A horizontally placed spring mass system has time period T. The same system is now placed on a car moving with acceleration a in horizontal direction. Then,

Options

A. time period will increase
B. time period will decrease
C. time period will remain constant
D. no conclusion can be drawn

Explanation

To solve the problem, we need to analyze how the time period of a spring-mass system is affected when it is placed in a car that is accelerating horizontally.

### Step-by-Step Solution:

1. **Understand the Time Period of a Spring-Mass System**:
The time period \( T \) of a simple harmonic oscillator (spring-mass system) is given by the formula:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
where \( m \) is the mass of the block and \( k \) is the spring constant.

2. **Identify the Forces Acting on the Mass**:
In a spring-mass system, the restoring force acting on the mass when it is displaced from its equilibrium position is given by Hooke's Law:
\[ F = -kx \]
where \( x \) is the displacement from the equilibrium position.

3. **Consider the Effect of Acceleration**:
When the spring-mass system is placed in a car that is accelerating with acceleration \( a \), we need to consider how this affects the system. The acceleration of the car does not change the spring constant \( k \) or the mass \( m \) of the block.

4. **Analyze the System in the Accelerating Frame**:
In the accelerating frame of the car, the effective force acting on the mass can be thought of as being influenced by the pseudo force due to the acceleration of the car. However, this does not alter the restoring force provided by the spring.

5. **Conclude on the Time Period**:
Since the time period \( T \) depends only on the mass \( m \) and the spring constant \( k \), and neither of these quantities changes when the system is placed in an accelerating car, we can conclude that the time period remains unchanged.

Therefore, the new time period \( T' \) when the system is in the accelerating car is:
\[ T' = T \]

### Final Answer:
The time period remains constant when the spring-mass system is placed in a car moving with acceleration \( a \).