Buoyant Force in Accelerating Fluids
Suppose a body is dipped inside a liquid of density \(
\rho _L
\) placed in an elevator moving with an acceleration \(
\mathop a\limits^ \to
\). The buoyant force F in this case becomes, \(
F = V\rho _L g_{eff}
\)
Here, \(
g_{eff} = \left| {\mathop g\limits^ \to - \mathop a\limits^ \to } \right|
\)
For example, if the lift is moving upwards with an acceleration a, the value of geff is g + a and if it is moving downwards with acceleration a, the geff is g – a. In a freely falling lift geff is zero (as a = g) and hence, net buoyant force is zero. This is why, in a freely falling vessel filled with some liquid, the air bubbles do not rise up (which otherwise move up due to buoyant force). The above result can be derived as follows.
Suppose a body is dipped inside a liquid of density in an elevator moving up with an acceleration a. As was done earlier also, replace the body into the liquid by the same liquid of equal volume. The replaced liquid is at rest with respect to the elevator. Thus, this replaced liquid is also moving up with an acceleration a together with the rest of the liquid.
The forces acting on the replaced liquid are,
i) the buoyant force F and
ii) the weight mg of the substituted liquid.
From Newton’s second law,
F – mg = ma (or) F = m(g + a)
Here, \(
m = V\rho _L
\)
\(
\therefore F = V\rho _L \left( {g + a} \right) = V\rho _L g_{eff}
\)
where geff = g + a
Note: In case of thrust force in accelerating fluids the thrust force (Buoyant force) contains two parts
i) \(
\mathop F\limits^ \to _1 = V\rho _L \mathop a\limits^ \to
\) and ii) \(
\mathop F\limits^ \to _2 = V\rho _L \mathop g\limits^ \to
\) , then the thrust force in magnitude and direction is given by \(\vec F\) where \(
\mathop F\limits^ \to = \mathop F\limits^ \to _1 - \mathop F\limits^ \to _2 = V\rho _L \mathop a\limits^ \to - V\rho _L \mathop g\limits^ \to
\), Here \(\vec g\) is always in the downward direction and V is the volume of the liquid displaced.
A body of volume V and density \(\rho_b\)is floating in a liquid of density \(\rho\) with volume Vin immersed in the liquid \(
\therefore V\rho _b = V_{in} \rho
\)