What force acts in the cross-section of a homogeneous rod of length l at a distance x from the end to which the force F is applied along the rod?

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Solutions of Savchenko Problems in Physics
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    <h3 id="back-link"><a href="/#2.1">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <p>
        $2.1.5.$ What force acts in the cross-section of a homogeneous rod of length $l$ at a distance $x$ from the end to which the force $F$ is applied along the rod?

    <h3>Solution</h3>
    <p>
        <center>

If we consider a rod of mass $m$ as a single whole, then it will move with acceleration $$a = \frac{F}{m}$$ Since the rod is inextensible, the acceleration of all its parts is the same and equal to $a$

Let's consider a small section of the rod of length $dx$ and mass $dm$. Since the rod is homogeneous $$dm = m \frac{dx}{l}$$ Let's write down Newton's second law for this section. $$a\\, dm = F(x+dx) - F(x)\quad(1)$$ Where $F(x+dx)$ and $F(x)$ are the interaction forces with neighbors

Let's integrate expression $(1)$ along the horizontal coordinate $x$: $$\int_x^l am\frac{dx}{l} = \int_x^l dF$$

$$ma\frac{l-x}{l}$$

$$F(x) = \fbox{$F(1-\frac{x}{l})$}$$

    <h4>Answer</h4>
    <p>
        $$T = F(1 − x/l)$$
    </p>


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