1. Why Energy Matters in SHM
In Simple Harmonic Motion, the object continuously speeds up, slows down, and reverses direction. All these changes happen because the energy keeps shifting between kinetic energy and potential energy. Understanding how this energy exchange works makes the entire motion feel much more intuitive.
I like to imagine SHM as a constant energy ‘conversation’ between KE and PE — when one increases, the other decreases, but together they always stay the same.
2. Types of Energy in SHM
There are two main forms of energy involved:
- Kinetic Energy (KE): energy due to motion.
- Potential Energy (PE): stored energy due to displacement from the mean position.
The total energy is the sum of these two and stays constant as long as there is no friction or damping.
3. Kinetic Energy in SHM
Definition: Kinetic energy is the energy an object has because of its motion.
In SHM, the object moves fastest at the mean position, so kinetic energy is maximum there. At the extreme positions, the velocity becomes zero, so kinetic energy is zero.
3.1. Mathematical Expression for KE
The instantaneous kinetic energy is:
\( K = \dfrac{1}{2} m v^2 \)
Using the velocity expression for SHM:
\( v(t) = A\omega \cos(\omega t + \phi) \)
we get:
\( K = \dfrac{1}{2} m A^2 \omega^2 \cos^2(\omega t + \phi) \)
3.2. Where KE Is Maximum and Minimum
Maximum KE: at the mean position (velocity is maximum).
Zero KE: at the extreme positions (velocity is zero).
4. Potential Energy in SHM
Definition: Potential energy in SHM is the energy stored due to displacement from the equilibrium position.
The farther the object is from the mean position, the greater the potential energy. At the mean position, PE is zero.
4.1. Mathematical Expression for PE
For a spring system, potential energy is:
\( U = \dfrac{1}{2} k x^2 \)
Using the displacement equation:
\( x(t) = A \sin(\omega t + \phi) \)
we get:
\( U = \dfrac{1}{2} k A^2 \sin^2(\omega t + \phi) \)
4.2. Where PE Is Maximum and Minimum
Maximum PE: at the extreme positions (displacement is maximum).
Zero PE: at the mean position (displacement is zero).
5. Total Mechanical Energy in SHM
A beautiful feature of SHM is that the total energy remains constant. It only switches back and forth between KE and PE.
Total energy is given by:
\( E = K + U = \dfrac{1}{2} k A^2 \)
This value depends only on the amplitude and not on time, which is why the system keeps oscillating smoothly (in the absence of friction).
5.1. Energy Distribution at Key Positions
- At extremes: PE is maximum, KE is zero.
- At mean position: KE is maximum, PE is zero.
- Between these two: both KE and PE share the total energy.
The smooth swapping between KE and PE is what keeps the motion rhythmic.
6. Graphical Understanding of Energy in SHM
The energy curves in SHM look like this conceptually:
- KE varies as a cos² wave.
- PE varies as a sin² wave.
- Total energy is a constant horizontal line.
Whenever cos² is maximum, sin² is zero, and vice versa — this perfectly reflects the swapping of energies.
7. Simple Real-Life Picture of Energy Exchange
If I picture a mass on a spring:
- At the topmost stretch: it has only potential energy.
- As it moves downward: potential energy decreases; kinetic energy increases.
- At the centre: it has only kinetic energy.
- As it moves up on the other side: kinetic energy decreases; potential energy builds up again.
This cycle repeats endlessly as long as no external forces drain the energy.