Question

Displacement between maximum potential energy position and maximum kinetic energy postion for a particle executing simple harmonic motion is

• A. $ \pm \, a/ 2$
• B. #NAME?
• C. $\pm$ a
• D. -1

Answer 1

The Correct Option is C

To solve the question about the displacement between the maximum potential energy position and the maximum kinetic energy position for a particle executing simple harmonic motion (SHM), we need to understand the characteristics of SHM.

In simple harmonic motion, the potential energy is maximum when the particle is at its extreme positions, i.e., at the amplitude positions. The kinetic energy is maximum when the particle passes through the mean position because velocity is maximum there.

Consider the following details:

1. In SHM, the equation of motion can be expressed as \( x = a \sin(\omega t + \phi) \) or \( x = a \cos(\omega t + \phi) \), where \( x \) is the displacement, \( a \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase constant.
2. The extreme positions (where potential energy is maximum) are at \( x = \pm a \).
3. The mean position (where kinetic energy is maximum) is at \( x = 0 \).

The displacement between these positions is calculated as follows:

The displacement between the maximum potential energy position \( x = \pm a \) and the maximum kinetic energy position \( x = 0 \) is simply the amplitude itself, i.e., \( |a - 0| = a \) or equivalently \(-a\). This results because you are measuring from either extreme end \( +a \) or \(-a \) to the center at \( 0 \).

Therefore, the correct answer, representing the displacement between the maximum potential energy position and the maximum kinetic energy position for a particle executing simple harmonic motion, is $\pm$ a.

Let's rule out incorrect options:

• The option \( \pm \, a/2 \): This would be incorrect because no standard SHM characteristic or calculation results in half the amplitude being relevant to both potential and kinetic maximums.
• The option \(-1\): This is irrelevant to the context of SHM amplitude measurement.

Thus, the option $\pm$ a is correctly justified.