Question

A slab of ice floats on a freshwater lake. What minimum volume must the slab have for a 45.0 kg woman to be able to stand on it without getting her feet wet?

 

Answer 1

Concepts and reason

The concept required to solve this question is the buoyant force. Initially, write the expression for the buoyant force of the water and the force due to her and the ice weight. Then, rearrange the expression for the volume. Finally, substitute the given values in the expression of the volume.

Fundamentals

A body experiences an upward force when it is immersed in a liquid. This force will be equal to the weight of the fluid displaced by the body when it is partially or fully immersed in the liquid. This upward force exerted by the fluid on the body is known as the Buoyancy force.

 

The expression of the buoyant force \(F_{\mathrm{B}}\) for the water is, \(F_{\mathrm{B}}=\rho_{\mathrm{w}} V g\)

Here, \(\rho_{\mathrm{w}}\) is the density of water, \(\mathrm{V}\) is the volume, and \(\mathrm{g}\) is the acceleration due to gravity. The total weight \(F_{\mathrm{T}}\) when the woman standing on the ice slab is given by, \(F_{\mathrm{T}}=\rho_{\mathrm{ice}} V g+m g\)

Here, \(\rho_{\text {ice }}\) is the density of the ice, \(\mathrm{V}\) is the volume, \(\mathrm{g}\) is the acceleration due to gravity, and \(\mathrm{m}\) is the mass of the

woman. The buoyant force is equal to the total weight and it is given as:

\(\rho_{\mathrm{w}} V g=\rho_{\mathrm{ice}} V g+m g\)

The buoyant force depends on the density of the fluid, the volume, and the acceleration due to gravity. For the floating of ice slab on the lake, the buoyant force should be equal to the weight of the object.

 

The buoyant force is equal to the total weight and it is given as:

\(\rho_{\mathrm{w}} V g=\rho_{\mathrm{ice}} V g+m g\)

Rearrange the expression for the V.

$$ \begin{array}{c} \rho_{\mathrm{w}} V g-\rho_{\mathrm{ice}} V g=m g \\ \left(\rho_{\mathrm{w}}-\rho_{\mathrm{ice}}\right) g V=m g \\ \left(\rho_{\mathrm{w}}-\rho_{\mathrm{ice}}\right) V=m \\ V=\frac{m}{\left(\rho_{\mathrm{w}}-\rho_{\mathrm{ice}}\right)} \end{array} $$

Substitute \(45.0 \mathrm{~kg}\) for \(\mathrm{m}, 1000 \mathrm{~kg} / \mathrm{m}^{3}\) for \(\rho_{\mathrm{w}},\) and \(920 \mathrm{~kg} / \mathrm{m}^{3}\) for \(\rho_{\mathrm{ice}}\) in the above equation.

\(V=\frac{45.0 \mathrm{~kg}}{\left(1000 \mathrm{~kg} / \mathrm{m}^{3}-920 \mathrm{~kg} / \mathrm{m}^{3}\right)}\)

\(=0.5625 \mathrm{~m}^{3}\)

The minimum volume for the slab is \(0.5625 \mathrm{~m}^{3}\).

The Archimedes principle states that the buoyant force on a body is equal to the weight of the fluid displaced by the body.

The minimum volume for the slab is 0.5625 m³ .