Q.Two identical springs of spring constant $k$ are attached to a block of mass $m$ and to fixed supports as shown in Fig. 13.14. Show that when the mass is displaced from its equilibrium position on either side, it executes a simple harmonic motion. Find the period of oscillations.

Concept: Simple Harmonic Motion — the net restoring force must be proportional to displacement and directed toward equilibrium.

Step 1 – Forces when displaced

Let the block be displaced by $x$ to the right. The left spring stretches by $x$, pulling left with force $kx$. The right spring compresses by $x$, pushing left with force $kx$. Both forces act in the same direction (toward equilibrium).

Step 2 – Net restoring force

The total force on the block is

The negative sign shows the force is opposite to displacement.

Step 3 – Equation of motion

From Newton’s second law: …

The key insight is that simple harmonic motion arises whenever the net restoring force on a body is directly proportional to its displacement from equilibrium and acts opposite to that displacement — that is, $F = -C x$ where $C$ is a positive constant. Once you identify that form, the angular frequency is $\omega = \sqrt{C/m}$ and the period follows immediately.

Here, the block is attached between two identical springs. When the block is at the equilibrium position, both springs are at their natural lengths (assuming no initial tension). If you displace the block to the right by a small distance $x$, the right spring gets compressed by $x$ and pushes left; the left spring gets stretched by $x$ and pulls left. Both forces act in the same direction — toward the equilibrium position.

Let’s work through it step by step.

1. Set up the forces. Take the equilibrium position as $x = 0$, with $x$ positive to the right.
   • Left spring: stretched by $x$, so it exerts a force $F_L = -k x$ (negative means to the left).
   • Right spring: compressed by $x$, so it exerts a force $F_R = -k x$ as well (also to the left). The net force on the block is the sum:

2. Recognise the SHM condition.

   The net force is proportional to displacement ($-2k x$) and opposite in direction. This is exactly Hooke’s law with an effective spring constant $k_{\text{eff}} = 2k$.

   For SHM: $F = -k_{\text{eff}} \, x$, where $k_{\text{eff}} = 2k$ here.

3. Write the equation of motion.

   Using Newton’s second law: …

Step 1: Displace the block by $x$ to the right of equilibrium; the left spring (attached to a fixed wall) stretches by $x$ and pulls left with force $kx$, while the right spring compresses by $x$ and pushes left with force $kx$ — both act in the same restoring direction. …