Chapter 9, Problem 4SP

A flat-bottomed wooden box is 2.8 m long and 1.3 m wide, with sides 1 m high. The box serves as a boat carrying five people as it floats on a pond. The total mass of the boat and the people is 1600 kg.

1. a. What is the total weight of the boat and the people in newtons?
2. b. What is the buoyant force required to keep the boat and its load afloat?
3. c. What volume of water must be displaced to support the boat and its load? (The density of water is 1000 kg/m³.)
4. d. How much of the boat is underwater? (What height of the sides must be submerged to produce a volume equal to that of part c?)

To determine

Total weight of boat and people.

Answer to Problem 4SP

The total weight of people and boat is $15,680\text{N}$

Explanation of Solution

Given info: length of boat is $2.8\text{m}$ , width of boat is $1.3\text{m}$, total mass of people and boat is $1600\text{kg}$

Write the equation to find the weight.

$W=Mg$

Here,

$W$ is the weight

$M$ is the mass

$g$ is the acceleration due to gravity

Substitute $1600\text{kg}$ for $M$ and $9.8\text{m}/{\text{s}}^{\text{2}}$ for $g$ to get $W$

$\begin{array}{r}W=1600\text{kg}\times \text{9}\text{.8}\text{m}/{\text{s}}^{\text{2}}\\ =15,680\text{N}\end{array}$

Conclusion:

The total weight of people and boat is $15,680\text{N}$

To determine

The buoyant force on the boat

Answer to Problem 4SP

The buoyant force is $15,680\text{N}$

Explanation of Solution

The buoyant force is equal to the weight of the water displaced.

Therefore buoyant force is $15,680\text{N}$

Conclusion:

The buoyant force is $15,680\text{N}$

To determine

The volume of water displaced.

Answer to Problem 4SP

The volume of water displaced by the boat is $1.6{\text{m}}^3$

Explanation of Solution

Write the equation to find the volume of water displaced.

$volume=\frac{densityofboat}{densityofwater}$

Write the equation to find the density of boat.

${\rho }_{boat}=\frac{M}{V}$

Here,

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

$M$ is the mass

$V$ is the volume

Write te equation to find the volume of boat.

$V=l\times b\times h$

Here,

$l$ is the length of boat

$b$ is the breadth

$h$ is the height

Substitute $2.4\text{m}$ for $l$ . $1.2\text{m}$ for $b$ and $1\text{m}$ for $h$ in equation to get the density

$\begin{array}{c}{V}_{boat}=2.4\text{m}\times \text{1}\text{.2}\text{m}\times \text{1}\text{m}\\ \text{=3}{\text{.36m}}^3\end{array}$

Substitute $2.88{\text{m}}^3$ for $V$ and $1600\text{kg}$ for $M$ in equation to find the ${\rho }_{boat}$

$\begin{array}{c}{\rho }_{boat}=\frac{1600\text{kg}}{2.88{\text{m}}^3}\\ =476.19\text{kg}/{\text{m}}^{\text{3}}\end{array}$

Substitute $555.55\text{kg}/{\text{m}}^{\text{3}}$ for $densityofboat$ and $1000\text{kg}/{\text{m}}^{\text{3}}$ for $densityofwater$ to get volume of water displaced

$\begin{array}{c}volume=\frac{476.19\text{kg}/{\text{m}}^{\text{3}}}{1000\text{kg}/{\text{m}}^{\text{3}}}\\ =1.6{\text{m}}^3\end{array}$

Conclusion:

The volume of water displaced by the boat is $1.6{\text{m}}^3$

To determine

How much of boat is underwater.

Answer to Problem 4SP

$0.44\text{m}$ of boat is under water.

Explanation of Solution

Volume of water displaced is equal to the volume of boat in the water.

$l\times b\times h=1.6{\text{m}}^3$

Substitute $2.8\text{m}\text{for l}$ , $1.2\text{m}$ for $b$ in equation and simplify to get $h$

$\begin{array}{c}h=\frac{1.6{\text{m}}^3}{2.8\text{m}\times \text{1}\text{.2}\text{m}}\\ =0.476\text{m}\end{array}$

Conclusion:

$0.44\text{m}$ of boat is under water.