# Effect of Temperature on Upthrust

## Effect of Temperature on Upthrust

In this article we are going to discuss about the Effect of Temperature on Upthrust as the temperature of a body which is changed.

We know that whenever a body is immersed in a liquid, partially or completely, its weight appears to be less in comparison to its weight in air.

This is because of the force applied by the liquid on the body in vertically upward direction (opposite to the direction of weight), which is called UPTHRUST or BUOYANT FORCE, given by :-

F = Vs ρL g     ……….(1)

Where,

Vs = Volume of the solid body immersed in the liquid

ρL = Density of the liquid   &

g = acceleration due to gravity.

The apparent weight (weight when immersed in the liquid), Wapp of the body is given by :-

Wapp = W – F     ……….(2)

Where W is the weight of the body as measured in air.

Now as we increase the temperature, Vs increases ( volume expansion )  and ρL decreases

So upthrust F, may increase or may decrease or may remains the same, depending on the factor which dominates [from equation (1)].

Let us discuss these factors one by one. From equation (1) :-

$\displaystyle F={{V}_{S}}{{\rho }_{L}}g$     ……….(3)

Now let the temperature changes by ΔT, then new upthrust F’

$\displaystyle {{F}^{'}}=V_{_{S}}^{'}\rho _{_{L}}^{'}g$     ……….(4)

Dividing equation (4) by equation (3), we get :-

$\displaystyle \Rightarrow \frac{{{F}^{'}}}{F}=\frac{V_{_{S}}^{'}\rho _{_{L}}^{'}g}{{{V}_{S}}{{\rho }_{L}}g}$

$\displaystyle \Rightarrow \frac{{{F}^{'}}}{F}=\frac{V_{_{S}}^{'}\rho _{_{L}}^{'}}{{{V}_{S}}{{\rho }_{L}}}=\frac{{{V}_{S}}(1+{{\gamma }_{S}}\Delta T)}{{{V}_{S}}}\times \frac{1}{{{\rho }_{L}}}\frac{{{\rho }_{L}}}{(1+{{\gamma }_{L}}\Delta T)}$

$\displaystyle \Rightarrow {{F}^{'}}=F\frac{(1+{{\gamma }_{S}}\Delta T)}{(1+{{\gamma }_{L}}\Delta T)}$     ……….(5)

Now if [from eq. (5) and (2)]

(a) γS > γL , then F’ > F     ⇒ W’app <  Wapp

(b) γS < γL , then F’ < F     ⇒ W’app >  Wapp     and if

(c) γS = γL , then F’ = F     ⇒ W’app =  Wapp