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Find the dimensional formulae of the fol...

Find the dimensional formulae of the following quantities:
a. the universal constant of gravitation G,
b. the surface tension S,
c. the thermal conductivity k and
d. the coeficient of viscosity `eta`.
Some equation involving these quantities are
`F=(Gm_1m_2)/r^2, S= (rho g r h)/2,`
`Q=k(A(theta_2-theta_1)t)/d and F=- rho A (v_2-v_1)/(x_2-x_1)`
where the symbols have their usual meanings.

Text Solution

Verified by Experts

a. `F= G (m_1-m_2)/r^2`
` or , G=(Fr^2)/(m_1m_2)`
or, ` [G] =([F]L^2)/M^2 = (MLT^(-1))/M^2 = M^-1 L^3 T^(-2)`
` b. S= (rho g r h )/2`
or ` [S]=[rho][g]L^2 = M/L^3 L/T^(-2)`
c. `Q=k(A(theta_2-theta_1)t)/d`
` k=(Qd)/(A(theta_2-theta_1)t)`
Here Q is the heat energy having dimension `ML^2T^-1, theta_1-theta_1` is tem-erature, A is rea, d is thickness and t is true. Thus,
`[K] = (ML^2T^-2L)/(L^2 KT) = MLT^-3 K^-1`
d ` F=- eta A (v_2-v_1)/(x_2-x_1)`
or, `MLT^2= [eta] L^2 (L/T)/L = [eta] L^2/T`
or, ` [eta]= ML^(-1) T^(-1)`
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Knowledge Check

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