Week - 1 - Consistency of Units
CHALLENGE 1:
For conversion from SI unit to gm/mm/ms unit system:
The SI unit is represented by mass in 'kg', length in 'm' and time in 's'. Other quantities are calculated/derived from the fundamental quantities. Given in the problem are:
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
To convert the above mentioned quantities to gm/mm/ms system of units:
1 kg = 1000 gm
1 m = 1000 mm
1 s = 1000 ms
FORCE
Force of 1 N in SI units is given by
Force(N) = Mass(m) X Acceleration due to gravity(g)
1 N = 1 kg X 1 m/s2
Converting the above into gm/mm/ms form we get,
1N = 1000 gm * 1000 mm/ 10002 ms2
1 N = 1.00 gm mm/ms2
STRESS
The definition of stress is given by force acting normally per unit area. Stress is denoted by sigma (σ) and its SI unit is Pa
Stress = Force/Area
1 Pa = 1 N/1 m2
1 Pa = 1 gm mm /ms2 / 0.0012 ms2
1 Pa = 106 gm /mm s2
ENERGY
The definition of energy is given by amount of work done i.e force * displacement. The unit of energy is Nm.
Energy = Force * displacement
1 N-m = 1 N * 1m
1 N-m = 1.00 gm mm/ms2 * 0.001 mm
1 N-m = 10-3 gm-mm2/ms2
DENSITY
Density is defined as mass per unit volume. The unit of density is kg/m3 in SI unit.
Density = mass (kg) / volume(m3)
1 kg/m3 = 0.001 gm / (0.001)3
1 kg/m3 = 106 gm/mm3
YOUNG’S MODULUS
Young’s Modulus denotes the elastic strength of a material. The unit of Young’s modulus is GPa (Giga Pascal).
For the given material, the value of Young’s modulus is 210 GPa.
To convert to gm/mm/ms
210 GPa = 210 * 109 * 106 gm – mm/ms2 / mm2
210 GPa = 2.10 * 1018 gm-mm/ms2
VELOCITY
Velocity is displacement per unit time. Its unit in SI system is m/s. However given in the problem is 56.33 kmph.
56.33 kmph = 56.33 * 103 * 0.001 / 3600 * 0.001
56.33 kmph = 15.37 mm/ms
GRAVITY
Gravity is same as acceleration due to gravity and is calculated from the law of gravitational force equation. Its value on earth is estimated as 9.8 m/s2 in SI unit.
9.8 m/s2 = 9.8 * 0.001 / 0.0012
9.8 m/s2 = 9800 mm/ms2
For system of units, the conversion factors are given below in the table below:
| Mass | Length | Time | Force | Stress | Energy | Density | Young's Modulus | Velocity | Gravity | |
| kg m s | 1 kg | 1 m | 1 s | 1 N | 1 Pa | 1 Nm | 1 kg/m3 | 210GPa | 56.33 kmph | 9.8 m/s2 |
| CONVERSION FACTORS FROM SI | ||||||||||
| Kg cm ms | 1.00 | 0.01 | 0.001 | 1.00E+04 | 1.00E+08 | 1.00E+02 | 1.00E+06 | 2.10E+20 | 153.690 | 9.80E+04 |
| Kg cm s | 1.00 | 0.01 | 1.00 | 1.00E-02 | 1.00E+02 | 1.00E-04 | 1.00E+06 | 2.10E+14 | 0.154 | 0.09800 |
| gm mm ms | 0.001 | 0.001 | 0.001 | 1.00E+00 | 1.00E+06 | 1.00E-03 | 1.00E+06 | 2.10E+18 | 15.369 | 9.80E+03 |
| kg m s | 1.00 | 1.00 | 1.00 | 1.00E+00 | 1.00E+00 | 1.00E+00 | 1.00E+00 | 2.10E+12 | 15.369 | 9.80000 |
| g cm s | 0.001 | 0.01 | 1.00 | 1.00E-05 | 1.00E-01 | 1.00E-07 | 1.00E+03 | 2.10E+11 | 0.154 | 0.09800 |
| g cm us | 0.001 | 0.01 | 1.00E-06 | 1.00E+07 | 1.00E+11 | 1.00E+05 | 1.00E+03 | 2.10E+23 | 1.54E+05 | 9.80E+10 |
| g mm s | 0.001 | 0.001 | 1.00 | 1.00E-06 | 1.00E+00 | 1.00E-09 | 1.00E+06 | 2.10E+12 | 0.0154 | 0.00980 |
| ton mm s | 1000 | 0.001 | 1.00 | 1.00E+00 | 1.00E+06 | 1.00E-03 | 1.00E+12 | 2.10E+18 | 0.0154 | 0.00980 |
| slug ft s | 14.5939 | 0.3048 | 1.00 | 4.45E+00 | 4.79E+01 | 1.36E+00 | 5.15E+02 | 1.01E+14 | 4.6845 | 2.98704 |
| kg mm s | 1.00 | 0.001 | 1.00 | 1.00E-03 | 1.00E+03 | 1.00E-06 | 1.00E+09 | 2.10E+15 | 0.0154 | 0.00980 |
| g cm ms | 0.001 | 0.01 | 0.001 | 1.00E+01 | 1.00E+05 | 1.00E-01 | 1.00E+03 | 2.10E+17 | 153.6900 | 9.80E+04 |
| kg mm ms | 1.00 | 0.001 | 0.001 | 1.00E+03 | 1.00E+09 | 1.00E+00 | 1.00E+09 | 2.10E+21 | 15.3690 | 9.80E+03 |
Please follow the link for soft copy files of assignment
https://drive.google.com/drive/folders/16BgSuJIGae2qHzPW9OXYL8VIFFJ9OMYg?usp=sharing