If young modulus of a metal is 10^10 N/m^2, express it in a new system of units where the units of mass, length and time are 10 g, 100 cm, and 1/10 sec respectively.

							
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)