7.1 Defining Simple Harmonic Motion (SHM)

What is simple harmonic motion in AP Physics C?

Simple harmonic motion (SHM) is a special kind of repeating 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 $ma_x = -k\Delta x$, which means acceleration always opposes displacement. A mass on a spring is the standard model.

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Why This Matters for the AP Physics C: Mechanics Exam

Oscillations make up about 10 to 15 percent of the AP Physics C: Mechanics exam, and everything in Unit 7 builds on the definition you learn here. Getting the SHM condition right lets you recognize when a system oscillates, set up the correct equation of motion, and connect force to acceleration using Newton's second law.

This topic pulls together tools you already have, like free-body diagrams, Newton's second law, and energy reasoning, and applies them to oscillating systems. On the exam you may be asked to create diagrams of a system, compare a quantity at different points in the motion, apply the restoring-force model to make a claim, and justify that claim with physical reasoning. The free-response question that asks you to translate between representations often uses spring-block oscillators, so a solid grip on the basics here pays off directly.

Key Takeaways

• SHM is a special case of periodic motion, defined by a restoring force proportional to displacement.
• The restoring force always points back toward equilibrium and follows $F = -k\Delta x$.
• Equilibrium is the position where the net force on the object is zero.
• Newton's second law gives $ma_x = -k\Delta x$, which rearranges to $a = -\frac{k}{m}x$.
• Acceleration in SHM is largest where displacement is largest, and zero at equilibrium.
• The negative sign in the restoring-force equation is what makes the motion oscillate instead of run away.

Periodic Motion

Periodic motion repeats in regular cycles. SHM is a special case of periodic motion in which the restoring force is proportional to displacement from equilibrium. Not every repeating motion is SHM. The proportional restoring force is what sets SHM apart.

Restoring Force

The restoring force is what gives SHM its characteristic oscillating behavior. This force always acts opposite to an object's displacement from equilibrium, constantly pushing or pulling it back toward equilibrium.

• The magnitude of the restoring force increases linearly with distance from equilibrium.
• For a spring, this is written as $F = -kx$
• The negative sign means the force points opposite to the displacement.
• The constant $k$ is the spring constant, which measures the stiffness of the system.

For a spring, the spring constant $k$ represents how hard it is to stretch or compress the spring. A higher $k$ means a stiffer spring that needs more force to displace by the same amount.

When an object is released from a displaced position, the restoring force accelerates it toward equilibrium. Because of inertia, it overshoots equilibrium, so the restoring force then acts in the opposite direction. This continuous back-and-forth is the oscillation that defines SHM.

Equilibrium Position

The equilibrium position is the reference point for measuring displacement in an SHM system. It is the position where the net force on the object or system is zero. Other forces may still act, but they balance so there is no net force at equilibrium.

• At equilibrium, the net force on the object is zero.
• For a mass-spring system, equilibrium is where the spring is neither compressed nor stretched.

When an object is displaced from equilibrium, a restoring force appears that is proportional to that displacement. This force always acts to return the object to equilibrium, but because the object has momentum, it oscillates around this position instead of stopping there.

Newton's second law lets you describe the motion mathematically:

• Start with $F = ma$ and substitute the restoring force $F = -kx$
• This gives $ma = -kx$
• Rearranging: $a = -\frac{k}{m}x$

This shows that the acceleration of an object in SHM is proportional to its displacement but points in the opposite direction. This relationship is the basis for deriving the full equations of motion for SHM and understanding how the system behaves over time.

How to Use This on the AP Physics C: Mechanics Exam

Problem Solving

To find a restoring force, identify the displacement from equilibrium and the spring constant, then apply $F = -k\Delta x$. Keep displacement in meters and watch the sign so your force points the right way.

To connect force and acceleration, substitute the restoring force into Newton's second law to get $a = -\frac{k}{m}x$. This tells you acceleration is greatest at maximum displacement and zero at equilibrium.

Common Trap

A common slip is dropping the negative sign or treating the spring constant as negative. The constant $k$ is positive. The negative sign in the equation comes from the force pointing opposite to the displacement, not from $k$ itself.

Free Response

When a question asks you to compare quantities at different points in the motion, link them through the restoring-force model. At maximum displacement, the restoring force and acceleration are largest. At equilibrium, both are zero. Stating that reasoning clearly is exactly the kind of justification free-response graders look for.

Practice Problem: Restoring Force

Solution

Use the restoring-force equation: $F = -kx$

Where:

• $k$ is the spring constant (20 N/m)
• $x$ is the displacement from equilibrium (15 cm = 0.15 m)

Substituting: $F = -20 \times 0.15$ $F = -3 \text{ N}$

The negative sign shows the force acts opposite to the displacement. If the mass is displaced in the positive direction, the restoring force acts in the negative direction, pulling it back toward equilibrium.

The magnitude of the restoring force is 3 N, and it acts toward the equilibrium position.

Common Misconceptions

• All periodic motion is SHM. Only motion with a restoring force proportional to displacement counts as SHM. Many repeating motions do not meet this condition.
• The spring constant is negative. The constant $k$ is always positive. The minus sign in $F = -kx$ comes from direction, not from $k$.
• The object stops at equilibrium. The net force is zero at equilibrium, but the object is moving fastest there, so it passes through rather than stopping.
• Acceleration is constant in SHM. Acceleration changes with position because it depends on displacement through $a = -\frac{k}{m}x$. It is largest at the extremes and zero at equilibrium.
• The restoring force points the same way as the displacement. It always points back toward equilibrium, which is opposite to the displacement.

Related AP Physics C: Mechanics Guides

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