7.1 Defining Simple Harmonic Motion (SHM)

Simple harmonic motion is a type of periodic motion where objects oscillate back and forth. It occurs when a restoring force acts on an object displaced from equilibrium, causing it to accelerate back towards its resting position.

The key feature of simple harmonic motion is that the restoring force is proportional to displacement. In simple harmonic motion, the net restoring force satisfies Hooke's-law behavior and points toward equilibrium: $F_x = -k\Delta x$. Using Newton's second law, this gives $m a_x = -k\Delta x$. The negative sign shows that the acceleration and net force are always directed opposite the displacement from equilibrium.

A restoring force is a force directed opposite the object's displacement from equilibrium, so it tends to pull or push the object back toward the equilibrium position. The equilibrium position is the position where the net force on the object or system is zero. If the object is displaced from this position, a restoring force may act to bring it back.

This proportional relationship between force and displacement results in oscillations around an equilibrium position, like a mass on a spring or a swinging pendulum.

Simple harmonic motion

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Periodic motion

Periodic motion occurs when an object moves in a regular, repeating pattern over equal time intervals.

• Simple harmonic motion (SHM) is a specific type of periodic motion where an object oscillates back and forth repeatedly 🔁
• Occurs when a restoring force acts on an object displaced from its equilibrium position, causing it to accelerate back towards equilibrium
• Examples of periodic motion include a mass on a spring and a vibrating guitar string
• A pendulum is approximately simple harmonic only for small angular displacements, because for small angles the restoring torque is proportional to the angular displacement: $\tau \propto -\theta$. This is why small-angle pendulum motion can be modeled as SHM.

Key Characteristics of SHM

Understanding the defining features of SHM will help you recognize it in different physical situations.

• The restoring force is always proportional to displacement and opposite in direction: $F_x = -k\Delta x$
• Applying Newton's second law gives the defining equation of SHM: $m a_x = -k\Delta x$
• At equilibrium, displacement is zero, so the net force and acceleration are also zero
• At maximum displacement (the amplitude), the restoring force and acceleration are at their greatest magnitudes
• Velocity is maximum as the object passes through equilibrium and zero at the turning points (maximum displacement)
• For Topic 7.1, the key idea is conceptual: SHM is periodic motion in which the restoring force (and therefore acceleration) is proportional to displacement and opposite in direction. More detailed equations for position, period, and energy are developed in later topics.

Practice Problem: Identifying SHM

Solution:

(a) At the moment of release, the mass is displaced $\Delta x = 0.10$ m from equilibrium. Using the restoring force relationship: $F_x = -k\Delta x = -16\text{ N/m} \times 0.10\text{ m} = -1.6\text{ N}$

The net force is 1.6 N directed back toward the equilibrium position.

(b) Using Newton's second law: $a_x = \frac{F_x}{m} = \frac{-1.6\text{ N}}{0.25\text{ kg}} = -6.4\text{ m/s}^2$

The acceleration is $6.4\text{ m/s}^2$ directed toward equilibrium, opposite the displacement — exactly what we expect from $m a_x = -k\Delta x$.

(c) At the equilibrium position, the displacement is zero ($\Delta x = 0$): $F_x = -k\Delta x = -16\text{ N/m} \times 0\text{ m} = 0\text{ N}$

The net force is zero at equilibrium, which is consistent with the definition of the equilibrium position. Even though the force is zero here, the mass is moving at its maximum speed and will continue past equilibrium due to its inertia.