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Buoyancy is an upward force exerted by the fluid in the upward direction that opposes the weight of a fully or partially submerged body. It is equal to the weight of the water displaced. It is expressed in Newtons.

The volume of fluid displaced by a scuba diver and her gear is \({V_{{\rm{displaced}}}} = 69.6\;{\rm{L}}\).

The total mass of the scuba diver and her gear is \({m_{{\rm{diver}}}} = 72.8\;{\rm{kg}}\).

As discussed above,

\({F_{{\rm{buoyant}}}} = {W_{\scriptstyle{\rm{water}}\atop\scriptstyle{\rm{displaced}}}}\)

We know that weight of any matter is equal to the product of its and acceleration due to gravity. Therefore,

\(\begin{array}{l}{F_{buoyant}} ={m_{\scriptstylewater{\rm{water}}\atop\scriptstyle{\rm{displaced}}}}g\\{F_{{\rm{buoyant}}}} = {\rho _{{\rm{water}}}}{V_{{\rm{displaced}}}}g\end{array}\)

Here \({m_{\scriptstyle{\rm{water}}\atop\scriptstyle{\rm{displaced}}}},\;{\rho _{{\rm{water}}}},\;{V_{{\rm{displaced}}}}\) is the mass, density, and volume of water displaced, respectively, and \(g\) acceleration due to gravity.

Now using the density of seawater, i.e., \({\rho _{{\rm{water}}}} = 1.025 \times {10^3}\;{{{\rm{kg}}} \mathord{\left/{\vphantom {{{\rm{kg}}} {{{\rm{m}}^3}}}} \right.\\} {{{\rm{m}}^3}}}\), we have,

\(\begin{array}{l}{F_{{\rm{buoyant}}}} = \left( {1.025 \times {{10}^3}\;{{{\rm{kg}}} \mathord{\left/{\vphantom {{{\rm{kg}}} {{{\rm{m}}^3}}}} \right.\\} {{{\rm{m}}^3}}}} \right)\left( {69.6\;{\rm{L}}} \right)\left( {\frac{{1 \times {{10}^{ - 3}}\;{{\rm{m}}^3}}}{{1\;{\rm{L}}}}} \right)\left( {9.80\;{{\rm{m}} \mathord{\left/{\vphantom {{\rm{m}} {{{\rm{s}}^2}}}} \right.\\} {{{\rm{s}}^2}}}} \right)\\{F_{{\rm{buoyant}}}} = 699\;{\rm{N}}\end{array}\)

Hence, the buoyant force acting on the scuba diver and her gear is \(699\;{\rm{N}}\).

The weight of the diver is

\(\begin{array}{l}{W_{{\rm{diver}}}} = {m_{{\rm{diver}}}}g\\{W_{{\rm{diver}}}} = \left( {72.8\;{\rm{kg}}} \right)\left( {9.80\;{{\rm{m}} \mathord{\left/{\vphantom {{\rm{m}} {{{\rm{s}}^2}}}} \right.\\} {{{\rm{s}}^2}}}} \right)\\{W_{{\rm{diver}}}} = 713\;{\rm{N}}\end{array}\)

Since the buoyant force \(\left( {699\;{\rm{N}}} \right)\) is not as large as her weight, she will sink, although she will gradually sink since the two forces are almost the same.