Problem 889 - JEE Physics

Question

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) \([\mathrm{Er} / \mathrm{AR}]\)

Step-by-Step Solution

Answer

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

Step 1: 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: \[ C_{f} = 2\pi R \]

Step 2: 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. \[ \Delta L = C_{f} - C_{i} \] \[ \Delta L = 2\pi R - 2\pi r \] \[ \Delta L = 2\pi (R - r) \]

Step 3: 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: \[ \text{Strain} = \frac{\Delta L}{C_{i}} \] \[ \text{Strain} = \frac{2\pi (R-r)}{2\pi r} \] \[ \text{Strain} = \frac{R-r}{r} \]

Step 4: 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: \[ \text{Stress} = E \cdot \text{Strain} \] \[ \text{Stress} = E \cdot \frac{R-r}{r} \]

Step 5: 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).

Key Concepts

Young's Modulus

Young's modulus, denoted by the symbol \( E \), is a fundamental material property that describes the stiffness of a solid material. It relates the stress applied to a material to the resulting strain, measuring the material's ability to withstand changes in length when under length-wise tension or compression.
In simpler terms, it helps us understand how much a material will stretch or compress when a force is applied along its length.

• It is a constant that applies to a certain material.
• A high Young's modulus means the material is stiff and does not deform easily.
• It is defined by the formula: \( E = \frac{\text{Stress}}{\text{Strain}} \).

In this exercise, Young's modulus is used to calculate the stress on the steel ring due to strain from expansion.

Circumference

The concept of circumference is useful in understanding the shape and size changes of round objects like a steel ring. The circumference is the perimeter of a circle or any circular object.
It is calculated as the distance around the edge of a circle.

• The formula for circumference is \( C = 2\pi r \), where \( r \) is the radius of the circle.
• It represents the initial and final boundary lengths in our exercise scenario.

In our calculation, we consider two circumferences: the initial circumference of the smaller steel ring and the larger, final circumference after fitting the ring around the wooden disc. The difference between these circumferences gives us the change in length, essential for strain calculation.

Strain Calculation

Strain is a measure of deformation representing the amount of stretch or compression the material undergoes,relative to its initial length. This dimensionless quantity is crucial for materials science and engineering.
Strain is expressed by this formula: \( \text{Strain} = \frac{\Delta L}{L_i} \), where \( \Delta L \) is the change in length and \( L_i \) is the original length.

• In our exercise, the initial length is the original circumference of the steel ring (\( C_{i} \)).
• \( \Delta L \) is the increase in the circumference necessary to fit around the wooden disc, given by \( 2\pi(R-r) \).
• Strain is calculated as \( \frac{R-r}{r} \), simplifying how much the radius and thus the circumference needed to expand.

Understanding strain helps determine how much stress the material is subjected to during this deformation process.

Force Expansion

Force expansion involves determining the force needed to make the steel ring expand enough to fit around the wooden disc. To find this force, we first calculate the stress resulting from the strain on the steel ring.
Stress is force per unit area.

• Through Young's modulus, we obtained the relationship \( \text{Stress} = E \times \text{Strain} \).
• The stress was derived as \( E \times \frac{R-r}{r} \).
• Finally, to find the force (\( F \)) exerted over the ring's cross section, we solve \( \text{Stress} = \frac{F}{A'} \).

Thus the calculation gives \( F = A'E \times \frac{R-r}{r} \), explaining the transition from stress to the force required for expansion. This expression helps us compute the exact force exerted on the ring to achieve the desired expansion.