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Chapter 4: Problem 424

The force constant of a wire is \(\mathrm{K}\) and that of the another wire is \(3 \mathrm{k}\) when both the wires are stretched through same distance, if work done are \(\mathrm{W}_{1}\) and \(\mathrm{W}_{2}\), then... (A) \(\mathrm{w}_{2}=3 \mathrm{w}_{1}^{2}\) (B) \(\mathrm{W}_{2}=0.33 \mathrm{~W}_{1}\) (C) \(\mathrm{W}_{2}=\mathrm{W}_{1}\) (D) \(\mathrm{W}_{2}=3 \mathrm{~W}_{1}\)

Short Answer

Expert verified
\(W_2 = 3\cdot W_1\)

Step by step solution

01

Write down the work done formula for each wire

For the wire with force constant K and the work done W1, the formula is: \(W_1 = \frac{1}{2}Kx^2\) For the wire with force constant 3K and the work done W2, the formula is: \(W_2 = \frac{1}{2}(3K)x^2\)
02

Divide both equations to find the relationship between W1 and W2

Now, to find the relationship between \(W_1\) and \(W_2\), we will divide the second equation by the first equation: \(\frac{W_2}{W_1} = \frac{(1/2)(3K)x^2}{(1/2)Kx^2}\)
03

Simplify the equation

Observe that \((1/2)x^2\) can be canceled out in both numerator and denominator, and we are left with: \(\frac{W_2}{W_1} = \frac{3K}{K}\)
04

Solve for the relationship between W1 and W2

Now, dividing both sides by K, we have \(\frac{W_2}{W_1} = 3\) Multiplying both sides by \(W_1\), we get: \(W_2 = 3\cdot W_1\)
05

Compare the result with the given options

Finally, we can compare our result to the given options. Our result matches option (D), which states that: \(W_2 = 3\cdot W_1\) Therefore, the correct answer is (D).

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