Class 11 > Unit # 11:Oscillations > Simple Harmonic Motion & Spring Mass System

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Simple Harmonic Motion and Motion Under Elastic Restoring Force

Simple Harmonic Motion (SHM)

Definition: If a body moves in a straight line such that its acceleration is always directed towards a fixed point on that line and its magnitude is directly proportional to the displacement from that point, then the body is said to execute simple harmonic motion.

Examples of SHM

• Motion of a mass attached to one end of a spring.
• Vibration of a string of a sitar.
• Motion of the bob of a simple pendulum.
• Motion of the pendulum of a clock.

Motion Under Elastic Restoring Force

Consider a mass '$m$' attached to one end of a spring placed on a horizontal smooth surface, while the other end is rigidly fixed. If the mass '$m$' is pulled to the right through a distance '$x_0$' and then released, it will vibrate back and forth about its mean position.

Hooke's Law

The magnitude of the restoring force exerted by the spring is directly proportional to the displacement '$x$':

$F \propto x \implies F = kx$

Where '$k$' is a constant of proportionality called the force constant (or spring constant). Since the spring exerts an equal and opposite force to restore its original shape, the restoring force is given by:

$\text{Restoring Force} = -kx$

Explanation & Energy Conversion

In equilibrium (fig. a), no force acts on the mass '$m$' as the spring suffers no extension. When pulled to the right by displacement '$x_0$' (fig. b), the work done is stored as potential energy. Upon release, the restoring force pulls the mass toward the left, converting potential energy into kinetic energy.

At the mean position, kinetic energy and speed are maximum, and due to inertia, the mass overshoots and moves to the left, compressing the spring. The motion decelerates until all kinetic energy converts back to potential energy at the extreme left position. This cyclical energy transformation sustains the oscillation.

Mathematical Proof of SHM

Let '$x$' be the displacement of the mass '$m$' at any instant. From Newton's second law of motion:

$F = ma$

Equating the restoring force with Newton's second law:

$ma = -kx$

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

Since both '$k$' (spring constant) and '$m$' (mass) are positive constants, their ratio $\frac{k}{m}$ is constant. Therefore:

$a \propto -x$

This equation proves that the acceleration of the mass is directly proportional to its displacement and is always directed towards the mean position. Hence, the motion of a body under an elastic restoring force is strictly Simple Harmonic Motion (SHM).

Fig: Mass-spring system demonstrating displacement, restoring force, and simple harmonic oscillations.

Video Lecture: Simple Harmonic Motion

Watch the detailed derivation and conceptual explanation of Simple Harmonic Motion and elastic restoring forces below:

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