Time taken to traverse half the amplitude in SHM

In simple harmonic motion (SHM), a particle oscillates back and forth about its mean position. The time taken for the particle to complete one full oscillation is known as the periodic time or time period. In this case, we are given that the particle has a periodic time of 6 seconds.

To find the time taken to traverse half the amplitude from the mean position, we need to understand the nature of SHM and its relationship with time.

Understanding Simple Harmonic Motion (SHM)

SHM is a type of periodic motion that can be represented by a sine or cosine function. It follows certain characteristics:

1. The motion is periodic, meaning it repeats itself at regular intervals.
2. The motion is symmetrical around the mean position, which is the equilibrium or rest position.
3. The motion is governed by a restoring force that is proportional to the displacement from the mean position and acts opposite to the direction of displacement.

Relationship between Time and Displacement in SHM

The displacement (x) of a particle undergoing SHM can be represented by the equation:

x = A * sin(2π/T * t)

where:
- x is the displacement from the mean position
- A is the amplitude of the motion
- T is the periodic time or time period
- t is the time

In SHM, the particle oscillates between the extreme positions, which are at a distance of 2A from the mean position. It takes half the periodic time to traverse half the amplitude from the mean position.

Calculation of Time Taken to Traverse Half the Amplitude

Given that the periodic time (T) is 6 seconds, we can find the time taken to traverse half the amplitude by dividing the periodic time by 2:

Time taken = T/2 = 6/2 = 3 seconds

Therefore, the time taken for the particle to traverse half the amplitude from the mean position is 3 seconds.

Summary

In summary, when a particle undergoes simple harmonic motion (SHM) with a periodic time of 6 seconds, it takes half the periodic time (3 seconds) to traverse half the amplitude from the mean position. This can be determined by understanding the nature of SHM and its relationship between time and displacement.