Week - 1 - Consistency of units

Consistency of Units

Aim: To convert the given measurement values to the following unit systems by showing the conversion factors.

Measurement Values:

Mass	1 kg
Length	1 m
Time	1 s
Force	1 N
Stress	1 Pa
Energy	1 Nm
Density	1 Kg/m3
Young’s Modulus	210 GPa
Velocity	56.33 Km/hr
Gravity	9.81 m/s2

Unit Systems:

1. Kg m s
2. Kg cm ms
3. Kg cm s
4. g mm ms
5. g cm s
6. g cm µs
7. g mm s
8. ton mm s
9. slug ft s
10. Kg mm s
11. g cm ms
12. Kg mm ms

Procedure:

Calculation for conversion from SI unit to “g mm ms” unit system:

Mass : 1 kg `=10^3 g`

Conversion factor `=10^3`

Length : 1 m`=10^3 mm`

Conversion factor`=10^3`

Time : 1 s `=10^3 ms`

Conversion factor `=10^3`

Force : 1 N`=1×kg×(m/s^2 )`

`=1×(10^3 g)×(10^3 mm)/(10^6 (ms)^2 )`

`=1 (g (mm))/(ms)^2`

Conversion factor`=1`

Stress : 1 Pa`=1×(N/(m)^2 )`

`=1×((g (mm))/((ms)^2 ))×(( 1)/(10^6 (mm)^2 ))`

`=10^(-6) ((g )/(mm (ms)^2 ))`

Conversion factor`=10^(-6)`

Energy : 1 Nm`=1×(g (mm)/(ms)^2 )×10^3 mm`

`=10^3 (g(mm^2 ) )/(ms)^2`

Conversion factor`=10^3`

Density : `1 (kg)/m^3=1×(10^3 g)/(10^9 mm^3 )`

`=10^(-6) ( g)/(mm)^3`

Conversion factor`=10^(-6)`

Youngs Modulus: 210 GPa `=(210×10^9)×10^(-6) (g )/((mm)( ms)^2)`

`=210×10^3 (g )/((mm)(ms)^2)`

Conversion factor `=10^-6`

Velocity: `56.33 (km)/(hr)=56.33((10^6 mm)/(3600(10^3)ms))`

`=15.6472 (mm)/(ms)`

Conversion factor`=1`

Gravity: `9.81 (m)/(s^2)=9.81× (10^3 mm)/(10^6 (ms)^2 )`

`=9.81×10^(-3) (mm)/(ms)^2`

Conversion factor `=10^-3`

Note: Detailed calculation for conversion of SI units to “g mm ms” is shown above and the remaining unit systems values are tabulated as shown below.

Measurement values of different unit systems:

S. No.	Unit System	Mass	Length	Time	Force	Stress	Energy	Density	Young’s Modulus	Velocity	Gravity
1	kg m s	1	1	1	`(1 kg(m))/s^2`	`(1 (kg))/((m)s^2)`	`(1 (kg)(m^2))/s^2`	`(1kg)/m^3`	`(210\times 10^9 (kg))/(m(s^2))`	`(15.64 m)/s`	`(9.81 m)/(s^2)`
2	Kg cm ms	1	102	103	`(10^-4(kg) (cm))/(ms)^2`	`(10^-8 (kg))/((cm)(ms)^2)`	`(10^-2 (kg) (cm^2))/(ms)^2`	`(10^-6 (kg))/(cm)^3`	`(210\times 10^1(kg))/((cm)(ms)^2)`	`(15.64 \times 10^-1 (cm))/(ms)`	`(9.81\times 10^-4 (cm))/(ms)^2`
3	Kg cm s	1	102	1	`(10^2 (kg )(cm))/(s^2)`	`(10^-2 (kg))/((cm)s^2)`	`(10^4 (kg)(cm^2))/s^2`	`(10^-6 (kg))/(cm)^3`	`(210\times 10^7 (kg))/((cm)(s)^2)`	`(15.64 \times 10^2(cm))/s`	`(9.81\times 10^2 (cm))/(s^2)`
4	g mm ms	103	103	103	`(1 g(mm))/(ms)^2`	`(10^-6 g)/((mm)(ms)^2)`	`(10^3 g (mm^2))/(ms)^2`	`(10^-6 g)/(mm)^3`	`(210\times 10^3 g)/((mm)(ms)^2)`	`(15.64 (mm))/(ms)`	`(9.81\times 10^-3 (mm))/(ms)^2`
5	g cm s	103	102	1	`(10^5 g (cm))/s^2`	`(10 g)/((cm)s^2)`	`(10^7 g(cm^2))/s^2`	`(10^-3 g)/(cm)^3`	`(210\times 10^10 g)/((cm)s^2)`	`(15.64 \times 10^2 (cm))/s`	`(9.81\times 10^2 (cm))/s^2`
6	g cm µs	103	102	106	`(10^-7 g (cm))/s^2`	`(10^-11 g)/((cm)(µs)^2)`	`(10^-5 g(cm^2))/(µs)^2`	`(10^-3 g)/(cm)^3`	`(210\times 10^-11 g)/((cm)(µs)^2)`	`(15.64 \times 10^-4 (cm))/(µs)`	`(9.81\times 10^-10 (cm))/((µs)^2)`
7	g mm s	103	103	1	`(10^6 g(mm))/s^2`	`(1 g)/((mm)s^2)`	`(10^9 g(mm^2))/s^2`	`(10^-6 g)/(mm)^3`	`(210\times 10^9 g)/((mm)s^2)`	`(15.64 \times 10^3(mm))/s`	`(9.81\times 10^3 (mm))/s^2`
8	ton mm s	103	103	1	`(1 (Ton )(mm))/s^2`	`(10^-6 (Ton))/((mm)s^2)`	`(10^3 (Ton)(mm^2))/s^2`	`(10^-12 (Ton))/(mm)^3`	`(210 \times 10^3 (Ton))/((mm)s^2)`	`(15.64 \times 10^3 (mm))/s`	`(9.81 \times 10^3 (mm))/s^2`
9	slug ft s	0.0685	3.28	1	`(0.224 (slug) (ft))/s^2`	`(0.021 (slug))/((ft)s^2)`	`(0.736 (slug)(ft^2))/s^2`	`(19 \times 10^-4 (Slug))/(ft^3)`	`(4.38 \times 10^9 (slug))/((ft)s^2`	`(51.32 (ft))/s`	`(32.2 (ft))/s^2`
10	kg mm s	1	103	1	`(10^3 (kg) (mm))/s^2`	`(10^-3 (kg))/((mm)s^2)`	`(10^6 (kg)(mm^2))/(s^2)`	`(10^-9 (kg))/(mm)^3`	`(210 \times 10^6 (kg))/((mm)s^2)`	`(15.64 \times 10^3(mm))/s`	`(9.81\times 10^3(mm))/s^2`
11	g cm ms	103	102	103	`(10^-1 g(cm))/(ms)^2`	`(10^-5 g)/((cm)(ms)^2)`	`(10 g(cm^2))/(ms)^2`	`(10^-3 g)/(cm)^3`	`(210\times 10^4 g)/((cm)(ms)^2)`	`(15.64 \times 10^-1 (cm))/(ms)`	`(9.81\times 10^-4 (cm))/(ms)^2`
12	kg mm ms	1	103	103	`(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\times 10^-3 (mm))/((ms)^2)`

Conclusion:

1. The calculation is shown for converting SI unit to 'g mm ms' unit system.
2. The measurement values of different unit system is tabulated.