A particle undergoes simple harmonic motion having time period T. The time taken to move one extreme to other position is

In simple harmonic motion (SHM), the time period \( T \) is the total time it takes for a particle to complete one full cycle of motion, which includes moving from one extreme position to the other and back again. To understand the time taken to move from one extreme to the opposite extreme, we can break down the motion into segments.

Understanding the Motion

In SHM, the particle oscillates back and forth around an equilibrium position. The motion can be visualized as follows:

• Starting at one extreme position (let's call it A).
• Moving through the equilibrium position (let's call it O).
• Reaching the opposite extreme position (let's call it B).

Time Taken for Half a Cycle

The time taken to move from one extreme position (A) to the other extreme position (B) is actually half of the time period \( T \). This is because a full cycle consists of moving from A to O and then from O to B, which takes the entire time period \( T \).

Mathematical Representation

To express this mathematically, we can say:

• Time taken to move from A to O = \( \frac{T}{2} \)
• Time taken to move from O to B = \( \frac{T}{2} \)

Thus, the total time for the complete cycle is:

T = Time from A to O + Time from O to B

Therefore, the time taken to move from one extreme position to the other is:

Time from A to B = \( \frac{T}{2} \)

Example for Clarity

Let’s say the time period \( T \) of a pendulum is 4 seconds. The time taken to move from one extreme position to the other would be:

Time from A to B = \( \frac{4 \text{ seconds}}{2} = 2 \text{ seconds} \)

This means it takes 2 seconds for the pendulum to swing from one side to the other.

Visualizing the Motion

Imagine a swing at a playground. When you push it to one side (A), it takes some time to come back to the center (O) and then to the other side (B). The time it takes to go from A to B is the same as the time it takes to go from B back to A, which is why we can confidently say that it takes half the time period to travel between the extremes.

In summary, the time taken for a particle undergoing simple harmonic motion to move from one extreme position to the opposite extreme is always half of the time period \( T \), represented as \( \frac{T}{2} \). This fundamental concept is crucial in understanding the dynamics of oscillatory systems.