Deduce the dimensional formula for the following quantities (i) Gravitational constant (ii) Young’s
modulus (iii) Coefficient of viscosity.

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$i) M a g n i t u d e o f g r a v i t a t i o n a l f o r c e i s g i v e n b y, F = \frac{G m_{1} m_{2}}{r^{2}}; w h e r e G \to G r v a i t a t i o n a l c o n s \tan t, m_{1} \to m a s s o f 1^{s t} b o d y, m_{2} \to m a s s o f 2^{n d} b o d y, r \to s e p e r a t i o n b e t w e e n t h e t w o b o d i e s \Rightarrow G = \frac{F r^{2}}{m_{1} m_{2}} D i m e n s i o n s o f F = M L T^{- 2} D i m e n s i o n s o f r = L D i m e s n i o n s o f m a s s (m_{1}, m_{2}) = M T h e r f e o r e d i m e n s i o n a l f o r m u l a o f G = \frac{[M L T^{- 2}] [L^{2}]}{[M] [M]} = M^{- 1} L^{3} T^{- 2} i i) Y o u n g' s M o d u l u s i s g i v e n b y, Y = \frac{l i n e a r s t r e s s}{l i n e a r s t r a i n} L i n e a r s t r e s s = \frac{I n t e r n a l F o r c e}{A r e a} T h e r e f o r e, d i m e n s i o n s o f l i n e a r s t r e s s = \frac{M L T^{- 2}}{L^{2}} = M L^{- 1} T^{- 2} L i n e a r s t r a i n = \frac{c h a n g e i n l e n g t h}{o r i g i n a l l e n g t h} \Rightarrow l i n e a r s t r a i n i s d i m e s n i o n l e s s . T h e r e f o r e, d i m e n s i o n a l f o r m u l a o f Y o u n g' s M o d u l u s = M L^{- 1} T^{- 2}$

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