1.

What is the effective density of the maxture of water and milk when (i) they are taken with same masses (ii) they are taken with same volumes

Answer»

Solution :Let US say the DENSITIES of water and milk are `rho_(1) and RHO _(2)`
(i) When they are taken with same mass .m. and their VOLUMES are `V_(1) and V_(2),` the mass of water `m = rho _(1) V_(1) , V_(1) =(m)/(rho_(1))` and the mass of milk `m = rho_(2) V_(2), V_(2) = (m)/( rho _(2))`
Total mass of water and milk is `m + m 2m`
Total volume of the water and milk is `V _(1) + V _(2) = (m)/( rho _(1)) + (m)/( rho _(2))`
`= m ((1)/( rho_(1)) + (1)/( rho _(2 )))`
` = (m ( rho _(1) + rho _(2)))/(rho _(1) rho _(2))`
The effective density of the mixture `(rho _(eff))=` Total mass /Total volume
`= (2m)/( m (rho _(1) + rho _(2)) // rho _(1) rho _(2))`
`= ( 2)/(( rho _(1)+ rho _(2)) // rho_(1) rho _(2))`
` = (2 rho _(1) rho_(2))/( rho_(1) + rho _(2))`
(ii) when they are taken with same volume .V. and their masses are `m _(1) and m _(2)` the volume of water `V = m _(1) // rho _(1)`
That is `m _(1) = V rho _(1)`
and the volume of milk `V = m _(2) // rho _(2)`
Total mass of water and milk is `m _(1) + m _(2) = V rho _(1) + V rho _(2)`
`=V (rho _(1) + rho _(2))`
Total volume of water and milk is `V +V =2V`
The effective density of the mixture `(rho _(eff))=`Total mass/Total volume
`rho _(eff) =(V (rho _(1) + rho _(2))/( 2V)`
`= 1/2 (rho _(1) + rho _(2))`


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