the dimensional formula of Youngs modulus is ML^-1T^-2 with units as N/m²

ML^-1T^-2 here M in kgs, L in meters, T in secs

   which is 1000grams/[100 cms*(10*10^-1)² secs²]

in terms of 10 g, 100 cm, and 1/10 sec respectively we have

100*(10 grams) / (100 cms) * 100*(1/10 sec)²

so Youngs modulus of metal is 10^10 [(10 grams) / (100 cms)*(1/10 sec)²]

the old unit for young's modulus is N/m^2.=>Kg.m/s^(-2).1/m^2

                                                                          =>kg.m^(-1).s^(-2)

                                                                         =>1000g.(100cm)^(-1).(10.(1/10)s)^(-2)

                                                                         =>100.10g.(100cm)^(-1).(1/100).(1/10)s)^(-2)

                                                                         =>100.(1/100).[10g.(100cm)^(-1).(1/10)s)^(-2)]

                                                                         =>new unit of youngs modulus

        hence in new system also magnitude of youngs modulus is 10^10 new units.

Hi

1N = 1 kg.m/s^2
      = (1000 g)(100 cm) / (10 x 1/10 s)^2
      = (100 x 10g)(1 x 100cm) / (100 x (1/10s)^2)        in this step we are trying to convert in the required system of units
      = 1 (units of new system)
1m = 100 cm
      = 1 (units of new system)

this implies 1 N/m^2 = 1 (units in new system)
Young's modulus in new system = 10^10 units

Let the new units be

Dg = 10 g

m = 100 cm (is the same as meter) , and

ds = 1/10 s

Now, 1 N = 1 kg. m/s²

So we can write 10¹⁰ N/m² as

10¹⁰ kg m/(m²s²) = 10¹⁰ kg/(ms²) 

Now we use two key ideas:

1. Multiplying by 1 does not make any difference

2. Since 1 kg = 1000 g, we can also say that (1000g/1 kg) , or (100 Dg/1 kg) = 1

Similary,

(1 s/10 ds)  = 1.

Squaring both sides, we get

(1 s²/100 ds²) = 1

Hence, 10¹⁰ kg/(ms²) = ( 10¹⁰ kg/ms² )(1 s²/100 ds²) (100 Dg/1 kg)

= 10¹⁰ Dg/(m(ds)²) (cancelling out kg and s²)