A particle of mass 0.2 kg performs SHM of amplitude 0.1 M and period 3.14 seconds find its kinetic energy and potential energy when it is at a distance of 0.03 m from mean position

To find the kinetic energy (KE) and potential energy (PE) of a particle undergoing simple harmonic motion (SHM) at a specific distance from the mean position, we can use some fundamental equations related to SHM. Let's break this down step by step.

Understanding the Basics of SHM

In SHM, a particle oscillates around a mean position, and its motion can be described using amplitude, period, and displacement. The key parameters are:

• Mass (m): 0.2 kg
• Amplitude (A): 0.1 m
• Period (T): 3.14 s
• Displacement (x): 0.03 m from the mean position

Calculating Potential Energy

The potential energy in SHM can be calculated using the formula:

PE = (1/2) k x²

Where:

• k: the spring constant
• x: the displacement from the mean position

First, we need to find the spring constant (k). The relationship between the spring constant, mass, and period is given by:

k = (4π²m) / T²

Plugging in the values:

• m = 0.2 kg
• T = 3.14 s

Calculating k:

k = (4 * π² * 0.2) / (3.14)² ≈ 0.08 N/m

Now, substituting k and x into the potential energy formula:

PE = (1/2) * 0.08 * (0.03)² = (1/2) * 0.08 * 0.0009 ≈ 0.000036 J or 36 µJ

Calculating Kinetic Energy

The total mechanical energy (E) in SHM is constant and is the sum of kinetic and potential energy:

E = KE + PE

The total energy can also be expressed as:

E = (1/2) k A²

Calculating the total energy:

E = (1/2) * 0.08 * (0.1)² = (1/2) * 0.08 * 0.01 = 0.0004 J or 400 µJ

Now, we can find the kinetic energy using the total energy and the potential energy we calculated:

KE = E - PE = 0.0004 J - 0.000036 J ≈ 0.000364 J or 364 µJ

Summary of Results

At a distance of 0.03 m from the mean position, the energies are:

• Potential Energy (PE): 36 µJ
• Kinetic Energy (KE): 364 µJ

This analysis shows how energy is distributed in a particle undergoing SHM at a specific displacement, illustrating the interplay between kinetic and potential energy in oscillatory motion.