Consider a cylindrical container of cross-section area A length h and having coefficient of linear expansion `alpha_(c)`. The container is filled by liquid of real expansion coefficient `gamma_(L)` up to height `h_(1)`. When temperature of the system is increased by `Deltatheta` then
(a). Find out the height, area and volume of cylindrical container and new volume of liquid.
(b). Find the height of liquid level when expansion of container is neglected.
(c). Find the relation between `gamma_(L)` and `alpha_(c)` for which volume of container above the liquid level
(i) increases
(ii). decreases
(iii). remains constant.
(d). On the surface of a cylindrical container a scale is attached for the measurement of level of liquid of liquid filled inside it. If we increase the temperature of the temperature of the system by `Deltatheta`, then
(i). Find height of liquid level as shown by the scale on the vessel. Neglect expansion of liquid.
(ii). Find the height of liquid level as shown by the scale on the vessel. Neglect expansion of container.

Answer

(a) `h_(f) = h { 1+ alpha_(c) Delta theta}`
`A_(f) =A{1+2 alpha_(c) Delta theta}`
`V_(f) = Ah {1+3 alpha_(c) Delta theta}`
volume of liquid `V_(w) =Ah_(1) (1+gamma_(L) Delta theta)`
(b) `h_(f) = h_(1) {1+gamma_(L)Delta theta}`
(c ) (i) `3h alpha_(a) gt h_(1) gamma_(l)`
(ii) `3h alpha_(c) lt h_(1) gamma_(l)`
(iii) `3h alpha_(c) = h_(1) gamma_(l)`
(d) `DeltaV = Ah (gamma_(L) - 3 alpha_(c)) Delta theta`
(i) `h_(f) = h_(1) (1-3 alpha_(c) Delta theta)`
(ii) `h_(f) = h_(1) (1+ gamma_(L) Delta theta)`
(iii) `(1) gamma_(L) gt 2 alpha_(c)`
(2) `gamma_(L) lt 2 alpha_(c)`
(3) `gamma_(L) = 2 alpha_(c)`

Explanation

(a) `h_(f) = h {1 +alpha_(c) Delta theta}`
`A_(f) = A {1+2 alpha_(c) Delta theta}`
`v_(f) = Ah {1+ 3 alpha_(c) Delta theta}`
volume of liquid `V_(w) = Ah_(1) (1+gamma_(L)Delta theta)`
(b) `h_(f) = h_(1) {1+gamma_(L)Delta theta}`
(c ) (i) `3h alpha_(c) gt h_(1) gamma_(l)` (ii) `3halpha_(c) lt h_(1) gamma_(l)` (iii) `3h alpha_(c) = h_(1) gamma_(l)`
(d) `DeltaV = Ah (gamma_(L) - 3 alpha_(c)) Delta theta`
(e) (i) `h_(f) = h_(1) (1-3 alpha_(c) Delta theta)` (ii) `h_(f) = h_(1) (1+ gamma_(L)Delta theta)`
(iii) `(1) gamma_(L) gt2 alpha_(c) (2) gamma_(L) lt 2 alpha_(c) (3) gamma_(L) = 2 alpha_(c)`