AP Physics 1 Unit 7 ReviewOscillations

AP Physics 1 Unit 7 covers oscillations, the back-and-forth motion of systems like a block on a spring or a pendulum swinging through small angles. The single biggest idea is simple harmonic motion (SHM), which happens whenever a restoring force is proportional to displacement from equilibrium, so the math of the motion is a sine or cosine function. You analyze SHM with the same tools you already have, force analysis and energy conservation, and the unit makes up 5-8% of the AP exam. It is short, but the graph-reading and energy-tracking skills it tests show up everywhere.

What this unit covers

What makes motion "simple harmonic"

• Periodic motion is anything that repeats. SHM is the special case where the net force on the object obeys $ma_x = -k\Delta x$, meaning force is proportional to displacement and points back toward equilibrium.
• A restoring force always pushes opposite the displacement. Pull a spring-block right, the spring pulls left. Swing a pendulum left, gravity's tangential component pulls right.
• The equilibrium position is where the net force is zero. The object never stays there during oscillation; it has maximum speed there and coasts through.
• A pendulum is only approximately SHM, and only for small angles. At small angles the gravitational restoring force is roughly proportional to displacement, so the SHM equations apply.

Period and frequency

• Period $T$ is the time for one full cycle. Frequency $f$ is cycles per second (hertz). They are reciprocals, $T = 1/f$.
• A spring-block oscillator has period $T_s = 2\pi\sqrt{m/k}$. More mass means a longer period. A stiffer spring (bigger $k$) means a shorter period.
• A simple pendulum has period $T_p = 2\pi\sqrt{\ell/g}$. Longer string means longer period. Stronger gravity means shorter period.
• Notice what is missing from both equations. Amplitude does not appear, and for the pendulum, mass does not appear. Doubling amplitude changes how far and how fast the object moves, but not how long a cycle takes. This is one of the most-tested facts in the unit.

Graphs and the sinusoidal description

• Displacement in SHM follows $x = A\cos(2\pi f t)$ or $x = A\sin(2\pi f t)$, depending on whether the clock starts at maximum displacement or at equilibrium.
• Displacement, velocity, and acceleration are all sinusoidal, but offset from each other. When displacement is at a maximum, velocity is zero and acceleration is at its maximum magnitude (pointing toward equilibrium). When displacement is zero, speed is at its maximum and acceleration is zero.
• You should be able to look at one graph and sketch the others. The velocity graph peaks where the position graph crosses zero, and the acceleration graph is the position graph flipped upside down.
• Changing the amplitude rescales the height of all three graphs but leaves the period untouched.

Energy in an oscillator

• Total mechanical energy is $E_{total} = U + K$, and with no friction it stays constant the whole time.
• Energy sloshes back and forth between forms. At maximum displacement, all the energy is potential ($U = \frac{1}{2}k(\Delta x)^2$ for a spring) and kinetic energy is at its minimum. At equilibrium, kinetic energy ($K = \frac{1}{2}mv^2$) is at its maximum and spring potential energy is at its minimum.
• Total energy depends on amplitude. Double the amplitude of a spring oscillator and the total energy quadruples, because $U \propto (\Delta x)^2$.
• Energy bar charts and energy-vs-position graphs are the standard representations here. The $U$ curve is a parabola in position, the $K$ curve is an upside-down parabola, and they always add to the same flat total.

Unit 7, Oscillations at a glance

Why Unit 7, Oscillations matters in AP Physics 1

Oscillations is where the whole first half of the course gets reused on one problem. A single spring-block question can ask for a free-body diagram, a kinematics-style graph reading, and an energy conservation calculation, all in the same scenario. It is also the course's best test of whether you can move between representations, since the same motion gets described with equations, graphs, bar charts, and words.

• SHM is the model behind real measurement tools, like weighing astronauts in microgravity by timing their oscillation on a spring device, where mass comes from $T_s = 2\pi\sqrt{m/k}$ instead of a scale.
• The unit reinforces the conservation-of-energy theme. A frictionless oscillator is the cleanest possible closed system, so it is a favorite setting for energy reasoning.
• Proportional reasoning gets a workout here. Questions constantly ask what happens to $T$ if you quadruple the mass or halve the length, and the square roots in the period equations punish sloppy scaling.

How this unit connects across the course

• Kinematics (Unit 1) gave you position, velocity, and acceleration graphs. SHM graphs are those same graphs, just sinusoidal, and the same rules apply (velocity is the slope of position, acceleration is the slope of velocity).
• Force and Translational Dynamics (Unit 2) introduced Hooke's law and free-body diagrams. SHM is what happens when you let a Hooke's-law force run the show, and $ma_x = -k\Delta x$ is just Newton's second law applied to a spring.
• Work, Energy, and Power (Unit 3) gave you spring potential energy and conservation of mechanical energy. Unit 7 applies both continuously, tracking the K-U tradeoff through every point of the cycle.
• Torque and Rotational Dynamics (Unit 5) connects through the pendulum, where gravity's torque about the pivot is what restores the bob toward equilibrium. Thinking of the pendulum rotationally explains why the tangential force component is what matters.

Key equations and processes

• $T = 1/f$ converts between period and frequency. If a system completes 2 cycles per second, each cycle takes 0.5 s.
• $ma_x = -k\Delta x$ is the defining condition for SHM. The minus sign encodes "restoring," and it tells you acceleration is largest at maximum displacement.
• $T_s = 2\pi\sqrt{m/k}$ gives the period of a spring-block oscillator. Use it for proportional reasoning (quadruple $m$, double $T$).
• $T_p = 2\pi\sqrt{\ell/g}$ gives the period of a simple pendulum at small angles. Use it to find $g$ from pendulum timing data, a classic lab setup.
• $x = A\cos(2\pi f t)$ or $x = A\sin(2\pi f t)$ describes position over time. Cosine if the object starts at maximum displacement, sine if it starts at equilibrium.
• $E_{total} = U + K$ with $U = \frac{1}{2}k(\Delta x)^2$ and $K = \frac{1}{2}mv^2$. Set energy at one point equal to energy at another to find speeds or positions anywhere in the cycle.
• Process to know: identify equilibrium, find amplitude, then use energy conservation between the endpoint (all U) and equilibrium (all K) to get maximum speed.

Unit 7, Oscillations on the AP exam

Oscillations is 5-8% of the exam, so expect a handful of multiple-choice questions and possibly an oscillating system inside a free-response question. Here is what you actually do with this content:

• Translate between representations. A question gives you a position-vs-time graph and asks for the velocity or acceleration graph, or asks at which labeled time the kinetic energy is maximum.
• Reason proportionally. The square roots in the period equations make "what happens to T if..." questions a staple. Practice doubling and quadrupling variables until the scaling is automatic.
• Justify with the model. Paragraph-style responses ask you to explain why a pendulum's period does not depend on mass, or why amplitude does not affect period, using the SHM condition and the period equations as evidence.
• Design and analyze experiments. Lab-based questions use pendulum or spring data, asking you to linearize (plot $T^2$ vs $\ell$ to get a straight line through $g$), identify sources of uncertainty, or evaluate whether data supports the SHM model.
• Apply energy conservation quantitatively. Given $k$, $m$, and amplitude, find the maximum speed, or find the displacement where kinetic and potential energy are equal.

Essential questions

• What conditions must a force satisfy for a system to undergo simple harmonic motion, and why do those conditions produce sinusoidal motion?
• Why does the period of an oscillator depend on the system's properties (mass, spring constant, length) but not on its amplitude?
• How does energy move between kinetic and potential forms during a cycle, and what stays constant?
• How can timing an oscillation let you measure something you cannot measure directly, like an astronaut's mass or the local value of $g$?

Key terms to know

• Simple harmonic motion (SHM): periodic motion produced when the restoring force is proportional to displacement from equilibrium.
• Restoring force: a force directed opposite to an object's displacement, always pushing it back toward equilibrium.
• Equilibrium position: the location where the net force on the oscillating object is zero.
• Amplitude: the maximum displacement from equilibrium during a cycle.
• Period: the time for one complete oscillation, measured in seconds.
• Frequency: the number of oscillations per second, measured in hertz; the reciprocal of period.
• Hooke's law: the spring force relationship $F = -k\Delta x$, where force is proportional to stretch or compression.
• Spring constant: the stiffness of a spring in N/m; larger $k$ means a stronger force per meter of stretch.
• Simple pendulum: a mass on a light string that approximates SHM when displaced by a small angle.
• Spring potential energy: energy stored in a deformed spring, equal to $\frac{1}{2}k(\Delta x)^2$.
• Mechanical energy: the sum of a system's kinetic and potential energies, constant in frictionless SHM.
• Sinusoidal function: a sine or cosine curve, the mathematical shape of position, velocity, and acceleration in SHM.

Common mix-ups

• Maximum speed happens at equilibrium, but maximum acceleration happens at the endpoints. Students flip these constantly. Remember that acceleration follows force, and the restoring force is biggest where displacement is biggest.
• Amplitude does not affect period. A bigger swing covers more distance, but the object also moves faster, and the two effects cancel exactly in SHM.
• Pendulum period ignores mass; spring period ignores amplitude and gravity. Mixing up which variables matter for which system is an easy way to lose points on proportional-reasoning questions.
• Velocity is zero at the turnaround points, but the object is not "at rest" in the everyday sense. Acceleration there is at its maximum, so the object immediately speeds back up toward equilibrium.