Week - 1 - Consistency of Units

Objective:

To convert the given measurements in the below mentioned 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 .

A system of units:

The system of units is the complete set of units, both fundamental units and derived units, for all kinds of physical quantities. Each system is named with reference to the fundamental units on which it is based.

The quantities which we can measure directly or indirectly are known as physical quantities, the physical quantities are divided into two categories as :

• Fundamental Quantities
• Derived Quantities

Fundamental Quantities:

The physical quantities that do not depend upon the other quantities are the fundamental quantities

Derived Quantities:

The physical quantities that depends on the fundamental quantities are called derived quantities

Now to covert the given measurements in to the asked unit system as follows:

1.Conversion of Kg,m,s

• Mass = 1kg
• Length = 1m
• Time = 1s
• Force = 1N `implies``(kg*m)/(s^2)`

Conversion Factor = 1

• Stress = 1Pa `implies``(kg)/(m*s^2)`

Conversion Factor = 1

• Energy = 1Nm `implies``(kg*(m^2))/(S^2)`

Conversion Factor = 1

• Density = 1 `(kg)/(m^3)`

Conversion Factor = 1

• Youngs modulus = 210 Gpa `implies``210*10^9``(kg)/(m*s^2)`

Conversion Factor = 10^9

• Velocity = 56.33 kmph = 15.647 m/s

Conversion Factor = 1

• Gravity = 9.81 `(m)/(s^2)`

Conversion Factor = 1

2. Conversion of Kg,cm,ms

• Mass = 1kg
• Length 1m = `10^2`Cm
• Time 1s = `10^3`ms
• Force = 1N `implies``(kg*m)/(s^2)`

= `10^2``(kg*cm)/(10^6(ms^2))`

= `10^-4``(kg*cm)/(10^6(ms^2))`

Conversion Factor = 10^-4

• Stess = 1Pa `implies``(kg)/(m*s^2)`

= `10^-8``(kg)/(cm*ms^2)`

Conversion Factor = 10^-8

• Energy = 1Nm `implies``(kg*(m^2))/(S^2)`

= 10^4 `(kg*(cm^2))/10^6(ms^2)`

= 10^-2 `(kg*(cm^2))/(ms^2)`

Conversion Factor = 10^-2

• Density = 1 `(kg)/(m^3)`

= `(kg)/10^6(cm^3)`

=10^-6 `(kg)/(cm^3)`

Conversion Factor = 10^-6

• Youngs modulus = 210 Gpa `implies``210*10^9``(kg)/(m*s^2)`

= 2100kg/(cm*ms³)

• Velocity = 56.33 kmph = 15.647 m/s

= 15.647 * `10^2` cm/10^3 ms

= 1.5647 cm/ms

• Gravity = 9.81 `(m)/(s^2)`

= 9.8 * `10^-4``(cm)/(ms^2)`

3. Conversion of gm,mm,ms

• Mass 1kg = 1000g = `10^3`g
• Length 1m = `10^3`mm
• Time 1s = `10^3`ms
• Force = 1N `implies``(kg*m)/(s^2)`

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

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

Conversion Factor = `10^2`

• Stess = 1Pa `implies``(kg)/(m*s^2)`

= `10^3``(g)/`10^9`(mm*ms^2)`

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

Conversion Factor = 10^-6

• Energy = 1Nm `implies``(kg*(m^2))/(S^2)`

= 10^9 `(g*(mm^2))/(10^6(ms^2))`

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

Conversion Factor = 10^3

• Density = 1 `(kg)/(m^3)`

= 10^3``(g)/10^9(cm^3)`

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

Conversion Factor = 10^-6

• Youngs modulus = 210 Gpa `implies``210*10^9``(kg)/(m*s^2)`

= 210*10^3` g/(mm*ms³)

• Velocity = 56.33 kmph = 15.647 m/s

= 15.647 * `10^3` mm/10^3 ms

= 15.647 mm/ms

• Gravity = 9.81 `(m)/(s^2)`

= 9.8 * `10^-3``(mm)/(ms^2)`

The converted values of the rest of the unit System are Plotted as a table below:

Unit system (SI unit)	Mass	Length	Time	Force	Stress	Energy	Density	Youngs 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)`	2100kg/(cm*ms³)	1.5647 cm/ms	9.8 * 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*s^2)`	10^4 `(kg(cm^2))/(s^2)`	10^-6 `(kg)/(cm^3)`	210*e⁷kg/(cm*s²)	1564.7cm/s	9.8*10² cm/s²
g,mm,ms	`10^3` g	`10^3` mm	`10^3` ms	`1`(g*mm)/(ms)^2`	`10^-6``(g)/(m*ms^2)`	10^3 `(g(mm^2))/(ms^2)`	10^-6 `(g)/(m^3)`	210e³g/(mm*ms²)	15.647mm/ms	9.8*10⁻³ mm/ms²
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*e⁹ kg/ m*s²	1564.7 m/s	9.81m/s²
g,cm,s	`10^3` g	`10^2` cm	1 s	`10^5``(g*cm)/(s)^2`	`10``(g)/(cm*s^2)`	10^7 `(g(cm^2))/(s^2)`	10^-3 `(g)/(cm^3)`	210*e¹⁰g/(cm*s²)	1564.7cm/s	9.8*10² cm/s²
g,cm,us	`10^3` g	`10^2` cm	`10^6` us	`10^-7``(g*cm)/(us)^2`	`10^-11``(g)/(cm*us^2)`	10^-5 `(g(cm^2))/(us^2)`	10^-3 `(g)/(cm^3)`	2.10g/cm*us²	15.647*10⁻⁴cm/us	9.8*10⁻¹⁰ cm/us²
g,mm,s	`10^3`g	`10^3` mm	1 s	`10^6``(g*mm)/(s)^2`	`1``(g)/(mm*s^2)`	10^9 `(g(mm^2))/(s^2)`	10^-6 `(g)/(mm^3)`	2.10e¹¹g/mm*s²	15647mm/s	9.8*10³mm/s²
ton,mm,s	`10^-3`g	`10^3` mm	1 s	`1``(ton*mm)/(s)^2`	`10^-6``(ton)/(mm*s^2)`	10^3 `(ton(mm^2))/(s^2)`	10^-12 `(ton)/(mm^3)`	210 e³ ton/(mm*s²)	15647 mm/s	9.8*10³ mm/s²
slug,ft,s	0.068 slug	3.28 ft	1 s	0.22468 `(slug*ft)/(s)^2`	0.0208`(slug)/(ft*s^2)`	0.7366 `(slug(ft^2))/(s^2)`	1.94*10^-3`(slug)/(ft^3)`	4.368 e⁹ slug/(ft*s²)	51.322 ft/s	32.144ft/s²
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)`	210e⁶kg/mm*s²	15647mm/s	9.8*10³mm/s²
g,cm,ms	`10^3`g	`10^2` cm	`10^3` ms	`10^-1``(g*cm)/(ms)^2`	`10^-5``(g)/(cm*ms^2)`	10 `(g(cm^2))/(ms^2)`	10^-3 `(g)/(cm^3)`	210e⁴g/cm*ms²	1.5647cm/ms	9.8*10⁻⁴cm/ms²
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)`	210kg/(mm*ms²)	15.647mm/ms	9.8*10⁻³mm/ms²

Conclusion:

The given values are coverted into specific unit system as asked inthe question and the values are tabulated.