Relation between the Time Period of Two Simple Harmonic Motions

Two simple harmonic motions can be represented by two curves. The time period of a simple harmonic motion is the time taken for one complete cycle of oscillation. In order to understand the relation between the time periods of two simple harmonic motions, we need to consider the following points:

1. Definition of Simple Harmonic Motion:
Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction. It can be represented mathematically as: F = -kx, where F is the restoring force, k is the force constant, and x is the displacement from the equilibrium position.

2. Factors Affecting Time Period:
The time period of a simple harmonic motion depends on two factors:

- Mass: The mass of the object undergoing the simple harmonic motion affects the time period. A larger mass will result in a longer time period.
- Force Constant: The force constant (k) represents the stiffness of the restoring force. A larger force constant will result in a shorter time period.

3. Mathematical Relation:
The time period (T) of a simple harmonic motion can be mathematically represented as: T = 2π√(m/k), where T is the time period, m is the mass, and k is the force constant.

4. Comparing Two Simple Harmonic Motions:
In order to compare the time periods of two simple harmonic motions represented by two curves, we need to consider the mass and force constant of both systems. If the mass and force constant are the same for both systems, the time periods will also be the same. This means that the two curves will have the same shape and frequency of oscillation.

However, if the mass or force constant differs between the two systems, the time periods will also differ. The curve with a larger mass or smaller force constant will have a longer time period, while the curve with a smaller mass or larger force constant will have a shorter time period.

Conclusion:
The time period of two simple harmonic motions represented by two curves depends on the mass and force constant of each system. If the mass and force constant are the same, the time periods will be the same. Otherwise, the time periods will differ, with the curve having a larger mass or smaller force constant having a longer time period.