The period of a particle executing SHM is T. There is a Point P at a distance x from mean position O. When the particle passes P towards OP it has speed V then the time in which it return to P again? (Ans in terms of T, V,x)

To find the time it takes for a particle executing simple harmonic motion (SHM) to return to point P after passing it, we need to consider the relationship between the period of the motion, the speed of the particle, and its position relative to the mean position O.

Understanding Simple Harmonic Motion

In simple harmonic motion, a particle oscillates back and forth around a mean position. The motion is periodic and can be described using various parameters, such as amplitude, period, and speed at any given point. The period (T) is the time taken to complete one full cycle of motion.

Key Variables

• T: The period of the SHM.
• V: The speed of the particle as it passes point P.
• x: The distance from the mean position O to point P.

Analyzing the Motion

When the particle is at point P, it is moving towards the mean position O. The time taken to return to point P can be analyzed in two parts: the time it takes to go from P to O and then back from O to P.

Calculating Time from P to O

Given that the particle is moving toward O with speed V, the time taken to reach O from point P can be calculated using the formula:

Time from P to O = Distance / Speed = x / V

Calculating Time from O to P

After reaching O, the particle will move back toward P. The total distance from O back to P is again x. In a complete cycle of SHM, the time taken to return from O to P is half of the period T because it takes T to go from one extreme to the other (A to -A) and back through the mean position. Therefore:

Time from O to P = T / 2

Bringing It All Together

To find the total time taken for the particle to return to point P after passing it, we simply add the two time intervals together:

Total time to return to P = Time from P to O + Time from O to P

Substituting the values we calculated:

Total time to return to P = (x / V) + (T / 2)

Final Expression

So, the time in which the particle returns to point P again, expressed in terms of T, V, and x, is:

Time = (x / V) + (T / 2)

This formula gives you a clear understanding of how long it takes for the particle to return to point P after passing through it in its oscillatory motion. It effectively combines the principles of SHM with the specifics of the particle's speed and position.