Question

Solve the initial-value problem in Problem 20 when a cross section of a uniform piece of wood is the triangular region given in Figure 3.R.7. Assume again that k=1. How long does it take to cut through this piece of wood?

Short answer

$t=\frac{1}{2}$

Step-by-step solution

Given data

We have a long uniform piece of wood with a triangular cross section is cut through perpendicular to its length by a vertical saw blade which moves through the section of piece of wood at the rate $\frac{dx}{dt}=k\frac{1}{w\left(x\right)}$which is $\frac{dx}{dt}=\frac{1}{w\left(x\right)}‐‐‐\left(1\right)$

Where$w\left(x\right)$ is the vertical line that the blade cut in the section (width of the wood in contact with the blade).

Form differential equation

Solve this differential equation as the following technique :

First, using Pythagoras rule in the figure shown below, we can have

$w\left(x\right)=\sqrt{s^2-x^2}‐‐‐‐\left(2\right)$

Substitute form equation (2) into equation (1), then we have

$\frac{dx}{dt}=\frac{1}{\sqrt{s^2-x^2}}$

Solve differential equation

Since this differential equation is separable, then we can solve it as

$\begin{array}{l}\int \sqrt{s^2-x^2}dx=\int dt\\ \int \sqrt{1-x^2}dx=\int dt\\ \int \sqrt{1-x^2}dx=t+C\end{array}$

But here the value $x=\frac{\sqrt{2}}{2}$.

Apply initial condition

Now, to find the value of constant $C$, we have to apply the initial condition $x\left(t=0\right)$, then we have

$C=0$

After that, substitute with the value of $C$into equation (4), then we have

$t=\frac{\sqrt{2}}{2}x$

is the time needed to cut the half of piece of wood at any position of $x$.

Find time

Now, to find the time at which the piece of wood is being cut, we have to find first the half of time needed to do that by substituting with $x=\frac{\sqrt{2}}{2}$into equation (5) as

$$\begin{gathered} t=\frac{\sqrt{2}}{2}\text{'}\frac{\sqrt{2}}{2} \\ =\frac{1}{4} \end{gathered}$$

Then we have

$$\begin{gathered} t=2\text{'}\frac{1}{4} \\ =\frac{1}{2} \end{gathered}$$

is the time needed to cut the piece of wood.