An elastic spring under tension of $3 \mathrm{~N}$ has a length $a$. Its length is $b$ under tension $2 \mathrm{~N}$. For its length $(3 a-2 b)$, the value of tension will be _______ N.

<p>To determine the tension for the elastic spring's length of $(3a - 2b)$, we can use Hooke's law which states that the force exerted by a spring is proportional to the extension or compression of the spring from its natural length. Mathematically, Hooke's law is given by:</p> <p> <p>$F = k \cdot \Delta x$</p> </p> <p>where:</p> <ul> <li>$F$ is the force exerted by the spring.</li> <li>$k$ is the spring constant (a measure of the stiffness of the spring).</li> <li>$\Delta x$ is the displacement from the natural length.</li> </ul> <p>Let's denote the natural (unstretched) length of the spring as $l_0$. Given that the spring's length is $a$ under a tension of $3 \mathrm{~N}$, and its length is $b$ under a tension of $2 \mathrm{~N}$, we can write the equations as:</p> <p> <p>$3 \mathrm{~N} = k \cdot (a - l_0)$</p> </p> <p> <p>$2 \mathrm{~N} = k \cdot (b - l_0)$</p> </p> <p>Next, we need to find the length displacement $(3a - 2b)$ in terms of the natural length $l_0$. We express the displacements as:</p> <p> <p>$x = (3a - 2b) - l_0$</p> </p> <p>To solve for this, let's express $a - l_0$ and $b - l_0$ from the given conditions:</p> <p> <p>$a - l_0 = \frac{3 \mathrm{~N}}{k}$</p> </p> <p> <p>$b - l_0 = \frac{2 \mathrm{~N}}{k}$</p> </p> <p>Substitute these into the displacement equation:</p> <p> <p>$$ x = (3a - 2b) - l_0 = 3 \left( \frac{3 \mathrm{~N}}{k} + l_0 \right) - 2 \left( \frac{2 \mathrm{~N}}{k} + l_0 \right) - l_0 $$</p> </p> <p>On simplifying, we get:</p> <p> <p>$x = 3 \frac{3 \mathrm{~N}}{k} + 3 l_0 - 2 \frac{2 \mathrm{~N}}{k} - 2 l_0 - l_0$</p> </p> <p> <p>$x = \frac{9 \mathrm{~N}}{k} + 3 l_0 - \frac{4 \mathrm{~N}}{k} - 3 l_0$</p> </p> <p> <p>$x = \frac{5 \mathrm{~N}}{k}$</p> </p> <p>Finally, by Hooke's Law, the force corresponding to this displacement is:</p> <p> <p>$F = k \cdot x = k \cdot \frac{5 \mathrm{~N}}{k} = 5 \mathrm{~N}$</p> </p> <p>Therefore, the tension required for the elastic spring's length $(3a - 2b)$ is <strong>5 N</strong>.</p>