If the velocity of light (c), gravitational constant (G), and Plancks' constant (h) are chosen as fundamental units, then find the dimensions of mass in new system.
Let m∝[cxGyhz]
or m=K[cxGyhz]
By substituting the dimension of each quantity in both the sides,
[M1L0T0]=K[LT−1]x[M−1L3T−2]y[ML2T−1]z
M=[M−y+zLx+3y+2zTx−2y−z]
By equating the power of M, L and T in both the sides :
−y+z=1,x+3y+2z=0,−x−2y−z=0
By solving above three equations,
x=1/2,y=−1/2,andz=1/2.
∴m∝c1/2G−1/2h1/2