Force Law for Simple Harmonic Motion

The Connection Between Force and Displacement in SHM

Simple harmonic motion is not just a pattern of motion — it is defined by a specific relationship between the force acting on a particle and its displacement from equilibrium. The textbook section 13.6 establishes this force law, which is the physical cause behind the sinusoidal motion we have already studied.

When a particle executes SHM, its acceleration at any instant is given by $a(t) = -\omega^2 x(t)$, where $x(t)$ is the displacement from the mean position. From Newton's second law, $F = ma$, the force acting on the particle must be:

Since $m$ and $\omega$ are constants for a given system, we can write $k = m\omega^2$, where $k$ is a positive constant. This gives the force law for SHM:

The negative sign is crucial: it tells us that the force always points towards the equilibrium position. When the particle is displaced to the right ($x > 0$), the force is to the left ($F < 0$), pulling it back. When displaced to the left ($x < 0$), the force is to the right ($F > 0$). This is why the force is called a restoring force — it always acts to restore the particle to its equilibrium position.

The Spring-Block System: A Concrete Example

The simplest physical system that obeys this force law is a block attached to a spring. For an ideal spring that obeys Hooke's law, the restoring force when the spring is stretched or compressed by a distance $x$ from its natural length is:

where $k_s$ is the spring constant (or force constant) of the spring. Comparing this with $F = -kx$, we see that for a spring-block system, $k = k_s$. The angular frequency of oscillation is then:

and the time period is:

Properties of the Force Law

The textbook lists three important properties that follow from $F = -kx$. Each one is derived directly from this equation.

›Proof

Property (I): The force is always directed towards the equilibrium position.

Let the equilibrium position be $x = 0$. For any displacement $x$:

• If $x > 0$ (particle to the right), then $F = -k x < 0$, so force points left (towards $x=0$).
• If $x < 0$ (particle to the left), then $F = -k x > 0$, so force points right (towards $x=0$).
• If $x = 0$, then $F = 0$.

In every case, the force vector points from the particle's position back to the origin. This is the restoring nature of the force.

›Proof

Property (II): The force is proportional to the displacement.

From $F = -kx$, the magnitude of the force is $|F| = k|x|$. Doubling the displacement doubles the force magnitude; halving the displacement halves it. This linear relationship is what makes the motion "simple" — the differential equation it produces has sinusoidal solutions. If the force were proportional to $x^2$ or $\sqrt{x}$, the motion would not be simple harmonic.

›Proof

Property (III): The constant $k$ determines the "stiffness" of the system.

For a given displacement $x$, a larger $k$ means a larger restoring force. This makes the system "stiffer" — it resists displacement more strongly. Since $\omega = \sqrt{k/m}$, a larger $k$ gives a higher angular frequency and shorter time period. A smaller $k$ gives a weaker restoring force, lower frequency, and longer period. The constant $k$ is called the force constant or spring constant of the system.

The Differential Equation of SHM

Combining the force law with Newton's second law gives the equation of motion. Starting from $F = ma$ and $F = -kx$:

Rearranging:

Substituting $\omega^2 = k/m$ gives the standard differential equation for SHM: