The length of a light string is 1.4 m when the tension on it is 5 N . If the tension increases to 7 N , the length of the string is 1.56 m . The original length of the string is__________m.

<p>To find the original length of the string, we use the relationship between tension, elasticity constant ($ K $), and the change in length of the string.</p> <p><p>Given the equation for tension:</p> <p>$ \mathrm{T} = \mathrm{K}(\ell - \ell_0) $</p> <p>where $\ell$ is the length of the string under tension and $\ell_0$ is the original length.</p></p> <p><p>When the tension is 5 N, the equation becomes:</p> <p>$ 5 = \mathrm{K}(1.4 - \ell_0) $</p></p> <p><p>When the tension increases to 7 N, the equation is:</p> <p>$ 7 = \mathrm{K}(1.56 - \ell_0) $</p></p> <p><p>By setting up a ratio from the two equations, we have:</p> <p>$ \frac{5}{1.4 - \ell_0} = \frac{7}{1.56 - \ell_0} $</p></p> <p><p>Solving this proportion gives us the original length $ \ell_0 $:</p> <p>$ \ell_0 = 1 \, \mathrm{m} $</p></p> <p>Thus, the original length of the string is 1 meter.</p>