Challenge

Week - 1 - Consistency of Units

YYogya Khanna30 May 2021

Objective:

To write the given measurements according to the given system of units and to mention the conversion factors for the same.

Value of the following measurements are given as follows:

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

System of units:

Kg cm ms
Kg cm s
gm mm ms
kg m s
g cm s
g cm us
g mm s
ton mm s
slug ft s
kg mm s
g cm ms
kg mm ms

Procedure:

As asked in the question a detailed calculation for conversion from SI unit to gm/mm/ms unit system is shown below.

Mass

1 kg = 1000 grams

Length

1 m = 1000 mm

Time

1 second = 1000 milli-seconds

Force

F=m⋅a=kg⋅ms2F=m⋅a=kg⋅ms2

1kg⋅ms2=103g⋅103mm106ms21kg⋅ms2=103g⋅103mm106ms2 = 1g⋅mmms21g⋅mmms2

Therefore,

1N=1g⋅mmms21N=1g⋅mmms2

Stress

Stress = Force / Unit Area

1 Pa = 1 N/m^2

1Pa=1g⋅mmms2/106mm21Pa=1g⋅mmms2/106mm2

Therefore,

1Pa=10−6gmm⋅ms21Pa=10-6gmm⋅ms2

Energy

1Nm= 1g⋅mmms2⋅(103)mm1Nm= 1g⋅mmms2⋅(103)mm

Therefore

1Nm=103 g⋅mm2ms21Nm=103 g⋅mm2ms2

Density

1kgm3=103g109mm31kgm3=103g109mm3

Therefore:

1kgm3=10−6gmm31kgm3=10-6gmm3

Young's Modulus

210GPa=210⋅109Pa=210⋅109⋅10−6 g⋅mmms2/mm2210GPa=210⋅109Pa=210⋅109⋅10-6 g⋅mmms2/mm2

Therefore:

210GPa= 210⋅103gmm⋅ms2210GPa= 210⋅103gmm⋅ms2

Velocity

56.33kmh=56.33⋅106mm3.6⋅106ms56.33kmh=56.33⋅106mm3.6⋅106ms

Therefore

56.33kmh=15.64722mmms56.33kmh=15.64722mmms

Gravity

9.8ms2=9.8⋅103mm106ms29.8ms2=9.8⋅103mm106ms2

Therefore:

9.8ms2=9.8⋅10−3mmms29.8ms2=9.8⋅10-3mmms2

The quantities in their corresponding unit systems are shown below:

Units	Mass	Length	Time	Force	Stress	Energy	Density	Young's Modulus	Velocity	Gravity
Kg cm ms	1 kg	100 cm	1000 ms	10−4kg⋅cmms210-4kg⋅cmms2	10−8kgcm⋅ms210-8kgcm⋅ms2	10−2(kg)⋅cm2ms210-2(kg)⋅cm2ms2	10−6 kg(cm)310-6 kg(cm)3	2100kgcm⋅ms22100kgcm⋅ms2	156.4⋅10−2cmms156.4⋅10-2cmms	9.8⋅10−4(cmms2)9.8⋅10-4(cmms2)
Kg cm s	1 kg	100 cm	1 s	102kg⋅cms2102kg⋅cms2	10−2kgcm⋅s210-2kgcm⋅s2	104(kg)⋅cm2s2104(kg)⋅cm2s2	10−6 kg(cm)310-6 kg(cm)3	210⋅107 kgcm⋅s2210⋅107 kgcm⋅s2	1564.722cms1564.722cms	980cms2980cms2
g mm ms	1000 g	1000 mm	1000 ms	1 g⋅mmms21 g⋅mmms2	10−6gmm⋅ms210-6gmm⋅ms2	103g⋅mm2ms2103g⋅mm2ms2	10−6 g(mm)310-6 g(mm)3	21010^3 g/(mmms^2)`	15.647mmms15.647mmms	9.8⋅10−3mmms29.8⋅10-3mmms2
kg m s	1 kg	1 m	1 s	1kg⋅ms21kg⋅ms2	1 kgm⋅s21 kgm⋅s2	1(kg)⋅m2s21(kg)⋅m2s2	1 kgm31 kgm3	210⋅109kgm⋅s2210⋅109kgm⋅s2	15.64ms15.64ms	9.8ms29.8ms2
g cm s	1000 g	100 cm	1 s	105g⋅cms2105g⋅cms2	10 gcm⋅s210 gcm⋅s2	107g⋅cm2s2107g⋅cm2s2	10−3g(cm)310-3g(cm)3	210⋅1010gcm⋅s2210⋅1010gcm⋅s2	1564.7cms1564.7cms	980cms2980cms2
g cm μμs	1000 g	100 cm	106106 μμs	10−7 g⋅cms210-7 g⋅cms2	10−11 gcm⋅μs210-11 gcm⋅μs2	10−5g⋅cm2μs210-5g⋅cm2μs2	10−3gcm310-3gcm3	210⋅10−2gcm⋅μs2210⋅10-2gcm⋅μs2	15.647⋅10−4cmμs15.647⋅10-4cmμs	9.8 ⋅(10−10)cmμs29.8 ⋅(10-10)cmμs2
g mm s	1000 g	1000 mm	1 s	106 g⋅mms2106 g⋅mms2	1gmm⋅s21gmm⋅s2	109g⋅mm2s2109g⋅mm2s2	10−6 gmm310-6 gmm3	210⋅109gmm⋅s2210⋅109gmm⋅s2	15647.2mms15647.2mms	9.8 ⋅103ms29.8 ⋅103ms2
ton mm s	0.001 ton	1000 mm	1 s	1ton⋅mms21ton⋅mms2	10−6tonmm⋅s210-6tonmm⋅s2	103ton⋅mm2s2103ton⋅mm2s2	10−12 tonmm310-12 tonmm3	210⋅103tonmm⋅s2210⋅103tonmm⋅s2	51.33fts51.33fts	9.8⋅103mms29.8⋅103mms2
slug ft s	0.068 slug	3.28 ft	1 s	0.224slug⋅fts20.224slug⋅fts2	0.020slugft⋅s20.020slugft⋅s2	0.731slug⋅ft2s20.731slug⋅ft2s2	1.92⋅10−3 slugft31.92⋅10-3 slugft3	4.35⋅109slugft⋅s24.35⋅109slugft⋅s2	15647.2mms15647.2mms	32.144fts232.144fts2
kg mm s	1 kg	1000 mm	1 s	1000kg⋅mms21000kg⋅mms2	10−3kgmm⋅s210-3kgmm⋅s2	106(kg)⋅mm2s2106(kg)⋅mm2s2	10−9 kgmm310-9 kgmm3	210⋅106kgmm⋅s2210⋅106kgmm⋅s2	15647.22mms15647.22mms	9.8⋅103ms29.8⋅103ms2
g cm ms	1000 g	100 cm	1000 ms	0.1 g⋅cmms20.1 g⋅cmms2	10−5 gcm⋅ms210-5 gcm⋅ms2	10g⋅cm2ms210g⋅cm2ms2	10−3gcm310-3gcm3	210⋅104gcm⋅ms2210⋅104gcm⋅ms2	1.564 cms21.564 cms2	9.8⋅10−4cms29.8⋅10-4cms2
kg mm ms	1 kg	1000 mm	1000 ms	10−3kg⋅mmms210-3kg⋅mmms2	10−9kgmm⋅ms210-9kgmm⋅ms2	1(kg)⋅mm2ms21(kg)⋅mm2ms2	10−9 kgmm310-9 kgmm3	210kgmm⋅ms2210kgmm⋅ms2	15.647mmms15.647mmms	9.8⋅10−3mmms29.8⋅10-3mmms2

Conversion units:

To convert the given value in (kg m s) to the required units, the conversion values are shown below:

Units	Mass	Length	Time	Force	Stress	Energy	Density	Young's Modulus	Velocity	Gravity
Kg cm ms	1	100	1000	0.0001	0.00000001	0.01	0.000001	0.00000001	0.027765	0.0001
Kg cm s	1	100	1	100	0.01	10000	0.000001	0.01	27.77777	100
g mm ms	1000	1000	1000	1	0.000001	1000	0.000001	0.000001	0.277774	0.001
kg m s	1	1	1	1	1	1	1	1	0.27765	1
g cm s	1000	100	1	100000	10	10000000	0.001	10	27.77738	100
g cm μμs	1000	100	1000000	1E-07	1E-11	0.00001	0.001	1E-11	2.78E-05	1E-10
g mm s	1000	1000	1	1000000	1	1000000000	0.000001	1	277.7774	1000
ton mm s	0.001	1000	1	1	0.000001	1000	1E-12	0.000001	0.911237	1000
slug ft s	0.068	3.28	1	0.224	0.02	0.731	0.00192	0.020714286	277.7774	3.28
kg mm s	1	1000	1	1000	0.001	1000000	1E-09	0.001	277.7777	1000
g cm ms	1000	100	1000	0.1	0.00001	10	0.001	0.00001	0.027765	0.0001
kg mm ms	1	1000	1000	0.001	1E-09	1	1E-09	0.000000001	0.277774	0.001

Result:

The given measurements were converted according to the given system of units and the mention the conversion factors for the same are mentioned.
Calculation for one conversion is shown extensively.

Week - 1 - Consistency of Units

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Ship work like this
in your own course.