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Chapter 3: Problem 87

A canoe is represented by a right semicircular cylinder, with \(R=1.2 \mathrm{ft}\) and \(L=17 \mathrm{ft} .\) The canoe floats in water that is \(d=1 \mathrm{ft}\) deep. Set up a general algebraic expression for the total mass (canoe and contents) that can be floated, as a function of depth. Evaluate for the given conditions. Plot the results over the range of water depth \(0 \leq d \leq R\).

Short Answer

Expert verified
The total mass that the canoe can float will be calculated using the formula \( M= \rho_{w} \cdot \pi \cdot L \cdot R^2 \cdot g \cdot [d/R - sin(d/R)] \). Plot this function over the range of water depth \( 0 \leq d \leq R \).

Step by step solution

01

Identify Variables

Let \(M\) be the total mass the canoe can float, and \(d\) the depth it is submerged. The Radius of the canoe is \(R = 1.2\) ft and length is \(L = 17\) ft. Also, the density of water, \(\rho_w = 62.4\) lb/ft鲁. The equation of buoyancy is: \( M = \rho_w \cdot V \cdot g \),where \(V\) is the volume of water displaced, and \(g\) is the gravity.
02

Calculate Water Displacement Volume

The volume of water, \( V \), displaced by the canoe is equivalent to the volume of the submerged section of the canoe. For a circle cross-section, the area is given by \(A = \pi \cdot R^2 \cdot (d/R) - sin(d/R) = \pi \cdot R^2 \cdot [d/R - sin(d/R)] \). The total volume of the submerged section of the canoe is \( V=A \times L = \pi \cdot L \cdot R^2 \cdot [d/R - sin(d/R)] \).
03

Evaluate the Mass of the Canoe

To find the total mass that the canoe can float (itself and its contents) we substitute \( V \) into the buoyancy equation: \( M= \rho_{w} \cdot \pi \cdot L \cdot R^2 \cdot g \cdot [d/R - sin(d/R)] \). Plug in the given values and simplify to find \( M \).
04

Plotting the Results

Plot the above equation with \(d\) changing from 0 to \( R \).

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Most popular questions from this chapter

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