Class 10 > Unit # 10: General Waves Properties > Simple Harmonic Motion (SHM)

Simple Harmonic Motion (S.H.M)

Video Lecture

Definition

Simple Harmonic Motion (S.H.M): To and fro motion of a body in which acceleration is directly proportional to displacement and always directed towards the mean position is known as Simple Harmonic Motion.

Examples of S.H.M

• Body attached to a spring horizontally on an ideal smooth surface.
• Motion of a simple and compound pendulum.
• Motion of a swing.
• Motion of a ball inside a bowl.

---

Motion of Mass Attached to a Spring

Consider the motion of a mass \(m\) attached to a spring placed horizontally on a frictionless surface. Initially, at the mean position \((x = 0)\), there is no net force acting on mass \(m\) because the spring suffers no extension or compression.

Figure 1: Mass-spring system executing Simple Harmonic Motion across mean and extreme positions.

Now, when the spring is pulled to the right through a displacement \(x\), a restoring force is applied by the spring pulling it back toward the mean position.

The work done in pulling the spring through distance \(x\) is stored as elastic potential energy in the spring. When released, the restoring force moves mass \(m\) back towards the left, converting potential energy into kinetic energy.

At the mean position, the potential energy becomes zero and kinetic energy reaches maximum (maximum speed). Due to inertia, the mass does not stop at the mean position but continues moving to the left, compressing the spring. As it compresses, the motion retards until it comes to a temporary stop at the extreme left position, where all energy is converted back into potential energy. This process repeats continuously, oscillating the energy between kinetic and potential forms.

---

Mathematical Derivation

According to Hooke's Law, the restoring force \(F\) exerted by the spring is directly proportional to the displacement \(x\) from the mean position:

\[ F \propto -x \] \[ F = -k x \quad \text{--- (i)} \]

Where \(k\) is the spring constant (or force constant), and the negative sign indicates that the restoring force is always directed opposite to the displacement (towards the mean position).

According to Newton's Second Law of Motion:

\[ F = m a \quad \text{--- (ii)} \]

Equating equations (i) and (ii):

\[ m a = -k x \] \[ a = -\left(\frac{k}{m}\right) x \]

Since mass \(m\) and spring constant \(k\) are constant for a given system, the ratio \(\frac{k}{m}\) is constant:

\[ a \propto -x \]

Conclusion: The acceleration \(a\) of the mass is directly proportional to its displacement \(x\) and is always directed towards the mean position. This proves that the motion of a mass attached to a spring is Simple Harmonic Motion (S.H.M).