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Chapter 7: Problem 906

If the force constant of a wire is \(\mathrm{k}\). What is the work done in increasing the length of the wire by \(\ell\) ? (A) \([\mathrm{k} \ell / 2]\) (B) \(\mathrm{k} \ell\) (C) \(\left[\mathrm{k} \ell^{2} / 2\right]\) (D) \(\mathrm{k} \ell^{2}\)

Short Answer

Expert verified
The short answer is: \(Work =\frac{k\ell^2}{2}\), which corresponds to option (C).

Step by step solution

01

Start with Hooke's Law

Hooke's law states that the force acting on an object is directly proportional to its elongation or compression. In this case, we have: Force = k × Δx where k is the force constant, and Δx is the change in length.
02

Integrate to find the work done

To find the work done in increasing the length of the wire by ℓ, we will integrate the force over the distance. Using the expression for force from Hooke's Law: Work = ∫ (k × Δx) d(Δx) We will integrate this equation with respect to Δx from 0 to ℓ.
03

Perform the integration

Now, we perform the integration: Work = ∫₀ᴸ (k × Δx) d(Δx) The force constant k can be treated as a constant during integration. So, we have: Work = k ∫₀ᴸ (Δx) d(Δx) Integrating, we get: Work = k [Δx²/2] |_₀ᴸ
04

Calculate the work done

Applying the limits of integration, we get: Work = k [ℓ²/2 - 0²/2] = kℓ²/2 The work done in increasing the length of the wire by ℓ is: \(Work =\frac{k\ell^2}{2}\) Comparing with the given options, we can see that the correct answer is: (C) \([\mathrm{k} \ell^{2} / 2]\)

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