91Ó°ÊÓ

Crossing a River. You and your team come to a slow flowing river that you need to cross. The nearest bridge is 20 miles to the north, too dangerous and too far to trek with your group. You explore the area down river and discover an abandoned shed with a stash of 55 -gallon drums (all empty and weighing 35.0 lb each) and a stack of 10 -foot planks: You will build a raft. The six members of your team have a combined weight of 925 pounds (assuming everyone was truthful). You also have a four-wheeler (all-terrain vehicle), which weighs 450 lb, and other gear, which adds another 315 lb. Your simple raft design is as follows: a platform of planks with barrels strapped to the bottom. You estimate that the planks weigh about \(45.0 \mathrm{lb}\) each, and you will need 20 of them to make a platform that can accommodate everything for the one-way trip. You measure the dimensions of the cylindrical drums and find they have diameter \(D=22.5\) inches and height \(h=33.5\) inches (note: not exactly \(^{455}\) gallons"). (a) What is the minimum number of barrels that you will need so that the raft will float when fully loaded? (b) What is the minimum number of barrels you will need if you want the platform to be at least 6 inches above the water when fully loaded?

Step by step solution

01

Calculate Total Weight

First, we need to calculate the total weight that the raft needs to support, including the people, equipment, and the raft itself. The total weight consists of the six people (925 lb), the four-wheeler (450 lb), other gear (315 lb), and the weight of the planks. The weight of the planks is calculated as 20 planks times 45 lb each, giving 900 lb. Thus, the total weight is:\[ 925 \, \text{lb} + 450 \, \text{lb} + 315 \, \text{lb} + 900 \, \text{lb} = 2590 \, \text{lb} \]

02

Determine Barrel's Buoyancy

Each barrel needs to provide enough buoyancy to support the raft's load. Considering that each barrel has a diameter of 22.5 inches and a height of 33.5 inches, we calculate the volume of a barrel using the formula for the volume of a cylinder: \[ V = \pi \left(\frac{D}{2}\right)^2 h \]Convert the dimensions to feet for consistency, where 1 inch = 0.0833333 feet:\[ D = 22.5 \times 0.0833333 = 1.875 \, \text{feet} \]\[ h = 33.5 \times 0.0833333 = 2.79167 \, \text{feet} \]Thus, the volume is:\[ V = \pi \left(\frac{1.875}{2}\right)^2 \times 2.79167 \approx 7.73 \, \text{cubic feet} \]A cubic foot of water weighs approximately 62.4 lb, so one barrel can displace:\[ 7.73 \times 62.4 \approx 482.35 \, \text{lb} \]

03

Calculate Number of Barrels for Floating

To float, the raft must displace a weight of water equal to its own weight (archimedes' principle). Using the total required buoyancy of 2590 lb and the buoyancy provided by one barrel (482.35 lb):\[ N = \frac{2590}{482.35} \approx 5.37 \]Since the number of barrels must be an integer, we round up to the nearest whole number, which is 6 barrels.

04

Calculate Barrels for Desired Clearance

To have the platform 6 inches above water, the raft must be able to support 6 more inches of its volume in water. Converting 6 inches to feet yields 0.5 feet. The volume above water per barrel remains:\[ \frac{1.875}{2}^2 \times 0.5 \approx 1.38 \, \text{cubic feet} \text{above water} \]Multiply by the weight of water:\[ 1.38 \times 62.4 = 86.11 \, \text{lb} \text{ extra}, \text{ per barrel}\]Thus, for additional clearance:\[ \frac{86.11 \times 6 \text{ clear barrels}}{482.35} \approx 1.07 \]Adding these barrels to our current working number \((6 + 1)\) gives us:\[ N' = 7 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Archimedes' Principle

When thinking about floating objects, Archimedes' principle plays a crucial role. This principle states that any object, fully or partially submerged in a fluid, experiences an upward force equal to the weight of the fluid displaced by the object. This buoyant force acts against the force of gravity, allowing things to float.
It’s essential to understand the balance between these two forces.
Here’s how it applies to our river-crossing exercise:

• The raft, with its weight including people and equipment, needs to displace an equal weight of water to float.
• Each barrel attached underneath contributes to this displacement, thus adding to the buoyant force.
• For the raft to lift, the total buoyant force (or the weight of the displaced water) must at least equal the total weight of the raft, ensuring it doesn’t sink.

By utilizing empty barrels filled with air, which have a lower density than water, we effectively increase the raft's buoyancy.

Force Balance

It’s crucial to consider the balance of forces to ensure that the raft remains afloat and stable. The force balance in this scenario involves a comparison between two critical components:

• The downward gravitational force, which pulls the raft towards the water, is equal to the total weight of the raft (including all its load).
• The upward buoyant force acts against this and increases with the water displaced by the submerged volume of the barrels.

For equilibrium, which keeps the raft floating, these forces must be balanced:

• The buoyant force must match the gravitational pull.
• If the buoyant force is greater, the raft will rise until a new balance is found.
• If it's less, the raft could sink further until equilibrium is restored.

For the raft to stay stable above the waterline, it’s important to have the right number of barrels that not only meet but slightly exceed the weight, ensuring safety and clearance from the water.

Cylindrical Volume Calculation

To determine the buoyancy of each barrel, it is crucial to calculate its volume, as this directly affects how much water it can displace. This calculation is done using the formula for the volume of a cylinder: \[ V = \pi \left(\frac{D}{2}\right)^2 h \]Here, \(D\) is the diameter, and \(h\) is the height.
In our scenario:

• The diameter is 22.5 inches, and the height is 33.5 inches.
• These dimensions are converted into feet for uniformity: 1 inch equals 0.0833333 feet.
• Thus, the calculations provide the effective volume of each barrel in cubic feet.

A precise conversion allows the evaluation of how much weight each barrel can support through the water it displaces. This volume, along with the weight of water (62.4 lbs per cubic foot), provides the basis for calculating the total buoyancy offered by each barrel.
This understanding helps us know how many barrels are needed to counterbalance the total weight of the raft and meet the desired clearance above the water.