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AP Physics 1: Algebra-Based · Unit 7 of 8

Oscillations

5–8% of the exam6 lessons · 81 min46 terms
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What this unit covers

The topics below follow the published Physics 1 course framework for Unit 7. This unit is worth 5–8% of the exam, so budget your time against that rather than against how long the unit takes to teach.

SpringsPendulumsPeriod & frequencyEnergy in SHM

Lessons in this unit

Formulas in Unit 7

Hooke’s law and spring period
F = −kx · T = 2π√(m/k)
k is in N/m. A stiffer spring (larger k) gives a shorter period; a heavier mass gives a longer one.
Pendulum period
T = 2π√(L / g)
L is the string length; g = 10 m/s². Mass and (small) amplitude do not appear — only length changes the period on a given planet.
Period, frequency, angular frequency
f = 1 / T · ω = 2πf = 2π / T
T in seconds, f in hertz (Hz), ω in rad/s. Doubling the frequency halves the period.
Energy in SHM
E_total = ½kA² = ½mv_max²
All potential at the extremes, all kinetic at equilibrium. Setting the two equal gives v_max = A√(k/m).
The defining relation
F = −kx → a = −(k/m)x
The minus sign is the physics: force and acceleration always point back toward equilibrium, opposite the displacement.
Maxima in SHM
v_max = ωA = A√(k/m) · a_max = ω²A = (k/m)A
Both scale with amplitude, which is why a wider swing is both faster at the bottom and harder pulled at the ends.
Periods
Spring: T = 2π√(m/k) · Pendulum: T = 2π√(L/g)
Spring period depends on mass but not g; pendulum period depends on g but not mass. Exactly reversed.

Every term in Unit 7

All 46 terms we publish for Oscillations, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Mass-spring period
T = 2π√(m/k). Independent of amplitude — a wider swing simply moves faster.
Simple pendulum period
T = 2π√(L/g), valid for small angles. Independent of mass and, for small amplitude, of amplitude.
Energy in SHM
Total energy is constant; it is all elastic PE at maximum displacement and all KE at equilibrium, exchanging continuously between them.
Velocity and acceleration in SHM
Speed is maximum at equilibrium where acceleration is zero; acceleration is maximum at the extremes where speed is zero.
Amplitude independence
Period does not depend on amplitude in ideal SHM, which is what makes pendulums useful as clocks.
Determining k from a graph
The slope of a force vs displacement graph for a spring is the spring constant.
Why T is independent of amplitude
A larger amplitude means a larger restoring force and hence greater speed, and the two effects cancel exactly for a linear restoring force.
Period from a T² graph
Plotting T² against m for a mass-spring system gives a straight line of slope 4π²/k, which is how k is measured experimentally.
Phase in SHM
Displacement, velocity and acceleration are each a quarter-cycle out of step: velocity peaks where displacement is zero, acceleration where displacement is extreme.
Pendulum on another planet
T = 2π√(L/g), so a pendulum swings more slowly where g is smaller. Timing one is a way to measure g.
Simple harmonic motion
Motion where the restoring force is proportional to displacement and directed toward equilibrium, F = −kx. Produces sinusoidal position, velocity and acceleration.
Restoring force
The force pulling the system back toward equilibrium. Its proportionality to displacement is what makes the motion harmonic rather than merely oscillatory.
Amplitude
The maximum displacement from equilibrium. It does NOT affect the period of a mass-spring system or a small-angle pendulum — the single most tested fact in this unit.
Period and frequency
T is the time for one full cycle; f = 1/T is cycles per second. A full cycle returns to the same position AND the same direction of motion.
Period of a mass on a spring
T = 2π√(m/k). More mass is slower, a stiffer spring is faster, and amplitude does not appear.
Period of a simple pendulum
T = 2π√(L/g). Longer is slower, stronger gravity is faster, and MASS does not appear — the pendulum counterpart of the amplitude trap.
Small-angle approximation
A pendulum is only simple harmonic for small angles, where sin θ ≈ θ. At large amplitude the period lengthens and the motion is no longer SHM.
Where speed is maximum
At equilibrium, where displacement is zero and all the energy is kinetic.
Where acceleration is maximum
At maximum displacement, where the restoring force is largest — and where speed is zero. Speed and acceleration peak at opposite points.
Maximum speed in SHM
From ½kA² = ½mv²_max, v_max = A√(k/m) = Aω. Proportional to amplitude, unlike the period.
Phase
Where in the cycle the motion is. Two oscillators with the same period can be in phase, out of phase, or anywhere between.
Position, velocity and acceleration graphs
Velocity leads position by a quarter cycle and acceleration is exactly opposite position. Acceleration is always the mirror of the displacement graph.
Why acceleration is opposite displacement
Because a = −(k/m)x. The negative sign is the restoring direction, and it is what makes the graphs mirror images.
Period of a spring on a different planet
Unchanged — g does not appear in T = 2π√(m/k). The pendulum period does change, which is the discrimination the question is after.
Vertical mass-spring system
Gravity shifts the equilibrium position but not the period, because the restoring force about the new equilibrium is still −kx.
Damping
Energy loss to friction or drag, shrinking the amplitude over time while leaving the period nearly unchanged for light damping.
Driven oscillation and resonance
Driving a system at its natural frequency transfers energy most efficiently, producing large amplitude. The reason a pushed swing needs correct timing rather than force.
Determining k experimentally
Hang known masses and measure extension: the slope of a force-extension graph is k. Or time oscillations and use T = 2π√(m/k).
Slope of T² against m
For a spring, T² = (4π²/k)m, so a graph of T² against m is a straight line of slope 4π²/k. Linearizing like this is what the experimental-design question expects.
What makes an oscillation simple harmonic
A restoring force proportional to displacement, F = −kx. That one condition produces the sinusoid, the amplitude-independent period, and everything else in the unit.
Acceleration is the displacement graph inverted
Since a = −(k/m)x, the two are 180° out of phase. Where x peaks positive, a peaks negative; where x is zero, a is zero.
Velocity leads displacement by a quarter cycle
Velocity is the slope of the displacement graph, and the slope of a sine is a cosine. Speed and acceleration therefore never peak at the same moment.
The two extremes of an oscillation
At the endpoints: x maximum, v zero, a maximum. At equilibrium: x zero, v maximum, a zero. Momentarily at rest does not mean zero net force.
v_max and a_max
v_max = ωA and a_max = ω²A, both proportional to amplitude. A wider swing is faster at the bottom and pulled harder at the ends.
Energy trades twice per cycle
The oscillator passes equilibrium twice and reaches an extreme twice, so an energy-versus-time graph runs at double the frequency of the displacement graph.
What is absent from the period formulas
Amplitude is absent from both, mass from the pendulum, and g from the spring. What is missing is tested as often as what is present.
A spring keeps the same period on the Moon
T = 2π√(m/k) has no g in it. Hanging it vertically shifts the equilibrium by mg/k but leaves the restoring force about that point as −kx with the same k.
Squaring a period to linearize it
T = 2π√(L/g) is a curve against L, but T² against L is a line of slope 4π²/g. Whenever a period formula has a square root, square the period.
Limits of the pendulum formula
A small-angle approximation good to about 15°. Beyond that the restoring force stops being proportional to displacement and the true period runs longer than predicted.
Frequency and angular frequency differ by 2 pi
omega = 2 pi f. Substituting f where omega belongs is off by a factor of about 6.28 and is the most common numerical error in this unit.
A hertz is one per second
Frequency counts cycles per second. A period of 0.25 s is a frequency of 4 Hz, and period and frequency are always reciprocals.
SHM position as a cosine
x = A cos(omega t) starts at maximum displacement; x = A sin(omega t) starts at equilibrium moving fastest. Choosing between them is choosing where the clock starts.
Energy in SHM scales with amplitude squared
E = 1/2 k A^2, so doubling the amplitude quadruples the energy while leaving the period untouched. Amplitude and period are independent in simple harmonic motion.
A stiffer spring oscillates faster
omega = sqrt(k/m), so raising k raises the frequency and shortens the period. Adding mass does the opposite.
Period grows as the square root of mass
Quadrupling the mass on a spring doubles the period, it does not quadruple it. The square root is why the T^2 against m graph is the straight one.
The equilibrium position is where the net force is zero
For a vertical spring that point is already stretched by mg/k. Oscillation is measured from there, not from the spring's natural length.

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Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

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Questions about this unit

How much of the AP Physics 1: Algebra-Based exam is Unit 7?

Unit 7, Oscillations, is worth 5–8% of the Physics 1 multiple-choice section according to the published course framework. Across all 8 units that makes it a middling share, roughly what an even split across units would give.

What topics are covered in Physics 1 Unit 7?

Oscillations covers Springs, Pendulums, Period & frequency and Energy in SHM. We publish 46 terms with definitions for this unit, all of them on this page.

How should I study Physics 1 Unit 7?

Read the 6 lessons below first — about 80 minutes — then drill the 46 terms in cram mode until you can produce each definition from memory rather than just recognize it. Recognition is what makes a unit feel finished when it is not. Finish with practice questions and read the explanation for every one you get right by elimination as well as the ones you miss.

All 8 units of AP Physics 1: Algebra-Based

  1. Unit 1 · Kinematics
  2. Unit 2 · Force and Translational Dynamics
  3. Unit 3 · Work, Energy, and Power
  4. Unit 4 · Linear Momentum
  5. Unit 5 · Torque and Rotational Dynamics
  6. Unit 6 · Energy and Momentum of Rotating Systems
  7. Unit 7 · Oscillations
  8. Unit 8 · Fluids

Unit names, topics and exam weights follow the published College Board course framework for AP Physics 1: Algebra-Based. AP® is a trademark registered by the College Board, which does not endorse this site.