Energy in Simple Harmonic Motion (SHM) Experiment for Schools, Teachers, and Students
In Simple Harmonic Motion (SHM), the total energy of the system remains constant and is the sum of the potential energy (due to displacement) and the kinetic energy (due to velocity) at any instant.
Total Mechanical Energy in SHM:
The total energy (E) of a particle undergoing SHM is given by:
E = 1/2 k A²
where:
- k: Force constant (N/m),
- A: Amplitude of oscillation (maximum displacement).
The total energy remains constant throughout the motion, assuming no energy loss due to friction or damping.
Kinetic Energy in SHM:
The kinetic energy (K) of the particle at displacement x is given by:
K = 1/2 m v²
Using the relation v = ω √(A² – x²), where ω = √(k / m), the kinetic energy becomes:
K = 1/2 k (A² – x²)
Here:
- A: Amplitude,
- x: Instantaneous displacement,
- k: Force constant.
At the mean position (x = 0), the kinetic energy is maximum:
K_max = 1/2 k A²
At the extreme positions (x = ± A), the kinetic energy is zero.
Potential Energy in SHM:
The potential energy (U) of the particle at displacement x is due to the restoring force and is given by:
U = 1/2 k x²
At the extreme positions (x = ± A), the potential energy is maximum:
U_max = 1/2 k A²
At the mean position (x = 0), the potential energy is zero.
Total Energy as the Sum of K and U:
The total energy at any displacement x is:
E = K + U
Substitute for K and U:
E = 1/2 k (A² – x²) + 1/2 k x²
Simplifying:
E = 1/2 k A²
This shows that the total energy remains constant and is independent of the displacement.
Energy Distribution in SHM:
- At the mean position (x = 0):
K = E and U = 0
- At the extreme positions (x = ± A):
K = 0 and U = E
- At any intermediate position:
K = 1/2 k (A² – x²) and U = 1/2 k x²
Graphical Representation of Energy in SHM:
- Kinetic Energy (K) vs Displacement (x): The kinetic energy is maximum at x = 0 and zero at x = ± A. The graph of K is parabolic:
K = 1/2 k (A² – x²)
- Potential Energy (U) vs Displacement (x): The potential energy is zero at x = 0 and maximum at x = ± A. The graph of U is also parabolic:
U = 1/2 k x²
- Total Energy (E) vs Displacement (x): The total energy remains constant and is independent of x:
E = 1/2 k A²
Observations:
- The total energy (E) is constant in ideal SHM.
- The kinetic energy and potential energy vary periodically and are complementary.
- At any point in the motion, K + U = E.
- The energy distribution depends on the displacement x and reaches its extremes at the mean and extreme positions.
Applications:
- Energy analysis in mechanical oscillators, like spring-mass systems.
- Understanding the energy transformation in pendulums.
- Basis for studying molecular vibrations and quantum oscillators.
