Chapter 9, Problem 4SP

(a)

To determine

The total weight of boat and people in newtons.

Answer to Problem 4SP

The total weight of boat and people in newtons is $13720\text{N}$.

Explanation of Solution

Given info: The length of the wooden box is $2.5\text{ m}$, its width is $1.5\text{ m}$ and the height of the box is $1\text{m}$. The total mass of the boat and the people is $1400\text{kg}$.

Write the expression to find the weight of the boat and people when the mass is given.

$W=Mg$

Here,

$W$ is the weight of the boat and the people

$M$ is the mass of the boat and the people

$g$ is the acceleration due to gravity

Substitute $1400\text{kg}$ for $M$ and $9.8\text{m}/{\text{s}}^2$ for $g$ to find the weight of the boat and the people.

$\begin{array}{c}W=\left(1400\text{kg}\right)\left(9.8\text{m}/{\text{s}}^2\right)\\ =13720\text{N}\end{array}$

Conclusion:

Therefore, the total weight of boat and people in newtons is $13720\text{N}$.

(b)

To determine

The buoyant force required to keep the boat and the people afloat.

Answer to Problem 4SP

The buoyant force required to keep the boat and the people afloat is $13720\text{ N}$.

Explanation of Solution

Write the expression to find the buoyant force.

$F_b=W$

Here,

$F_b$ is the buoyant force

Substitute $13720\text{ N}$ for $W$ in the above equation.

$F_b=13720\text{ N}$

Conclusion:

Therefore, the buoyant force required to keep the boat and the people afloat is $13720\text{ N}$.

(c)

To determine

The volume of the water displaced to support the boat and the people in it.

Answer to Problem 4SP

The volume of the water displaced to support the boat and the people in it is $1.4{\text{m}}^3$.

Explanation of Solution

Write the expression to find the volume of the boat.

$V=lbh$

Here,

$l$ is the length of the boat

$b$ is the width of the boat

$h$ is the height of the boat

Substitute $2.5\text{ m}$ for $l$, $1.5\text{ m}$ for $b$ and $1\text{m}$ for $h$ in the above equation.

$\begin{array}{c}V=\left(2.5\text{ m}\right)\left(1.5\text{ m}\right)\left(1\text{m}\right)\\ =3.75{\text{ m}}^3\end{array}$

Write the expression to find the density of the boat.

${\rho }_b=\frac{M}{V}$

Here,

$M$ is the mass of the boat

$V$ is the volume of the boat

${\rho }_b$ is the density of the boat

Substitute $1400\text{kg}$ for $M$ and $3.75{\text{ m}}^3$ for $V$ in the above equation.

$\begin{array}{c}{\rho }_b=\frac{1400\text{kg}}{3.75{\text{ m}}^3}\\ =373.33\text{kg}/{\text{m}}^3\end{array}$

Write the expression to find the volume of the water displaced.

$V_w=\frac{{\rho }_b}{\rho }V$

Here,

${\rho }_b$ is the density of boat

$\rho$ is the density of water

Substitute $373.33\text{kg}/{\text{m}}^3$ for ${\rho }_b$, $3.75{\text{ m}}^3$ for $V$ and $1000\text{kg}/{\text{m}}^3$ for $\rho$ in the above equation.

$\begin{array}{c}{V}_w=\frac{373.33\text{kg}/{\text{m}}^3}{1000\text{kg}/{\text{m}}^3}\left(3.75{\text{ m}}^3\right)\\ =1.4{\text{m}}^3\end{array}$

Conclusion:

Therefore, the volume of the water displaced to support the boat and the people in it is $1.4{\text{m}}^3$.

(d)

To determine

The height of the boat underwater.

Answer to Problem 4SP

The height of the boat underwater is $0.373\text{ m}$.

Explanation of Solution

Write the expression to equate the volume of boat immersed to the volume of the water displaced.

$V_w=lb{h}_n$

Here,

$h_n$ is the new height of the boat underwater

Substitute $2.5\text{ m}$ for $l$, $1.5\text{ m}$ for $b$ and $1.4{\text{m}}^3$ for $V_w$ in the above equation and rearrange to find the height of the boat underwater.

$\begin{array}{c}{h}_n=\frac{V_w}{lb}\\ =\frac{1.4{\text{m}}^3}{\left(2.5\text{ m}\right)\left(1.5\text{ m}\right)}\\ =0.373\text{ m}\end{array}$

Conclusion:

Therefore, the height of the boat underwater is $0.373\text{ m}$.