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Chapter 7: Problem 901

A load \(W\) produces an extension of \(1 \mathrm{~mm}\) in a thread of radius \(r\). Now if the load is made \(4 \mathrm{~W}\) and radius is made \(2 \mathrm{r}\) all other things remaining same the extension will becomes.......... (A) \(4 \mathrm{~mm}\) (B) \(16 \mathrm{~mm}\) (C) \(1 \mathrm{~mm}\) (D) \(0.25 \mathrm{~mm}\)

Short Answer

Expert verified
The new extension becomes \(\boxed{0.25 \, \text{mm}}\).

Step by step solution

01

Identify the given information

In the problem, we are provided with the following information: 1. Initial load: \(W\) 2. Initial extension: \(1 \, \text{mm}\) 3. Initial radius: \(r\) We are also given the changes in load and radius: 1. New load: \(4W\) 2. New radius: \(2r\) The problem asks for the new extension after these changes have been made.
02

Understand the Young's modulus formula

The Young's modulus of a material is defined as the ratio of stress to strain. Mathematically, this can be formulated as: \(Y = \frac{F/A}{\Delta L/L}\) Where: \(Y\) - Young's modulus \(F\) - Applied force/load (in our case, it is \(W\)) \(A\) - Cross-sectional area of the thread \(\Delta L\) - Extension \(L\) - Original length of the thread Since the problem assumes all other things remain the same, it implies that \(Y\) and \(L\) remain constant. Thus, our focus will be on the relationship between the load, radius, and extensions.
03

Write the formula for the initial condition

From the given information, for the initial condition, we have: \(Y = \frac{W / \pi r^{2}}{1 / L}\) (Y is constant in this case, we will equate this with the equation of the modified parameters) Now, we have to find an expression for the new condition when the load is changed to \(4W\) and the radius is changed to \(2r\).
04

Write the formula for the new condition

For the new condition, we have: \(Y = \frac{4W / \pi (2r)^{2}}{\Delta L' / L}\) Where: \(\Delta L'\) - The extension in the new condition
05

Equate the Young's modulus for both conditions and solve for the new extension

As the Young's modulus remains constant throughout this process, we can equate the two expressions we derived in Steps 3 and 4: \(\frac{W / \pi r^{2}}{1 / L} = \frac{4W / \pi (2r)^{2}}{\Delta L' / L}\) Now, we solve for the new extension \(\Delta L'\): \(\frac{W}{\pi r^{2}} \cdot \frac{L}{1} = \frac{4W}{\pi (2r)^{2}} \cdot \frac{L}{\Delta L'}\) Simplifying the equation and canceling out the common terms: \(\frac{1}{r^{2}} = \frac{4}{(2r)^{2}} \cdot \frac{1}{\Delta L'}\) \(\frac{1}{r^{2}} = \frac{4}{4r^{2}} \cdot \frac{1}{\Delta L'}\) Now, solve for \(\Delta L'\): \(\Delta L' = \frac{1}{4}\) Thus, the new extension becomes \(\boxed{0.25 \, \text{mm}}\), which corresponds to the option (D).

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