Energy In Simple Harmonic Motion

Potential energy of a simple harmonic oscillator

The potential energy,


or
or

Step 1: Potential Energy and Kinetic Energy in SHM.
The total energy in SHM is: Let the amplitude of the particle be , and the position at any instant is . The potential energy is: where is the spring constant. The kinetic energy is: Step 2: At .
Substitute : Simplify: Step 3: Ratio of to .
The ratio is: Simplify: Thus: Step 4: Final Answer.
The ratio of to is:

Concept: The potential energy of a particle performing Simple Harmonic Motion (SHM) is given by where
• = mass of the particle
• = angular frequency
• = displacement The frequency is related to angular frequency by

Step 1: Compare the given potential energy with the SHM formula. Given Standard form Thus,

Step 2: Substitute the mass. Mass Substitute into the equation:

Step 3: Find the frequency. For the given approximation in the problem,

Concept: For SHM starting from the mean position, displacement is Potential Energy: Kinetic Energy: Thus,

Step 1: Substitute . Angular frequency relation: Thus,

Step 2: Find displacement.

Step 3: Compute energy ratio. Thus,

Concept: For a particle executing simple harmonic motion, the potential energy is expressed as where is the force constant of the system. By comparing the given potential energy expression with this standard form, the value of the force constant can be determined. The angular frequency of oscillation is given by and the frequency of oscillation is These relations connect the restoring force constant, mass of the particle, and frequency of oscillation.

Step 1: Compare the given potential energy with the standard form.
The given expression is The standard expression is Comparing the coefficients of :

Step 2: Convert the mass into SI units.
The mass of the particle is given as

Step 3: Calculate the angular frequency.
Using

Step 4: Determine the frequency.


Step 5: Final result.
Thus, the frequency of oscillation of the particle is

Concept: In Simple Harmonic Motion (SHM):

• Displacement: (when motion starts from mean position)
• Angular frequency:
• Potential Energy:
• Total Energy:
• Kinetic Energy:

Thus, and
Step 1: {Find the displacement at .} Since and Thus,
Step 2: {Find potential energy fraction.} Thus,
Step 3: {Find kinetic energy.}
Step 4: {Find the required ratio.}