A uniform rope of mass m hangs from a ceiling and a block of mass m is attached to the free end of rope.
Statement-1: The speed of a transverse wave in rope is different at different points.
Statement-2: The tension in the rope is different at diferent points.

Options

A. Statement-1 is true, statement-2 is true, statement-2 is a correct explanation for statement-1
B. Statement-1 is true, statement-2 is true, statement-2 is not a correct explanation for statement-1
C. statement-1 is true, statement-2 is false
D. statement-1 is false, statement-2 is true

Answer

To solve the problem, we need to analyze the two statements regarding the uniform rope and the block attached to it.

### Step-by-step Solution:

1. **Understanding the Setup**:
- We have a uniform rope of mass \( m \) hanging from a ceiling.
- A block of mass \( m \) is attached to the free end of the rope.

2. **Analyzing Tension in the Rope**:
- The tension in the rope varies along its length due to the weight of the block and the weight of the rope itself.
- The tension \( T \) at a point \( x \) from the free end of the rope can be expressed as:
\[ T(x) = mg + \mu \cdot x \cdot g \]
where \( \mu \) is the mass per unit length of the rope, and \( g \) is the acceleration due to gravity.

3. **Understanding the Variation of Tension**:
- The total tension at any point \( x \) not only includes the weight of the block \( mg \) but also the weight of the rope segment below that point, which is \( \mu \cdot x \cdot g \).
- As \( x \) increases (moving up the rope), the tension increases because more of the rope's weight is included.

4. **Speed of Transverse Waves in the Rope**:
- The speed \( v \) of a transverse wave in the rope is given by the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
- Since the tension \( T \) varies with \( x \), the speed of the wave will also vary along the length of the rope.

5. **Evaluating the Statements**:
- **Statement 1**: "The speed of a transverse wave in the rope is different at different points."
- This statement is **true** because the tension varies with position, which affects the wave speed.
- **Statement 2**: "The tension in the rope is different at different points."
- This statement is also **true** as we derived that the tension depends on the position \( x \).

6. **Conclusion**:
- Both statements are true, and Statement 2 provides the correct explanation for Statement 1.

### Final Answer:
Both statements are true, and Statement 2 is the correct explanation for Statement 1.