For spring kL=constant and if you increasing leanth n time k become 1/ n time and T =2π√m/k so it become √n time.

Effect of Increasing the Length of a Spring on its Time Period

Introduction:
The time period of a spring is the time taken for one complete oscillation or cycle. It is the reciprocal of the frequency and is denoted by T. The time period depends on various factors, including the mass of the object attached to the spring, the spring constant, and the initial displacement. However, in this explanation, we will focus on the effect of increasing the length of a spring on its time period.

Relationship between Length and Time Period:
When the length of a spring is increased, it affects the time period in the following ways:

1. Inversely Proportional: The time period of a spring is inversely proportional to the square root of its effective length. This means that as the length of the spring is increased, the time period will decrease.

2. Mathematical Relationship: The relationship between the length (L) and time period (T) of a spring can be mathematically expressed as T ∝ √L or T = k√L, where k is a constant.

3. Explanation: When the length of a spring is increased, the restoring force acting on the mass decreases since the spring becomes less stiff. As a result, the time taken for one complete oscillation increases, leading to a longer time period. Conversely, if the length of the spring is decreased, the restoring force increases, resulting in a shorter time period.

4. Practical Example: Consider a simple pendulum, which consists of a mass (bob) attached to a spring. If we increase the length of the spring, the bob will take more time to complete one full swing. This demonstrates that increasing the length of the spring increases the time period.

Conclusion:
In conclusion, increasing the length of a spring has a direct effect on its time period. As the length is increased, the time period also increases. This relationship is mathematically expressed as T ∝ √L. It is important to note that the time period also depends on other factors, such as the mass and spring constant. However, for the specific case of changing the length of a spring, the inverse relationship between length and time period holds true.