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Chapter 7: Problem 907

Wire \(\mathrm{A}\) and \(\mathrm{B}\) are made from the same material. \(\mathrm{A}\) has twice the diameter and three times the length of \(\mathrm{B}\). If the elastic limits are not reached when each is stretched by the same tension, what is the ratio of energy stored in \(\mathrm{A}\) to that in \(\mathrm{B}\) ? (A) \(2: 3\) (B) \(3: 4\) (C) \(3: 2\) (D) \(6: 1\)

Short Answer

Expert verified
The ratio of energy stored in wire A to that in wire B is 12:1. So the correct answer is (D) \(6:1\).

Step by step solution

01

Write down the elastic potential energy formula

The elastic potential energy stored in a wire can be calculated as follows: \[E = \frac{1}{2} \cdot k \cdot x^2\] Here, \(E\) is the elastic potential energy, \(k\) is the spring constant, and \(x\) is the displacement.
02

Express the spring constant in terms of Young's modulus, length, and cross-sectional area

We know that spring constant \(k = \frac{F}{x}\), where \(F\) is the force, and \(x\) is the displacement. The force \(F\) can also be expressed in terms of stress and cross-sectional area, which gives us: \[F = \sigma A\] where \(\sigma\) is the stress and \(A\) is the cross-sectional area. The stress, in turn, can be expressed in terms of Young's modulus and strain, which is the relative length change: \[\sigma = E_Y \cdot \epsilon\] Here, \(E_Y\) denotes the Young's modulus, and \(\epsilon = \frac{x}{L}\) is the strain, with \(L\) representing the original length of the wire. Combining the above equations, we get: \[k = \frac{E_Y A}{L}\]
03

Determine the cross-sectional areas of wire A and wire B

Let the cross-sectional area of wire B be \(A_B\), and the diameter of wire A is twice that of wire B. Therefore, the cross-sectional area of wire A is: \[A_A = 4A_B\]
04

Determine the spring constants of wire A and wire B

We can use the formula we derived in Step 2 to find the spring constants of wire A and wire B: \[k_A = \frac{E_Y A_A}{L_A} =\frac{E_Y (4A_B)}{(3L_B)}\] \[k_B = \frac{E_Y A_B}{L_B}\]
05

Calculate the elastic potential energy of wire A and wire B

Using the elastic potential energy formula with the spring constants we determined in step 4 for both wires, we get: \[E_A = \frac{1}{2} \cdot k_A \cdot x^2 = \frac{1}{2} \cdot \frac{E_Y (4A_B)}{(3L_B)} \cdot x^2\] \[E_B = \frac{1}{2} \cdot k_B \cdot x^2 = \frac{1}{2} \cdot \frac{E_Y A_B}{L_B} \cdot x^2\]
06

Calculate the ratio of energy stored in wire A to that in wire B

Now we can find the desired ratio by dividing the elastic potential energy of wire A by that of wire B: \[\frac{E_A}{E_B} = \frac{\frac{1}{2} \cdot \frac{E_Y (4A_B)}{(3L_B)} \cdot x^2}{\frac{1}{2} \cdot \frac{E_Y A_B}{L_B} \cdot x^2} = \frac{4A_B(3L_B)}{A_B L_B} = \frac{12 A_B L_B}{A_B L_B}\] This simplifies to: \[\frac{E_A}{E_B} = 12\] Thus, the ratio of energy stored in wire A to that in wire B is 12:1. So the correct answer is (D) \(6: 1\).

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Most popular questions from this chapter

In a water fall the water falls from a height of \(100 \mathrm{~m}\). If the entire K.E. of water is converted in to heat the rise in temperature of water will be (A) \(0.23^{\circ} \mathrm{C}\) (B) \(0.46^{\circ} \mathrm{C}\) (C) \(2.3^{\circ} \mathrm{C}\) (D) \(0.023^{\circ} \mathrm{C}\)

Radius of a soap bubble is \(\mathrm{r}^{\prime}\), surface tension of soap solution is \(\mathrm{T}\). Then without increasing the temperature how much energy will be needed to double its radius. (A) \(4 \pi r^{2} T\) (B) \(2 \pi r^{2} T\) (C) \(12 \pi r^{2} T\) (D) \(24 \pi r^{2} T\)

A large number of water drops each of radius \(r\) combine to have a drop of radius \(\mathrm{R}\). If the surface tension is \(\mathrm{T}\) and the mechanical equivalent at heat is \(\mathrm{J}\) then the rise in temperature will be (A) \((2 \mathrm{~T} / \mathrm{rJ})\) (B) \((3 \mathrm{~T} / \mathrm{RJ})\) (C) \((3 \mathrm{~T} / \mathrm{J})\\{(1 / \mathrm{r})-(1 / \mathrm{R})\\}\) (D) \((2 \mathrm{~T} / \mathrm{J})\\{(1 / \mathrm{r})-(1 / \mathrm{R})\\}\)

If the young's modulus of the material is 3 times its modulus of rigidity. Then what will be its volume elasticity? (A) zero (B) infinity (C) \(2 \times 10^{10}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\) (D) \(3 \times 10^{10}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\)

What is the possible value of posson's ratio? (A) 1 (B) \(0.9\) (C) \(0.8\) (D) \(0.4\)
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