Energy of a Particle Executing Simple Harmonic Motion (SHM): Comprehensive NEET Physics Notes

1. Energy in Simple Harmonic Motion (SHM)

In simple harmonic motion (SHM), both kinetic and potential energies of the particle vary periodically over time. The total mechanical energy, which is the sum of kinetic and potential energies, remains constant throughout the motion, reflecting energy conservation. The interplay between these two types of energy is crucial to understanding SHM dynamics.

1.1 Potential Energy in SHM

The potential energy of a particle in SHM originates from the restoring force, which acts to bring the particle back to its equilibrium position. According to Hooke’s law, the restoring force is proportional to the displacement, given as:

$F=-kx$

where:

• $k$ is the spring constant, and
• $x$ is the displacement from equilibrium.

The potential energy at any displacement from the mean position is:

$U(x)=\frac{1}{2}k{x}^2$

At the extreme positions, where the displacement is maximum (i.e., $x=A$), the potential energy is also at its maximum:

$U_{\text{max}}=\frac{1}{2}k{A}^2$

1.2 Kinetic Energy in SHM

The kinetic energy of a particle in SHM is related to its velocity. The velocity is maximum at the equilibrium position and decreases to zero at the extreme positions. The expression for kinetic energy is:

$K(v)=\frac{1}{2}m{v}^2$

In SHM, the velocity varies with displacement, leading to:

$v=\omega \sqrt{A^2-x^2}$

Thus, the kinetic energy can be expressed as:

$K(x)=\frac{1}{2}m{\omega }^2(A^2-x^2)$

At the mean position (where $x=0$), all the energy is kinetic:

$K_{\text{max}}=\frac{1}{2}m{\omega }^2{A}^2$

1.3 Total Energy in SHM

The total mechanical energy of a particle executing SHM is the sum of its kinetic and potential energies. Importantly, this total energy remains constant, illustrating the conservation of mechanical energy in SHM:

$E=K+U=\frac{1}{2}m{\omega }^2{A}^2$

At any point in the motion, energy is divided between kinetic and potential forms, but the total remains constant.

2. Energy Distribution in SHM

As the particle moves through its SHM, energy shifts between potential and kinetic forms:

• At the extreme positions (where $x=\pm A$), the potential energy is maximum and kinetic energy is zero.
• At the equilibrium position (where $x=0$), the kinetic energy is maximum, and the potential energy is zero.

This exchange between kinetic and potential energy is periodic, with each form peaking alternately during the motion.

NEET Tip:

For NEET, understanding energy conservation in SHM is crucial. Many questions will focus on the conversion between kinetic and potential energies. Always remember, the total mechanical energy remains constant.

Real-Life Application:

The concept of SHM and its energy distribution is widely observed in systems like pendulums, springs, and even molecular vibrations. These principles are used in the design of clocks, musical instruments, and sensors.

Common Misconception:

Students often mistakenly think that kinetic and potential energies are equally divided at all times during SHM. In fact, they vary continuously, with one form reaching its maximum when the other is at its minimum.

Visual Aids

Diagram Suggestion: A graph showing potential energy, kinetic energy, and total mechanical energy as functions of displacement in SHM would help visualize energy changes over time. Label the points where energy is purely kinetic or purely potential.

Quick Recap:

• Potential Energy in SHM: $U(x)=\frac{1}{2}k{x}^2$
• Kinetic Energy in SHM: $K(x)=\frac{1}{2}m{\omega }^2(A^2-x^2)$
• Total Energy remains constant: $E=\frac{1}{2}m{\omega }^2{A}^2$
• Maximum kinetic energy occurs at the mean position ($x=0$), and maximum potential energy occurs at the extreme positions ($x=\pm A$).

3. Practice Problems with Solutions

1. A block of mass 2 kg is attached to a spring with a spring constant of 50 N/m. If the block is displaced by 10 cm from its equilibrium position, calculate its maximum potential energy and maximum kinetic energy.
2. • Solution:
   • • Maximum displacement: $A=0.1,\text{m}$
     • Maximum potential energy:
       $U_{\text{max}}=\frac{1}{2}k{A}^2=\frac{1}{2}\times 50\text{N/m}\times (0.1\text{m}{)}^2=0.25\text{J}$
     • Since total energy is conserved, maximum kinetic energy:
       $K_{\text{max}}=U_{\text{max}}=0.25\text{J}$
2. A particle of mass 1.5 kg is undergoing SHM with an angular frequency of 5 rad/s. What is the total mechanical energy if the amplitude of oscillation is 0.2 m?
3. • Solution:
   • • Total energy:$E=\frac{1}{2}m{\omega }^2{A}^2=\frac{1}{2}\times 1.5\text{kg}\times (5\text{rad/s}{)}^2\times (0.2\text{m}{)}^2=0.15\text{J}$

NEET Problem-Solving Strategy:

When dealing with SHM energy problems, start by identifying the maximum displacement (amplitude) and use conservation of energy to solve for unknowns such as velocity, displacement, or force.

NEET Exam Strategy:

In NEET, SHM problems often test your understanding of energy conservation. Focus on recognizing where the particle is in its motion to determine whether kinetic or potential energy is at its peak. Practice converting between energy forms to master this concept.

Glossary:

• Simple Harmonic Motion (SHM): A type of oscillatory motion where the restoring force is proportional to the displacement.
• Amplitude (A): The maximum displacement from the equilibrium position.
• Angular Frequency (ω): The rate of change of the phase of a sinusoidal waveform.
• Potential Energy (U): The energy stored in the system due to its position.
• Kinetic Energy (K): The energy possessed by the system due to its motion.
• Total Energy (E): The sum of the kinetic and potential energies, which remains constant in SHM.

Summary of Strengths:

• Clear explanation of key SHM energy concepts.
• Practice problems with detailed solutions to reinforce understanding.
• NEET-specific tips and strategies included.

Areas for Improvement:

• Include more visual aids such as energy vs. displacement graphs.
• Provide a glossary for important terms, aiding quick reference.
• Incorporate more practice questions with varying difficulty to challenge students.

Final Recommendations:

1. Add diagrams to visualize energy changes throughout SHM.
2. Include more NEET-style questions with solutions to help students practice.
3. Provide a formula sheet or quick-reference guide at the end for easy recall.