STATEMENT 1: When a negative charge -q is released at a distance R from the centre and along the axis of a uniformly and positvely charged fixed ring of radius R, the negative charge does oscillation but not SHM. STATEMENT 2: The force on negative charge is always towards the centre of the ring but it is not proportional to the displacement from the centre of the ring.

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 but statement-2 is false
D. Statement-1 is false, Statement-2 is true

Answer

To solve the given problem, we need to analyze the two statements provided regarding the behavior of a negative charge released near a uniformly positively charged ring.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a uniformly positively charged ring with radius \( R \).
- A negative charge \( -q \) is placed at a distance \( R \) along the axis of the ring from its center.

2. **Force on the Negative Charge**:
- The force acting on the negative charge due to the positive charge on the ring can be calculated using Coulomb's law.
- The force \( F \) on the charge \( -q \) at distance \( R \) from the center of the ring is given by:
\[ F = \frac{k \cdot Q \cdot (-q)}{R^2} \]
- Here, \( k \) is Coulomb's constant, and \( Q \) is the total charge of the ring.

3. **Direction of the Force**:
- The force is attractive, meaning it acts towards the center of the ring. Thus, the negative charge experiences a force directed towards the center.

4. **Oscillation of the Charge**:
- When the charge \( -q \) is released, it will move towards the center of the ring due to the attractive force.
- Upon reaching the center, it will continue moving due to inertia and will cross the center, moving to a distance \( R \) on the opposite side.
- The charge will then be pulled back towards the center again, leading to oscillatory motion.

5. **Nature of the Oscillation**:
- The oscillation is not simple harmonic motion (SHM) because, in SHM, the restoring force must be directly proportional to the displacement from the equilibrium position.
- In this case, the force is not proportional to the displacement from the center of the ring; instead, it varies with the square of the distance.

6. **Conclusion on Statements**:
- **Statement 1**: True - The negative charge does oscillate but not in SHM.
- **Statement 2**: True - The force is directed towards the center but is not proportional to the displacement.

### Final Answer:
Both statements are true, and statement 2 correctly explains statement 1.