7.1 Defining Simple Harmonic Motion (SHM)

Simple harmonic motion (SHM) is a special kind of periodic motion that happens when the restoring force on an object is proportional to how far it is displaced from equilibrium and points back toward equilibrium. The defining relationship is $m a_x = -k\Delta x$, which means the acceleration is largest at maximum displacement and zero at equilibrium.

Why This Matters for the AP Physics 1 Exam

Topic 7.1 sets up the entire Oscillations unit, which carries about 5 to 8 percent of the exam. Once you can identify SHM by its force and acceleration behavior, you can apply the force, energy, and momentum tools you already know to oscillating systems.

This topic is mostly conceptual, so expect to describe and justify why a motion counts as SHM rather than just plug into formulas. That kind of reasoning shows up on the free-response section, where you may need to connect free-body diagrams, energy bar charts, and graphs for a block on a spring. Getting comfortable with the restoring-force idea now makes later topics on period, position graphs, and energy much easier.

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Key Takeaways

• SHM is a special case of periodic motion where the restoring force is proportional to displacement: $F_x = -k\Delta x$.
• Newton's second law gives the defining equation $m a_x = -k\Delta x$, so acceleration always points opposite the displacement.
• The equilibrium position is where the net force is zero, so force and acceleration are zero there.
• At maximum displacement (the amplitude), the restoring force and acceleration reach their greatest magnitudes.
• Velocity is maximum at equilibrium and zero at the turning points.
• A pendulum acts as SHM only at small angles, because then the restoring torque is proportional to angular displacement.

What Simple Harmonic Motion Is

Periodic motion is any motion that repeats in a regular pattern over equal time intervals. SHM is a specific kind of periodic motion where an object oscillates back and forth around an equilibrium position.

The defining feature is the restoring force. A restoring force points opposite the object's displacement from equilibrium, so it always pushes or pulls the object back toward where the net force is zero. In SHM, the magnitude of that restoring force is proportional to the displacement:

$F_x = -k\Delta x$

Applying Newton's second law gives the equation that defines SHM:

$m a_x = -k\Delta x$

The negative sign is the important part. It tells you the acceleration and net force always point opposite the displacement, back toward equilibrium.

Key features to recognize SHM

• The restoring force is proportional to displacement and opposite in direction.
• 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.
• Velocity is maximum as the object passes through equilibrium and zero at the turning points.

A mass on a frictionless spring is the cleanest example. A pendulum is approximately SHM only for small angular displacements, because for small angles the restoring torque is proportional to the angular displacement:

$\tau \propto -\theta$

For Topic 7.1, the main goal is conceptual. The detailed equations for period, position, and energy come in later topics. Right now, focus on recognizing SHM by its force and acceleration behavior.

How to Use This on the AP Physics 1 Exam

Problem Solving

When you see a spring or pendulum scenario, identify the displacement first, then use $F_x = -k\Delta x$ to find the restoring force. Convert centimeters to meters before plugging in, and keep the negative sign so the direction stays correct.

Free Response

Be ready to explain why a motion is or is not SHM. The strongest answers tie the restoring force directly to displacement and state that acceleration points opposite the displacement. If a question gives you a block on a spring, you may need to connect a free-body diagram, an energy bar chart, and a position graph and explain how they agree with each other.

Common Trap

Do not assume the acceleration is constant. In SHM, acceleration changes with position because it depends on displacement through $m a_x = -k\Delta x$. Kinematics equations for constant acceleration do not apply here.

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.

Common Misconceptions

• "The force is zero at equilibrium, so the object stops there." The net force is zero at equilibrium, but the object is moving at its maximum speed and keeps going because of inertia.
• "Acceleration is constant in SHM." Acceleration depends on displacement, so it changes throughout the motion and is greatest at maximum displacement.
• "A pendulum is always SHM." A pendulum only behaves as SHM for small angular displacements, where the restoring torque is proportional to the angle.
• "Bigger amplitude means the force law changes." The relationship $F_x = -k\Delta x$ stays the same; a larger displacement just means a larger restoring force.
• "Restoring force and displacement point the same way." The restoring force always points opposite the displacement, which is what the negative sign in $m a_x = -k\Delta x$ shows.

Related AP Physics 1 Guides

• 7.3 Representing and Analyzing SHM
• 7.4 Energy of Simple Harmonic Oscillators
• 7.2 Frequency and Period of SHM