms^2)`
Conversion Factor = `10^-3`
Similarly, in other unit systems, the values are as follows:
| Mass | Length | Time | Force | Stress | Energy | Density | Young's Modulus | Velocity | Gravity | |
| kg cm ms | `1.kg` | `10^2.cm` | `10^3.ms` | `10^(-4).(kg.cm)/(ms^2)` | `10^(-8).(kg)/(cm.ms^2)` | `10^(-2) (kg.cm^2)/(ms^2)` | `10^(-6)(kg)/(cm^3)` | `2.1×10^3.(kg)/(cm.ms^2)` | `1.564.(cm)/(ms)` | `9.81×10^(-4).(cm)/(ms^2)` |
| kg cm s | `1.kg` | `10^2.cm` | `1.s` | `10^(2).(kg.cm)/(s^2)` | `10^(-2).(kg)/(cm.ms^2)` | `10^(4) (kg.cm^2)/(s^2)` | `10^(-6)(kg)/(cm^3)` | `210×10^7.(kg)/(cm.s^2)` | `1564.72.(cm)/(s)` | `981 (cm)/(s^2)` |
| gm mm ms | `10^3.gm` | `10^3.mm` | `10^3.ms` | `1.(gm.mm)/(ms)` | `10^(-6).(gm)/(mm.ms^2)` | `10^(3). (gm.mm^2)/(ms^2)` | `10^(-6)(gm)/(mm^3)` | `210×10^3.(gm)/(mm.ms^2)` | `15.64.(mm)/(ms)` | `9.81×10^(-3).(mm)/(ms^2)` |
| kg m s | `1.kg` | `1.m` | `1.s` |
`1.(kg.m)/(s^2)` |
`1.(kg)/(m.s^2)` | `1.(kg.m^2)/s^2` | `1.(kg)/(m^3)` | `210×10^9.(kg)/(m.s^2)` | `15.64.(m)/(s)` | `9.81.m/s^2` |
| g cm s | `10^3.gm` | `10^2.cm` | `1.s` | `10^5.(gm.cm)/(s^2)` | `10.(gm)/(cm.s^2)` | `10^7.(gm.cm^2)/(s^2)` | `10^-3.(gm)/(cm^3)` | `210×10^10.(gm)/(cm.s^2)` | `1564.72.(cm)/(s)` | `981 (cm)/(s^2)` |
| g cm us | `10^3.gm` | `10^2.cm` | `10^-6.mus` | `10^17.(gm.cm)/(mus^2)` | `10^13.(gm)/(cm.mus^2)` | `10^19.(gm.cm^2)/(mus^2)` | `10^-3.(gm)/(cm^3)` | `210xx10^22.(gm)/(cm.mus^2)` | `15.64xx10^8.(cm)/(mus)` | `981xx10^12.(cm)/(mus^2)` |
| g mm s | `10^3.gm` | `10^3.mm` | `1.s` | `10^6.(gm.mm)/(s^2)` | `1.(gm)/(mm.s^2)` | `10^9.(gm.mm^2)/(s^2)` | `10^-6.(gm)/(mm^3)` | `210xx10^9.(gm)/(mm.s^2)` | `15.64xx10^3.(mm)/(s)` | `9.81xx10^3.(mm)/(s^2)` |
| ton mm s | `10^-3.t` | `10^3.mm` | `1.s` | `1.(t.mm)/(s^2)` | `10^-6.(t)/(mm.s^2)` | `10^3.(t.mm^2)/(s^2)` | `10^-12.(t)/(mm^3)` | `210xx10^3.(t)/(mm.s^2)` | `15.64xx10^3.(mm)/(s)` | `9.81xx10^3.(mm)/(s^2)` |
| slug ft s | `0.0685218 slugs` | `3.28.ft` | `1.s` | `0.224.(slugs.ft)/(s^2)` | `0.02.(slugs)/(ft.s^2)` | `0.737.(slugs.ft^2)/(s^2)` | `1.94xx10^-3.(slugs)/(ft^3)` | `4.38xx10^9.(slugs)/(ft.s^2)` | `51.3.(ft)/(s)` | `32.1768.(ft)/(s^2)` |
| kg mm s | `1.kg` | `10^3.mm` | `1.s` | `10^3.(kg.mm)/(s^2)` | `10^-3.(kg)/(mm.s^2)` | `10^6.(kg.mm^2)/(s^2)` | `10^-9.(kg)/(mm^3)` | `210xx10^6.(kg)/(mm.s^2)` | `15.64xx10^3.(mm)/(s)` | `9.81xx10^3.(mm)/(s^2)` |
| g cm ms | `10^3.gm` | `10^2.mm` | `10^3.ms` | `10^-1.(gm.cm)/(s^2)` | `10^-5.(gm)/(cm.ms^2)` | `10.(gm.cm^2)/(ms^2)` | `10^-3.(gm)/(cm^3)` | `210xx10^-5.(gm)/(cm.ms^2)` | `1.564.(cm)/(ms)` | `9.81×10^(-4).(cm)/(ms^2)` |
| kg mm ms | `1.kg` | `10^3.mm` | `10^3.ms` | `10^-3.(kg.mm)/(ms^2)` | `10^-9.(kg)/(mm.ms^2)` | `1.(kg.mm^2)/(ms^2)` | `10^-9.(kg)/(mm^3)` | `210.(kg)/(mm.ms^2)` | `15.64.(mm)/(ms)` | `9.81×10^(-3).(mm)/(ms^2)` |
CONCLUSION
The conversion of the parameters of one unit system to another requires the use of a basic unit system viz. mass, length, time, etc. rather than a derived unit system viz. stress, force, pressure, etc.