On loading a spring with bob, its period of oscillation in a vertical plane is T. If this spring pendulum is tied with one end to the a friction less table and made to oscillate in a horizontal, plane, its period of oscillation will be :

Options

A. T
B. 2T
C. T/2
D. will not execute S.H.M.

Answer

To solve the problem, we need to analyze the motion of a spring pendulum in two different orientations: vertical and horizontal. The key is to understand how the forces acting on the bob affect the period of oscillation.

### Step-by-Step Solution:

1. **Understanding the Vertical Oscillation**:
- When the spring pendulum is oscillating in the vertical plane, the time period \( T \) is given by the formula:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
- Here, \( m \) is the mass of the bob and \( k \) is the spring constant.

2. **Analyzing the Horizontal Oscillation**:
- When the spring pendulum is tied to a frictionless table and oscillates in the horizontal plane, the forces acting on the bob change.
- The only forces acting on the bob in the horizontal direction are the restoring force from the spring and the inertial effects due to the mass of the bob.

3. **Restoring Force in Horizontal Motion**:
- The restoring force \( F_r \) when the bob is displaced from its equilibrium position is given by Hooke's law:
\[ F_r = -kx \]
- Here, \( x \) is the displacement from the mean position.

4. **Net Force and Acceleration**:
- According to Newton's second law, the net force acting on the bob is equal to the mass times the acceleration:
\[ F = ma \]
- For small oscillations, we can equate the restoring force to the mass times acceleration:
\[ -kx = m \frac{d^2x}{dt^2} \]
- Rearranging gives:
\[ \frac{d^2x}{dt^2} + \frac{k}{m}x = 0 \]
- This is the standard form of simple harmonic motion.

5. **Time Period in Horizontal Motion**:
- The solution to the differential equation indicates that the motion is simple harmonic with the same time period as in the vertical case:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]

6. **Conclusion**:
- Since the time period of oscillation in the horizontal plane is derived from the same parameters \( m \) and \( k \), it remains unchanged.
- Therefore, the time period of oscillation in the horizontal plane is also \( T \).

### Final Answer:
The period of oscillation in the horizontal plane is \( T \).