Energy in SHM
Analysis of kinetic energy, potential energy, and total mechanical energy of a particle executing SHM, including their conservation and graphical representation.
What is Energy in SHM?
A type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
Key formula / rule: Kinetic Energy
Key points
- To analyze the variation of kinetic energy during SHM.
- To analyze the variation of potential energy during SHM.
- To understand the conservation of total mechanical energy in SHM.
- To relate energy to amplitude, mass, spring constant, and angular frequency.
Common exam trap
Confusing maximum kinetic energy with total energy.
Definitions
- Term
Simple Harmonic Motion (SHM)
- Meaning
A type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
- Term
Amplitude (A)
- Meaning
The maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position.
- Term
Mean Position
- Meaning
The equilibrium position where the net force on the oscillating object is zero.
- Term
Extreme Positions
- Meaning
The positions of maximum displacement from the mean position.
Learning objectives
To analyze the variation of kinetic energy during SHM.
To analyze the variation of potential energy during SHM.
To understand the conservation of total mechanical energy in SHM.
To relate energy to amplitude, mass, spring constant, and angular frequency.
Formulae
- Name
Kinetic Energy
- Note
v is the instantaneous velocity.
- Expression
K = (1/2)mv2
- Name
Potential Energy
- Note
k is the spring constant, x is the displacement from mean position.
- Expression
U = (1/2)kx2
- Name
Total Mechanical Energy
- Note
Sum of kinetic and potential energy.
- Expression
E = K + U
- Name
Total Energy in terms of Amplitude
- Note
A is the amplitude of oscillation.
- Expression
E = (1/2)kA2
- Name
Total Energy in terms of Amplitude and Angular Frequency
- Note
m is mass, ω is angular frequency.
- Expression
E = (1/2)mω2A^2
- Name
Velocity in SHM
- Note
Used to find K at a given displacement x.
- Expression
v = ω√(A2 - x2)
Prerequisites
Understanding of displacement, velocity, and acceleration.
Basic concepts of kinetic and potential energy.
Definition and characteristics of periodic motion.
Understanding of Simple Harmonic Motion (SHM).
Common mistakes
Confusing maximum kinetic energy with total energy.
Assuming potential energy is always zero.
Forgetting that energy transformation is continuous, not instantaneous.
Incorrectly applying energy conservation in the presence of damping.
Keywords
SHM
Simple Harmonic Motion
Kinetic Energy
Potential Energy
Total Energy
Energy Conservation
Amplitude
Mean Position
Extreme Position
Angular Frequency
Practice preview
In simple harmonic motion (SHM), what is the nature of the total mechanical energy?…
easy
When a particle is at the extreme position in SHM, its potential energy is:…
easy
For a particle executing SHM, the total energy is E. What is the kinetic energy when the displacement is half of the amplitude?…
medium