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Step-by-Step Solution
Step 1: Understand the Problem
We have a wire with a natural length l_0 . When a tension of 100\,\mathrm{N} is applied, its length becomes l_1 . When a tension of 120\,\mathrm{N} is applied, its length becomes l_2 .
We also know that 10l_2 = 11l_1 .
We want to find the natural length l_0 given in the form \frac{1}{x} \, l_1 and determine the value of x .
Step 2: Express Tension Using Hooke's Law
According to Hooke's Law:
T = k\,\Delta l ,
where k is the force constant (spring constant) of the wire, and \Delta l is the extension (the increase in length from the natural length).
Therefore, for the two tensions, we have:
100 = k\,(l_1 - l_0), \quad
120 = k\,(l_2 - l_0).
Step 3: Form a Ratio
Divide the first equation by the second:
\frac{100}{120} = \frac{l_1 - l_0}{l_2 - l_0}
\quad \Longrightarrow \quad
\frac{5}{6} = \frac{l_1 - l_0}{l_2 - l_0}.
Step 4: Use the Given Relationship Between l_1 and l_2
From the problem statement:
10\,l_2 = 11\,l_1
\quad \Longrightarrow \quad
l_2 = \frac{11}{10}\,l_1.
Step 5: Substitute and Solve for l_0
From the ratio:
\frac{5}{6} = \frac{l_1 - l_0}{l_2 - l_0},
cross-multiplying gives:
5 \,(l_2 - l_0) = 6 \,(l_1 - l_0).
Expand both sides:
5\,l_2 - 5\,l_0 = 6\,l_1 - 6\,l_0.
Rearrange to isolate l_0 :
5\,l_2 - 6\,l_1 = -6\,l_0 + 5\,l_0
\quad \Longrightarrow \quad
5\,l_2 - 6\,l_1 = -\,l_0
\quad \Longrightarrow \quad
l_0 = 6\,l_1 - 5\,l_2.
Step 6: Substitute l_2 = \frac{11}{10}\,l_1
Substituting l_2 in the expression for l_0 :
l_0 = 6\,l_1 - 5\left(\frac{11}{10}\,l_1\right).
Simplify:
l_0 = 6\,l_1 - \frac{55}{10}\,l_1
= 6\,l_1 - \frac{11}{2}\,l_1.
Converting 6\,l_1 to a fraction with denominator 2:
6\,l_1 = \frac{12}{2}\,l_1, \quad
\text{so} \quad
l_0 = \frac{12}{2}\,l_1 - \frac{11}{2}\,l_1
= \frac{1}{2}\,l_1.
Step 7: Identify the Value of x
We found that
l_0 = \frac{1}{2}\,l_1
\ \Longrightarrow\
l_0 = \frac{1}{x}\,l_1.
Therefore,
\frac{1}{x} = \frac{1}{2}
\ \Longrightarrow\
x = 2.
Final Answer
The value of x is 2.
