Question

Caught in an avalanche, a skier is fully submerged in flowing snow of density$\mathbf{96}\mathbf{}\mathit{k}\mathit{g}\mathbf{/}{\mathit{m}}^{\mathbf{3}}$ . Assume that the average density of the skier, clothing, and skiing equipment is $\mathbf{1020}\mathbf{}\mathit{k}\mathit{g}\mathbf{/}{\mathit{m}}^{\mathbf{3}}$. What percentage of the gravitational force on the skier is offset by the buoyant force from the snow?

Short answer

The percentage of the gravitational force offset by the buoyant force is $9.4\%$.

Step-by-step solution

Listing the given quantities

• Density of snow${\rho }_{snow}=96kg/m^3$.
• Density of skier${\rho }_{skier}=1020kg/m^3$ .

Understanding the concept of buoyant force

The buoyant force is the upward force exerted by the liquid on the object that is fully or partially immersed in the fluid. The force is equal to the product of the density of the fluid, gravitational acceleration, and the volume of the immersed object.

By using the relation$\mathbf{}\mathit{F}\mathbf{=}\mathit{\rho }\mathit{g}\mathit{V}$,we can find the percentage of gravitational force offset by the buoyant force.

Formula:

$$\begin{gathered} F_g=mg \\ F=\rho gV \end{gathered}$$

Here,

$F_g$is the weight.

m is the mass of the object.

g is the gravitational acceleration.

F is buoyant force.

p is the density of the fluid.

(a) Calculation of the percentage of the gravitational force offset by the buoyant force

Gravitational force is given by

$\begin{array}{rcl}{F}_g& =& mg\\ & =& {\rho }_{skier}gV\\ & & \end{array}$

Buoyant force is,

$$\begin{gathered} F_b={\rho }_{snow}gV \\ \frac{F_b}{F_g}=\frac{{\rho }_{snow}gV}{{\rho }_{skier}gV} \\ \frac{F_b}{F_g}=\frac{{\rho }_{snow}}{{\rho }_{skier}} \end{gathered}$$

Substitute the given values in the above equation.

$\begin{array}{rcl}\frac{F_b}{F_g}& =& \frac{96{\text{kg/m}}^3}{1020{\text{kg/m}}^3}\\ & =& 0.094\\ & =& 9.4\%\\ & & \end{array}$

Percentage of the gravitational force offset by the buoyant force$9.4\%$