What is the differential equation of SHM?

Short Answer:

The differential equation of simple harmonic motion (SHM) expresses the relationship between displacement, time, and acceleration of a vibrating body. It is given by:

where  is the displacement,  is the time, and  is the angular frequency of vibration.

In simple words, this equation shows that the acceleration of a body in SHM is directly proportional to its displacement from the mean position and is always directed toward it. The equation helps describe mathematically how an object vibrates or oscillates in simple harmonic motion.

Detailed Explanation :

Differential Equation of SHM

The differential equation of simple harmonic motion (SHM) represents the mathematical relationship between the displacement of a vibrating particle and the restoring force (or acceleration) acting on it. It describes how the displacement of a body changes with time under the influence of a restoring force proportional to its displacement but acting in the opposite direction.

In SHM, when a body moves away from its mean or equilibrium position, a restoring force acts on it to bring it back. According to Hooke’s law, this restoring force is proportional to the displacement and acts in the opposite direction.

Here,

  •  = restoring force (N)
  •  = spring stiffness or force constant (N/m)
  •  = displacement from the mean position (m)
  • The negative sign indicates that the force acts opposite to the direction of displacement.

This fundamental law forms the basis of deriving the differential equation of SHM.

Derivation of Differential Equation of SHM

Consider a mass-spring system where a mass  is attached to one end of a light, elastic spring, and the other end is fixed to a rigid support. When the mass is displaced by a small distance  from its mean position, the spring exerts a restoring force .

According to Newton’s second law of motion:

Equating both expressions of force:

Rearranging, we get:

Let,

where  is the angular frequency of vibration.

Thus, the differential equation of simple harmonic motion becomes:

This is the standard form of the SHM equation.

Explanation of Terms

  1. Displacement (x):
    The distance of the vibrating particle from its mean position at any time . It can be positive or negative depending on the direction of motion.
  2. Acceleration ():
    The rate of change of velocity with respect to time. It is always directed opposite to displacement in SHM.
  3. Angular Frequency (ω):
    It represents how quickly the body oscillates in radians per second.
  1. Restoring Force:
    The force that brings the vibrating particle back to its equilibrium position. It is proportional to displacement but opposite in direction.
  2. Period (T):
    The time required to complete one full oscillation.
  1. Frequency (f):
    The number of oscillations per second.

Solution of Differential Equation

The general solution of the SHM differential equation:

is given by:

or equivalently,

where,

  • A = amplitude of vibration (maximum displacement),
  • ω = angular frequency,
  • t = time,
  • φ = phase angle, which determines the initial position of the particle.

This equation describes how the displacement varies with time during SHM.

Velocity and Acceleration from the Differential Equation

From the displacement equation:

  1. Velocity (v):
    The velocity is the first derivative of displacement with respect to time:

Velocity is maximum at the mean position and zero at the extreme positions.

  1. Acceleration (a):
    The acceleration is the second derivative of displacement:

or

This equation shows that acceleration is always directed toward the mean position and is proportional to displacement.

Physical Meaning of the Differential Equation

The differential equation of SHM expresses that:

  1. The acceleration of a vibrating body is directly proportional to its displacement from the mean position.
  2. The direction of acceleration is always opposite to displacement.
  3. The motion is periodic and repeats itself after equal intervals of time.
  4. The body vibrates at a frequency determined by its stiffness and mass.

This equation forms the foundation for understanding vibrations in various mechanical and electrical systems.

Examples of Systems Following the SHM Differential Equation

  1. Mass-Spring System:
    A weight suspended on a spring moves up and down according to the SHM equation.
  2. Simple Pendulum (for small angles):
    The motion of a pendulum is approximately SHM for small angular displacements.
  3. Tuning Fork:
    The vibrating prongs of a tuning fork follow SHM, producing sound waves.
  4. LC Electrical Circuit:
    The exchange of energy between an inductor and a capacitor in an LC circuit obeys the SHM differential equation.
  5. Vibration of Machine Components:
    Shafts, beams, and rods often exhibit motion that can be modeled using the SHM equation.
Conclusion

The differential equation of simple harmonic motion,

represents the fundamental relationship between acceleration and displacement in oscillatory motion. It shows that acceleration is directly proportional to the displacement and acts toward the equilibrium position. The solution to this equation describes the sinusoidal motion characteristic of SHM. This equation is widely used in mechanical, civil, and electrical engineering to study vibrations, resonance, and dynamic behavior of systems.