The net external force acting on the disk when its centre of mass is at displacement \(x\) with respect to its equilibrium position is (A) \(-k x\) (B) \(-2 k x\) (C) \(-\frac{2 k x}{3}\) (D) \(-\frac{4 k x}{3}\)

Simple harmonic motion (SHM) is typified by a repeating movement back and forth through an equilibrium position, where the maximum displacement on either side of this position is symmetrical. This motion is consistent, periodic, and can be observed in systems like pendulums and springs. What's fascinating about SHM is its predictability; the motion follows a sine or cosine wave over time, which allows for precise calculations of an object's position at any given time.

Objects in SHM experience a restoring force – more on this soon – and this force is directly proportional to the distance from the equilibrium position, but it acts in the opposite direction. When a spring or pendulum reaches its maximum displacement, the force pulling it back to the middle is strongest. Conversely, at the equilibrium position, this force is zero since there's no displacement to correct. This balance of forces is what creates the 'harmonic' part in SHM – a smooth and regular oscillation around a point of balance.

A restoring force, fundamentally, tends to bring a system back to its equilibrium state. In the context of spring-related motion, when a spring is stretched or compressed, the restoring force tries to bring the spring back to its original, unstressed length. This is what occurs in the exercise you're working on: the disk's displacement from the center is met with a force aiming to restore equilibrium.

The magic of the restoring force lies in its direction and magnitude. It's always directed opposite to the displacement, making it a 'kick back' towards the normal state. This relationship explains the negative sign in the force equation from Hooke's Law. As an educator explaining why the answer is '-kx' and not just 'kx', it's this simple principle that helps students remember it's the restoring nature of the force that requires the negative sign.

The spring constant, often denoted by the symbol 'k', represents the stiffness of a spring. Think of it as a quantifier of how much force you need to apply to a spring to stretch or compress it a certain amount. High values for k indicate a stiffer spring that's more difficult to manipulate, while a lower k means the spring is softer and requires less force for the same displacement.

In the context of the problem provided, the spring constant is crucial because it defines the relationship between the displacement and the restoring force. Hooke's Law is often written as F = -kx, where F is the restoring force, k is the spring constant, and x is the displacement from the equilibrium. This proportionality allows us to understand how much force results from a given displacement. When guiding students through problems such as the one at hand, emphasizing the physical meaning of the spring constant can enhance their grasp of how this abstract number translates into observable spring behavior.