Assertion : A spring block system is kept over a smooth surface as shown in figure. If a constant horizontal force `F` is applied on the block it will start oscillating simple harmonically.
Reason : Time period of oscillation is less then `2pi sqrt ((m)/(k))`.

Options

A. Both Assertion and Reason are true and the Reason is correct explanation of the Assertion.
B. Both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
C. Assertion is true, but the Reason is false.
D. Assertion is false but the Reason is true.

Explanation

To solve the question, we need to analyze both the assertion and the reason provided.

### Step-by-Step Solution:

1. **Understanding the Assertion**:
- The assertion states that a spring-block system on a smooth surface will oscillate harmonically when a constant horizontal force \( F \) is applied.
- Since the surface is smooth, we can ignore friction. The spring will stretch due to the applied force until it reaches a new equilibrium position.

2. **Equilibrium Position**:
- When the force \( F \) is applied, the spring stretches by a distance \( x_0 \) from its natural length.
- At this new equilibrium position, the force exerted by the spring \( (F_s) \) is equal to the applied force \( F \). Thus, we have:
\[ F = k \cdot x_0 \]
- This indicates that the system is in equilibrium at this position.

3. **Displacement from Equilibrium**:
- If the block is displaced from this equilibrium position by a distance \( x \), the spring will exert a restoring force.
- The net force acting on the block when displaced is:
\[ F' = F - k(x + x_0) \]
- Substituting \( F = k \cdot x_0 \) into the equation gives:
\[ F' = k \cdot x_0 - k(x + x_0) = -k \cdot x \]
- This shows that the net force is proportional to the displacement \( x \) and acts in the opposite direction, indicating simple harmonic motion.

4. **Equation of Motion**:
- According to Newton's second law, we can express the net force as:
\[ F' = m \cdot a = -k \cdot x \]
- This leads to the equation:
\[ a = -\frac{k}{m} \cdot x \]
- This is the standard form of simple harmonic motion, confirming that the assertion is true.

5. **Analyzing the Reason**:
- The reason states that the time period of oscillation is less than \( 2\pi \sqrt{\frac{m}{k}} \).
- The time period \( T \) for a simple harmonic oscillator is given by:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
- This indicates that the time period is exactly \( 2\pi \sqrt{\frac{m}{k}} \), not less than it. Therefore, the reason is false.

6. **Conclusion**:
- The assertion is true, and the reason is false. Thus, the correct option is that the assertion is true, but the reason is false.

### Final Answer:
The assertion is true, and the reason is false.