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Challenge

Week - 1 - Consistency of Units

AAlan Akkarakkaran31 Aug 2023

To determine the units of a physical quantity, we need to express it as a combination of fundamental units (mass, length, time). Using the given values, we can express different units in terms of these fundamental units.1.

SI (International System of Units):

  • MASS: 1 kg
  • LENGTH: 1 m
  • TIME: 1 s
  • FORCE: 1 N
  • STRESS: 1 Pa
  • ENERGY: 1 Nm
  • DENSITY: 1 kg/m^3
  • YOUNG'S MODULUS: 210 GPa = 210,000 MPa
  • VELOCITY: 56.33 km/h = 15.646 m/s
  • GRAVITY: 9.8 m/s^2

CGS (Centimeter-Gram-Second System):

  • MASS: 1 kg = 1000 g
  • LENGTH: 1 m = 100 cm
  • TIME: 1 s
  • FORCE: 1 N = 10^5 dyn
  • STRESS: 1 Pa = 10 dyn/cm^2
  • ENERGY: 1 Nm = 10^7 erg
  • DENSITY: 1 kg/m^3 = 0.001 g/cm^3
  • YOUNG'S MODULUS: 210 GPa = 2.1 × 10^11 dyn/cm^2
  • VELOCITY: 56.33 km/h = 15.646 m/s
  • GRAVITY: 9.8 m/s^2 = 980 cm/s^2

basic conversion :

Conversion from SI units to g/mm/ms units:

  • LENGTH: 1 m = 1000 mm
  • TIME: 1 s = 0.001 ms
  • FORCE: 1 N = 1000 g·mm/ms^2
  • STRESS: 1 Pa = 1 g/mm·ms^2
  • ENERGY: 1 Nm = 1000 g·mm^2/ms^2
  • DENSITY: 1 kg/m^3 = 0.001 g/mm^3
  • YOUNG'S MODULUS: 210 GPa = 210,000 MPa = 2.1 x 10^11 g/mm^2·ms^2
  • VELOCITY: 56.33 km/h = 15.646 m/s = 15,646 mm/s
  • GRAVITY: 9.8 m/s^2 = 9800 mm/s^2

Examples of conversion:

  • 2 meters to g/mm: 2 m x 1000 mm/m = 2000 mm
  • 5 seconds to ms: 5 s x 1000 ms/s = 5000 ms
  • 10 Newtons to g·mm/ms^2: 10 N x 1000 g/kg x 1 mm/m x 1 ms^2/s^2 = 10000 g·mm/ms^2
  • 3 Pa to g/mm·ms^2: 3 Pa x 1 g/kg x 1 mm/m x 1 ms^2/s^2 = 3 g/mm·ms^2
  • 20 Nm to g·mm^2/ms^2: 20 Nm x 1000 g/kg x (1000 mm/m)^2 x (0.001 ms/s)^2 = 20,000,000 g·mm^2/ms^2
  • 800 kg/m^3 to g/mm^3: 800 kg/m^3 x 0.001 g/kg x (1000 mm/m)^3 = 0.8 g/mm^3
  • 150 GPa to g/mm^2·ms^2: 150 GPa x (1000 MPa/GPa) x (1 g/kg) x (1 mm/m)^2 = 1.5 x 10^11 g/mm^2·ms^2
  • 10 m/s to mm/s: 10 m/s x 1000 mm/m = 10,000 mm/s
  • 5 m/s^2 to mm/s^2: 5 m/s^2 x 1000 mm/m = 5000 mm/s^2

Here are the conversions for the given measurements to the Kg cm ms system of units:

  1. MASS: 1000 g = 1 kg
  2. LENGTH: 100 cm = 1 m
  3. TIME: 1000 ms = 1 s
  4. FORCE: 1000 g cm/ms^2 = 1 N
  5. STRESS: 10.1972 g/(cm·ms^2) = 1 Pa
  6. ENERGY: 10^7 g cm^2/ms^2 = 1 Nm
  7. DENSITY: 1000 g/cm^3 = 1 kg/m^3
  8. YOUNG'S MODULUS: 2.1 x 10^11 g/(cm·ms^2) = 210 GPa
  9. VELOCITY: 15.64 cm/ms = 56.33 km/h
  10. GRAVITY: 980 cm/ms^2 = 9.8 m/s^2

Here are the conversions for the given measurements to the Kg cm s system of units:

  1. MASS: 1000 g = 1 kg
  2. LENGTH: 100 cm = 1 m
  3. TIME: 1 s = 1 s
  4. FORCE: 1000 g cm/s^2 = 1 N
  5. STRESS: 1000 g/(cm·s^2) = 1 Pa
  6. ENERGY: 1000 g cm^2/s^2 = 1 Nm
  7. DENSITY: 1000 g/cm^3 = 1 kg/m^3
  8. YOUNG'S MODULUS: 2.1 x 10^11 g/(cm·s^2) = 210 GPa
  9. VELOCITY: 0.5633 m/s = 56.33 cm/s
  10. GRAVITY: 980 cm/s^2 = 9.8 m/s^2

Here are the conversions for the given measurements to the gm mm ms system of units:

  1. MASS 1kg = 1000 gm
  2. LENGTH 1m = 1000 mm
  3. TIME 1s = 1000 ms
  4. FORCE 1N = 100000 dynes
  5. STRESS 1Pa = 1 N/m^2 = 10^4 dynes/cm^2
  6. ENERGY 1Nm = 10^7 ergs
  7. DENSITY 1Kg/m3 = 1000 gm/cm^3
  8. YOUNG'S Modulus 210GPa = 210x10^9 dynes/cm^2
  9. Velocity 56.33KMPH = 15.646 m/s
  10. GRAVITY 9.8m/s2 = 9800 mm/s^2

for,

kg m s:

  1. MASS 1kg = 1 kg
  2. LENGTH 1m = 1 m
  3. TIME 1s = 1 s
  4. FORCE 1N = 1 N
  5. STRESS 1Pa = 1 Pa
  6. ENERGY 1Nm = 1 Nm
  7. DENSITY 1Kg/m3 = 1 kg/m^3
  8. YOUNG'S Modulus 210GPa = 210 GPa
  9. Velocity 56.33KMPH = 15.646 m/s
  10. GRAVITY 9.8m/s2 = 9.8 m/s^2

g cm s:

  1. MASS 1kg = 1000 g
  2. LENGTH 1m = 100 cm
  3. TIME 1s = 1 s
  4. FORCE 1N = 10^5 dynes
  5. STRESS 1Pa = 10 dynes/cm^2
  6. ENERGY 1Nm = 10^7 ergs
  7. DENSITY 1Kg/m3 = 1000 g/cm^3
  8. YOUNG'S Modulus 210GPa = 2.1x10^11 dynes/cm^2
  9. Velocity 56.33KMPH = 15646 cm/s
  10. GRAVITY 9.8m/s2 = 980 cm/s^2

g cm us units:

  1. MASS: 1000g
  2. LENGTH: 100cm
  3. TIME: 10^6 us
  4. FORCE: 10^5 dyn
  5. STRESS: 10 dyn/cm^2
  6. ENERGY: 10^7 erg
  7. DENSITY: 1000 g/cm^3
  8. YOUNG'S MODULUS: 10^11 dyn/cm^2
  9. VELOCITY: 15.648 m/s
  10. GRAVITY: 980 gal (gal is the unit of acceleration in the cgs system, where 1 gal = 1 cm/s^2)

g mm s units:

  1. MASS: 1000g
  2. LENGTH: 1000mm
  3. TIME: 1s
  4. FORCE: 10^6 dyn
  5. STRESS: 10 dyn/mm^2
  6. ENERGY: 10^6 erg
  7. DENSITY: 1000 g/mm^3
  8. YOUNG'S MODULUS: 10^9 dyn/mm^2
  9. VELOCITY: 15648 mm/s
  10. GRAVITY: 9800 mm/s^2

ton mm s units:

  1. MASS: 1 tonne = 1000 kg
  2. LENGTH: 1000mm
  3. TIME: 1s
  4. FORCE: 101.97 kgf
  5. STRESS: 0.10197 kgf/mm^2
  6. ENERGY: 0.10197 kgf.m
  7. DENSITY: 1 kg/mm^3 = 0.001 tonne/mm^3
  8. YOUNG'S MODULUS: 101.97 kgf/mm^2
  9. VELOCITY: 15648 mm/s
  10. GRAVITY: 9.8 m/s^2 = 9800 mm/s^

slug ft s units:

  1. MASS: 2.20462 slugs
  2. LENGTH: 3.28084 ft
  3. TIME: 1s
  4. FORCE: 0.22481 lbf
  5. STRESS: 0.000145038 lbf/ft^2
  6. ENERGY: 0.737562 lbf.ft
  7. DENSITY: 0.0685218 slug/ft^3
  8. YOUNG'S MODULUS: 145.038 lbf/ft^2
  9. VELOCITY: 31.1825 ft/s
  10. GRAVITY: 32.174 ft/s^2

kg mm s units:

  1. MASS: 1000g
  2. LENGTH: 1000mm
  3. TIME: 1s
  4. FORCE: 1000 gf
  5. STRESS: 1 N/mm^2
  6. ENERGY: 1000 gf.mm
  7. DENSITY: 1000 g/mm^3
  8. YOUNG'S MODULUS: 1 N/mm^2
  9. VELOCITY: 15648 mm/s
  10. GRAVITY: 9.8 m/s^2 = 9800 mm/s^2

g cm ms units:

  1. MASS: 1000g
  2. LENGTH: 100cm
  3. TIME: 1000ms
  4. FORCE: 100000 dyn
  5. STRESS: 10 dyn/cm^2
  6. ENERGY: 10000000 erg
  7. DENSITY: 1000 g/cm^3
  8. YOUNG'S MODULUS: 10^11 dyn/cm^2
  9. VELOCITY: 15648 cm/s
  10. GRAVITY: 9800 cm/s^2

kg mm ms:

  1. MASS: 1000 g
  2. LENGTH: 1000 mm
  3. TIME: 1000 ms
  4. FORCE: 1000 gf
  5. STRESS: 1 N/mm^2
  6. ENERGY: 1000 gf.mm
  7. DENSITY: 1 g/mm^3
  8. YOUNG'S MODULUS: 210000 N/mm^2
  9. VELOCITY: 15648 mm/s
  10. GRAVITY: 9.8 m/s^2 = 9800 mm/s^2

Tabulation:

UNITS MASS LENGTH TIME FORCE STRESS ENERGY DENSITY YOUNG'S MODULUS VELOCITY GRAVITY
SI UNITS 1KG 1M 1SEC 1NEWTON 1 PASCAL 1 N-M 1 KG/M^3 210 GPA 56.33 KMPH 9.8 M/S^2
Kg, cm, ms 1 kg 10^2 10^3 10^-4 10^-8 10^-2 10^-6 210*10^1 1.564 cm/ms 9.8*10^-4
Kg, cm , s 1 kg 10^2 1 10^2 10^-2 10^4 10^-6 210*10^7 1.5*10^6 cm/s 9.8*10^2
gm, mm, ms 10^3 10^3 10^3 1 10^-6 10^3 10^-6 210*10^3 15.64 mm/ms 9.8*10^-3
kg, m, s 1 1 1 1 1 1 1 210*10^9 15.64 9.8
gm, cm, s 10^3 10^2 1 10^5 10 10^7 10^-3 210*10^8 15.64*10^2 9.8*10^2
gm, cm, us 10^3 10^2 10^6 10^-6 10^-11 10^-5 10^-3 210*10^-2 1.5*10^-3 9.8*10^-10
gm, mm, s 10^3 10^3 1 10^6 1 10^9 10^-6 210*10^9 1.5*10^3 9.8*10^3
ton, mm, s 10^-3 10^3 1 1 10^-6 10^3 10^-12 210*10^3 1.5*10^3 9.8*10^3
slug, ft, s 0.068 3.28 1 0.22304 0.0207 0.732 2.4 4.3*10^9 51.34 32.144
kg, mm, s 1 10^3 1 10^3 10^-3 10^6 10^-9 210*10^6 5.6*10^7 9.8*10^3
g, cm, ms 10^3 10^2 10^3 10^-1 10^-5 10^1 10^-3 210*10^4 1.564 9.8*10^-4
kg, mm, ms 1 10^3 10^3 10^-3 10^-9 1 10^-9 210 15.64 9.8*10^-3
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