Oscillations of Spring — Definition

Definition

Imagine a block of mass 'm' attached to one end of a spring, with the other end of the spring fixed to a rigid support. When the spring is in its natural, unstretched, and uncompressed state, the block is at its equilibrium position.

At this point, there's no net force acting on the block. Now, if you pull the block to one side, stretching the spring, the spring will try to pull the block back towards the equilibrium. If you push the block, compressing the spring, the spring will push it back towards equilibrium.

This force, which always tries to restore the block to its equilibrium position, is called the restoring force.

According to Hooke's Law, this restoring force ($F$) is directly proportional to how much you've stretched or compressed the spring (the displacement, $x$). Mathematically, this is written as $F = -kx$.

Here, 'k' is a constant called the spring constant, which tells us how stiff the spring is – a larger 'k' means a stiffer spring. The negative sign indicates that the restoring force always acts in the direction opposite to the displacement.

For example, if you pull the block to the right (positive $x$), the force acts to the left (negative $F$).

When you release the block after displacing it, this restoring force causes it to accelerate back towards equilibrium. As it approaches equilibrium, its speed increases. It doesn't stop at equilibrium, however, due to its inertia.

It overshoots, compressing the spring on the other side. Now the restoring force acts in the opposite direction, slowing it down, bringing it to a momentary stop, and then accelerating it back towards equilibrium again.

This back-and-forth motion, repeating over and over, is called oscillation.

Because the restoring force is directly proportional to the displacement, this specific type of oscillation is called Simple Harmonic Motion (SHM). In SHM, the acceleration of the object is also directly proportional to its displacement from equilibrium and is always directed towards the equilibrium point.

The time it takes for one complete back-and-forth cycle is called the period ($T$), and the number of cycles per second is the frequency ($f$). The maximum displacement from equilibrium is known as the amplitude ($A$).

For a spring-mass system, the period of oscillation depends only on the mass of the block ($m$) and the spring constant ($k$), given by the formula $T = 2pisqrt{\frac{m}{k}}$. It's crucial to note that the period does not depend on the amplitude of oscillation, as long as the spring remains ideal and Hooke's Law holds.