Expressing Derivative of Cable Tension Component

Question

I am looking at Hibbeler's section on a `Cable Subjected to a Distributed Load' in Engineering Mechanics - Statics, Ed13, as shown in the attached images.

I am getting stuck deriving equation (7-7). I derive the horizontal equilibrium equation just fine;

$$ |\vec{T} + \vec{\Delta T}| \cdot \cos(\theta + \Delta \theta) - |\vec{T}|\cdot \cos(\theta) = 0 $$

I rearrange to describe horizontal tension component as a function of $\Delta x$ (given $\Delta T$ and $\Delta \theta$ are ultimately functions of $\Delta x$);

$$ |\vec{T}|\cdot \cos(\theta) = |\vec{T} + \vec{\Delta T}|\cdot \cos(\theta + \Delta \theta) $$

Then I express the differential;

$$ \frac{\Delta(|\vec{T}|\cdot \cos(\theta))}{\Delta x} = \frac{|\vec{T} + \vec{\Delta T}|\cdot \cos(\theta + \Delta \theta)}{\Delta x} $$

Then when I go to express the derivative, I come up with a limit which does not exist.

$$ \frac{d(|\vec{T}|\cdot \cos(\theta))}{dx} = \lim_{\Delta x \to 0}\frac{|\vec{T} + \vec{\Delta T}|\cdot \cos(\theta + \Delta \theta)}{\Delta x} $$

Further to this, it is not obvious to me how the resultant force from the distributed load acting over $\Delta x$ can be defined as $w(x)\Delta x$ (in line 8) as the author suggests. I feel it should be defined as;

$$ \Delta F = \int_{x}^{x+\Delta x}w(x)dx $$

becuase the loading intensity function is not constant. Any pointers as to where my errors are? Appreciate any advice offered. Thanks.

Answer 1

You have

$$-T\cos \theta + (T+\Delta T) \cos( \theta + \Delta \theta)=0.$$

It is important to note that $T$ and $\theta$ are both depending on $x$. Dividing the previous expression by $\Delta x$ and rewriting $T+\Delta T = T(x+\Delta x)$ and $\theta + \Delta \theta = \theta(x+\Delta x)$, will lead to:

$$\frac{T(x+\Delta x)\cos [\theta(x+\Delta x)]-T(x)\cos[\theta(x+\Delta x)]}{\Delta x}=0.$$

Now, take the limit $\Delta x \to 0$. As the above is the definition of the derivative of $T(x)\cos[\theta(x)]$ we obtain:

$$\dfrac{d}{d x}\left[T(x)\cos[\theta (x)]\right]=0 \implies T(x)\cos[\theta (x)] = \text{const.}$$