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Chapter 7: Problem 889

A steel ring of radius \(\mathrm{r}\) and cross-section area ' \(\mathrm{A}^{\prime}\) is fitted on to a wooden disc of radius \(\mathrm{R}(\mathrm{R}>\mathrm{r})\) If young's modulus be \(\mathrm{E}\) then what is force with which the steel ring is expanded? (A) \(\mathrm{AE}(\mathrm{R} / \mathrm{r})\) (B) \(\mathrm{AE}[(\mathrm{R}-\mathrm{r}) / \mathrm{r}]\) (C) \((\mathrm{E} / \mathrm{A})[(\mathrm{R}-\mathrm{r}) / \mathrm{R}]\) (D) \([\mathrm{Er} / \mathrm{AR}]\)

Short Answer

Expert verified
The force required to expand the steel ring is: \( F = A'E\frac{R-r}{r} \).

Step by step solution

01

Determine the Circumferences

First, let's find the initial and final circumferences of the steel ring. The initial circumference of the steel ring is given by: \[ C_{i} = 2\pi r \] And the final circumference when it is fitted onto the wooden disc is given by: \[ C_{f} = 2\pi R \]
02

Find the Change in Length

Now let's find the change in length of the steel ring. The change in length is the difference between the final and initial circumferences. \[ \Delta L = C_{f} - C_{i} \] \[ \Delta L = 2\pi R - 2\pi r \] \[ \Delta L = 2\pi (R - r) \]
03

Calculate Strain

Next, we need to calculate the strain on the steel ring. Strain is defined as the change in length divided by the original length: \[ \text{Strain} = \frac{\Delta L}{C_{i}} \] \[ \text{Strain} = \frac{2\pi (R-r)}{2\pi r} \] \[ \text{Strain} = \frac{R-r}{r} \]
04

Calculate Stress

Now, let's calculate the stress using Young's modulus. Young's modulus is the ratio of stress to strain: \[ E = \frac{\text{Stress}}{\text{Strain}} \] We have the strain, so we can rearrange the formula to find the stress: \[ \text{Stress} = E \cdot \text{Strain} \] \[ \text{Stress} = E \cdot \frac{R-r}{r} \]
05

Calculate Force

Finally, let's calculate the force required to expand the steel ring. Stress is also defined as force divided by cross-sectional area: \[ \text{Stress} = \frac{F}{A'} \] Now, we can rearrange this formula to find the force F: \[ F = A' \cdot \text{Stress} \] Plugging in the stress value from Step 4, we get: \[ F = A'E\frac{R - r}{r} \] So the force required to expand the steel ring is: \[ F = A'E\frac{R-r}{r} \] which matches with option (B).

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Most popular questions from this chapter

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