Physics Formula: Simple Harmonic Motion

Introduction to Simple Harmonic Motion (SHM)

Simple Harmonic Motion (SHM) is a special type of periodic motion in which the restoring force is directly proportional to the negative of the displacement from equilibrium. It is a core concept in classical mechanics and forms the basis for understanding oscillations, waves, sound, and many physical systems.

Real-Life Examples of SHM

Spring-mass systems (both horizontal and vertical)
Pendulums (for small angular displacements)
Molecular vibrations in chemistry and biology
Alternating current in electrical circuits
Seismic waves in earthquakes

Mathematical Description of SHM

The displacement in SHM as a function of time is typically given by:

\[ x(t) = A \cos(\omega t + \phi) \]

Or alternatively:

\[ x(t) = A \sin(\omega t + \phi) \]

Depending on the initial conditions. Both forms are valid representations of SHM.

Important SHM Parameters

Amplitude (A): The maximum displacement from the equilibrium.
Angular Frequency (ω): Determines how fast the system oscillates in radians per second.
Time Period (T): Time taken for one complete oscillation.
Frequency (f): Number of oscillations per second, \( f = \frac{1}{T} \).
Phase Constant (φ): Determines the initial position and direction of motion.

Derivation of SHM from Newton’s Laws

Consider a mass attached to a spring. According to Newton's Second Law:

\[ F = ma = -kx \Rightarrow a = -\frac{k}{m}x \]

This differential equation:

\[ \frac{d^2x}{dt^2} + \omega^2 x = 0 \quad \text{where} \quad \omega = \sqrt{\frac{k}{m}} \]

has the general solution:

\[ x(t) = A \cos(\omega t) + B \sin(\omega t) \]

Which can also be written compactly as: \[ x(t) = A \cos(\omega t + \phi) \]

Energy Distribution in SHM

In SHM, the energy oscillates between potential and kinetic forms, but the total energy remains constant (in ideal conditions).

Graph of Energy vs. Time

Potential energy (PE) is maximum at extreme positions where velocity is zero.
Kinetic energy (KE) is maximum at the equilibrium position where displacement is zero.
Total energy (E) remains constant: \( E = \frac{1}{2}kA^2 \)

Graphical Insights

Graphs help visualize SHM concepts:

Displacement vs Time: Sinusoidal curve.
Velocity vs Time: Also sinusoidal but shifted by \( \frac{\pi}{2} \).
Acceleration vs Time: Sinusoidal, but with a negative cosine relationship.

Multiple SHM Systems

When multiple SHMs are superimposed, they result in complex motion. This is commonly observed in real mechanical systems and wave phenomena.

Case: Two SHMs of Same Frequency

If two SHMs with the same frequency but different amplitudes and phases are added:

\[ x(t) = A_1 \cos(\omega t + \phi_1) + A_2 \cos(\omega t + \phi_2) \]

This results in another SHM, whose amplitude and phase depend on the original amplitudes and phases.

Simple Pendulum as SHM

For small angles (less than about 15°), a simple pendulum performs SHM. The motion follows:

\[ \theta(t) = \theta_0 \cos(\omega t + \phi) \quad \text{with} \quad \omega = \sqrt{\frac{g}{L}} \]

And time period: \[ T = 2\pi \sqrt{\frac{L}{g}} \]

Vertical SHM

A mass hanging vertically from a spring also exhibits SHM. The extension in equilibrium shifts, but the motion remains sinusoidal with:

\[ \omega = \sqrt{\frac{k}{m}} \quad \text{and} \quad T = 2\pi \sqrt{\frac{m}{k}} \]

SHM in Circular Motion

Uniform circular motion is the projection of SHM on one axis. If a particle moves in a circle at constant speed, the projection on any diameter is simple harmonic.

This analogy helps to visualize SHM as circular motion in disguise.

Resonance in SHM

Resonance occurs when the frequency of a driving force matches the natural frequency of the system. This causes the amplitude to increase significantly.

Real-Life Resonance Examples

Vibrating strings in musical instruments
Tacoma Narrows Bridge collapse due to wind resonance
Resonance in microwave ovens and radio circuits

SHM and Damping

Damping is the effect of energy loss (e.g., through friction or air resistance). It causes the amplitude to gradually decrease over time.

Damped SHM equation: \[ x(t) = A e^{-\gamma t} \cos(\omega' t + \phi) \]

Where \( \gamma \) is the damping constant, and \( \omega' \) is the modified angular frequency.

Forced Oscillations

When an external periodic force is applied to a system, it undergoes forced oscillations. At resonance, the system can absorb maximum energy from the driver.

Complex SHM Applications

Simple Harmonic Motion is the building block for analyzing:

Mechanical vibrations in cars and machines
Sound waves (as longitudinal SHM)
Electrical circuits with capacitors and inductors (LCR circuits)
Quantum mechanics: quantum harmonic oscillator

Advanced Example Problem

Example 2: Simple Pendulum on the Moon

A pendulum is 1.5 m long. Find its period on the Moon where \( g = 1.63 \, \text{m/s}^2 \).

Solution:

\[ T = 2\pi \sqrt{\frac{L}{g}} = 2\pi \sqrt{\frac{1.5}{1.63}} \approx 2\pi (0.96) \approx 6.03 \, \text{seconds} \]

Common SHM Mistakes to Avoid

Assuming all periodic motion is SHM (e.g., square waves are not SHM)
Confusing angular frequency with angular velocity
Forgetting that SHM only applies for small angle approximations in pendulums
Mixing up phase and phase difference

Tips for Solving SHM Problems

Always identify the type of system: spring, pendulum, or compound.
Find the natural frequency or time period from given values.
Use energy conservation if dealing with maximum speed or displacement.
Understand the phase relationships between displacement, velocity, and acceleration.

Conclusion

Simple Harmonic Motion is not just a textbook topic — it's a cornerstone of physics with wide-ranging applications. From the simplest mass-spring systems to complex molecular vibrations and electrical circuits, SHM principles help us model and predict real-world phenomena. Mastering the SHM formulas and concepts prepares students for advanced topics in wave physics, acoustics, electromagnetism, and quantum theory.

Practice Questions

What is the time period of a 2 kg mass attached to a spring with \( k = 80 \, \text{N/m} \)?
Find the total mechanical energy of a SHM system with \( A = 0.05 \, \text{m} \) and \( k = 200 \, \text{N/m} \).
If a pendulum has a period of 2.5 s, what is its length?
Derive the expression for velocity in SHM from the displacement equation.

References

Halliday, Resnick & Walker, Fundamentals of Physics
Serway & Jewett, Physics for Scientists and Engineers
OpenStax Physics Textbook

Formula Quest Mania