Unit 7: Oscillations

What Unit 7 covers

Unit 7 covers oscillations: motion that repeats around an equilibrium position, like a block bouncing on a spring or a pendulum swinging. It carries 5-8% of the multiple-choice section, tied with Unit 6 for the smallest weight in AP Physics 1, but it earns its spot because it ties force, motion graphs, and energy together in one package.

The unit has four topics. Topic 7.1 defines simple harmonic motion (SHM), 7.2 gives the period and frequency formulas, 7.3 covers graphs and other representations, and 7.4 tracks energy through a cycle. Everything builds on two things you already know: the spring force from Unit 2 and energy conservation from Unit 3. If either of those is shaky, review it first. Unit 7 is mostly those ideas applied to repeating motion. The oscillators in this unit are also what generate periodic waves, an AP Physics 2 topic covered in wave speed, frequency, and wavelength.

Topic 7.1: Defining simple harmonic motion

An object is in simple harmonic motion when the net force on it is a restoring force proportional to displacement: push it twice as far from equilibrium and it gets pulled back twice as hard. For a spring, that is $F_s = -k\Delta x$, where the minus sign means the force always points back toward equilibrium.

Two model systems dominate the unit:

• A mass on an ideal spring, which is exact SHM.
• A simple pendulum, which is approximately SHM only for small angles.

The exam likes to test the definition itself. A ball bouncing off the floor repeats, but the force on it is not proportional to its displacement, so the motion is periodic without being simple harmonic. Expect questions that hand you a force-versus-position graph and ask whether the motion qualifies: the graph must be a straight line through the origin with negative slope. The simple harmonic motion guide works through more examples.

Topic 7.2: Frequency and period

Two period formulas do almost all the work in this unit, and both are on the equation sheet:

with $T = 1/f$ connecting period and frequency. Read them closely, because the exam tests what is not in them as much as what is:

• Amplitude appears in neither formula. Pull the spring farther and the block moves faster over a longer path; the period stays the same.
• The pendulum period does not depend on mass. Only length and $g$ matter.
• The dependence is a square root. Quadruple the mass on a spring and the period doubles; merely doubling the mass multiplies the period by $\sqrt{2} \approx 1.41$.

Ranking questions (order these oscillators by period) come straight from these proportionalities, no calculator needed.

Topic 7.3: Representing and analyzing SHM

Topic 7.3 is about reading and drawing SHM graphs. Position, velocity, and acceleration versus time are all sine-shaped curves with the same period, shifted relative to one another. The physical picture keeps them straight:

• At the extremes ($x = \pm A$): speed is zero, and the restoring force and acceleration are at their maximum, pointing toward equilibrium.
• At equilibrium ($x = 0$): speed is maximum and acceleration is zero.

So the velocity graph crosses zero wherever the position graph peaks, and the acceleration graph is the position graph flipped upside down, because $a$ is proportional to $-x$.

Given any one graph, you should be able to sketch the other two, read off amplitude and period, and mark where kinetic energy peaks. That skill feeds directly into the Translation Between Representations free-response question, which rewards consistent graphs and clear reasoning over plugged-in numbers.

Topic 7.4: Energy of simple harmonic oscillators

Energy turns most SHM problems into two-line calculations. A frictionless oscillator trades spring potential energy $U_s = \frac{1}{2}k(\Delta x)^2$ and kinetic energy $K = \frac{1}{2}mv^2$ back and forth while the total stays constant:

• At the extremes, the energy is all potential: $E = \frac{1}{2}kA^2$.
• At equilibrium, it is all kinetic: $E = \frac{1}{2}mv_{max}^2$.
• Anywhere in between, $\frac{1}{2}k(\Delta x)^2 + \frac{1}{2}mv^2$ equals that same total.

Setting the two extreme cases equal gives maximum speed without touching kinematics. This is ordinary conservation of energy from Unit 3 applied to a spring system, not a new law. Energy-versus-position graphs show up often: $U_s$ is a parabola, $K$ is the same parabola flipped, and their sum is a horizontal line at $E$.

How Unit 7 is tested and where to practice

On the multiple-choice section, Unit 7 leans on proportional reasoning (what happens to $T$ if $k$ doubles), graph reading, and energy accounting. On the free-response section, oscillations can appear in any of the four question types: Mathematical Routines, Translation Between Representations, Experimental Design and Analysis, and Qualitative/Quantitative Translation. An experimental design prompt might ask you to find $k$ or $g$ from measured periods, so know that plotting $T^2$ against mass (or pendulum length) turns the square-root relationship into a straight line. A calculator is allowed on both sections.

To practice:

• Work through the simple harmonic motion guide for the full problem-solving routine.
• Keep the AP Physics 1 formula sheet open until the period formulas are automatic.
• Then move on to Unit 8: Fluids, the last unit of the course.