Consider a uniform wire of length `l`. Cross-sectional area A, Young's modulus of the material of the wave is Y. Some information related to the wire is given in column-1 and dependence of the result is given in column-2. Then match the approprite choice between the columns and match the list given in options
`{:("Column-I"" ""Column-II"),("(A) Let us suspend wire vertically from a"" ""(p) Young'sModulus" ),("rigud support and attach a mass m at"),("its lower end. If the mass is slightly pulled"),("down and released, it executes S.H.M"),("of a time period which will depend on"),("(B) Work done in stretching the wire up to length"" ""(q) elongation (x)"),(l+x "will depend on"),("(C) If the given is fixed between two rigid"" ""(r) length (l)"),("supports and its temperature is decreased"),("thermal stress that develops in the wire"),("will depend on"),("(D) If the wire is pulled at its ends equal and"" ""(s) area of cross-section (A) "),("opposite forces of magnitude F so that it"),("undergoes an elongation x, according to"),("Hook's law, F=kx, where k is the force"),("constant.Force constant (k) of the wire"),("will depend on"):}`

Answer

To solve the problem, we need to match the items in Column-I with the corresponding dependencies in Column-II. Let's analyze each part step by step.

### Step 1: Analyze Part (A)
**Statement:** Let us suspend the wire vertically from a rigid support and attach a mass \( m \) at its lower end. If the mass is slightly pulled down and released, it executes SHM of a time period which will depend on...

**Solution:**
The time period \( T \) of the simple harmonic motion (SHM) can be expressed as:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
where \( k \) is the force constant. From the relationship of Young's modulus, we know:
\[ k = \frac{YA}{L} \]
Thus, substituting for \( k \):
\[ T = 2\pi \sqrt{\frac{mL}{YA}} \]
This shows that the time period depends on:
- Length \( l \)
- Area of cross-section \( A \)
- Young's modulus \( Y \)

**Matching:** (A) → (P), (R), (S)

### Step 2: Analyze Part (B)
**Statement:** Work done in stretching the wire up to length \( l + x \) will depend on...

**Solution:**
The work done \( W \) in stretching the wire can be calculated using:
\[ W = \int_0^x F \, dx \]
where \( F = \frac{YA}{L} x \). Thus,
\[ W = \int_0^x \frac{YA}{L} x \, dx = \frac{YA}{L} \cdot \frac{x^2}{2} = \frac{YAx^2}{2L} \]
This indicates that the work done depends on:
- Young's modulus \( Y \)
- Area of cross-section \( A \)
- Length \( l \)
- Elongation \( x \)

**Matching:** (B) → (P), (Q), (R), (S)

### Step 3: Analyze Part (C)
**Statement:** If the wire is fixed between two rigid supports and its temperature is decreased, the thermal stress that develops in the wire will depend on...

**Solution:**
The thermal stress \( \sigma \) can be expressed as:
\[ \sigma = Y \cdot \alpha \Delta T \]
where \( \alpha \) is the coefficient of linear expansion and \( \Delta T \) is the change in temperature. The stress depends on Young's modulus \( Y \) but does not depend on the area or length explicitly in this context.

**Matching:** (C) → (P)

### Step 4: Analyze Part (D)
**Statement:** If the wire is pulled at its ends with equal and opposite forces of magnitude \( F \) so that it undergoes an elongation \( x \), according to Hook's law, \( F = kx \), where \( k \) is the force constant. The force constant \( k \) of the wire will depend on...

**Solution:**
From the earlier derivation, we have:
\[ k = \frac{YA}{L} \]
Thus, \( k \) depends on:
- Young's modulus \( Y \)
- Area of cross-section \( A \)
- Length \( l \)

**Matching:** (D) → (P), (R), (S)

### Final Matching Summary:
- (A) → (P), (R), (S)
- (B) → (P), (Q), (R), (S)
- (C) → (P)
- (D) → (P), (R), (S)