Simple harmonic motion (SHM) is oscillation about an equilibrium where the restoring force (and acceleration) is proportional to minus the displacement: F ∝ −x. Mathematically it satisfies d^2x/dt^2 = −ω^2 x and solutions are x = A cos(ωt + φ) or x = A sin(ωt). How to know something is doing SHM: check that the acceleration is proportional to −x (not −v) and that displacement is sinusoidal. Useful quick checks: zeros and extrema line up (v = 0 at displacement extrema; a = 0 at displacement zeros), a = −ω^2 x, vmax = Aω, amax = Aω^2. Period depends on ω (or system parameters) but not amplitude. AP exam note: you’re expected to recognize/use x = A cos(ωt+φ) and d^2x/dt^2 = −ω^2 x, but you don’t need to prove the solution. For a short study refresher see the Topic 7.3 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/3-representing-and-analyzing-shm/study-guide/nHLSGDEgw3L81KKh). For extra practice, try problems at (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Because SHM is defined by the differential equation d²x/dt² = −ω²x, the motion needs a function whose second derivative is the negative of itself times a constant. Sine and cosine do exactly that: if x = A cos(ωt + φ) then d²x/dt² = −ω²A cos(ωt + φ) = −ω²x. That’s why the CED gives x = A cos(2πft) or x = A sin(2πft) as the standard forms (Topic 7.3, EK 7.3.A.1–7.3.A.3). Practically, sin/cos make it easy to read amplitude A, angular frequency ω (so v_max = Aω and a_max = Aω²), and phase φ to match initial conditions. Zeros and extrema of x, v, and a line up cleanly because derivatives of sin/cos are just shifted sin/cos functions (useful for qualitative graphs on the exam). You’re not expected to prove the solution in AP Physics C (Boundary Statement), but you should recognize and use these sinusoidal forms. For more review and practice, see the Topic 7.3 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/3-representing-and-analyzing-shm/study-guide/nHLSGDEgw3L81KKh) and Unit 7 resources (https://library.fiveable.me/ap-physics-c-mechanics/unit-7).

Think of three separate ideas: - Amplitude (A) is the maximum displacement from equilibrium—how far it swings. It appears in x = A cos(2πft) or x = A cos(ωt + φ). Changing A changes how big the motion is but NOT the period (CED 7.3.A.5). - Frequency (f) is how many cycles happen per second (Hz). It’s related to the period T by f = 1/T. In the CED you see x = A cos(2π f t) (CED 7.3.A.1). - Angular frequency (ω) measures how fast the oscillator’s phase advances in radians/sec: ω = 2π f. Using ω simplifies calculus: x = A cos(ωt + φ), the SHM differential equation is d²x/dt² = −ω² x (CED 7.3.A.2–3). Useful quick facts: v_max = Aω and a_max = Aω². Example: if f = 2 Hz then ω = 4π rad/s. For extra practice and AP-aligned review, check the Topic 7.3 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/3-representing-and-analyzing-shm/study-guide/nHLSGDEgw3L81KKh) and more problems at (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Differentiate x(t) = A cos(ωt + φ). Velocity is the first derivative, acceleration is the second. - v(t) = dx/dt = −A ω sin(ωt + φ). So v_max = Aω (when |sin| = 1). - a(t) = dv/dt = −A ω^2 cos(ωt + φ) = −ω^2 x(t). So a is proportional to −x (restoring acceleration) and a_max = A ω^2 (when |cos| = 1). Sign/phase notes: when x is maximum (cos = ±1) v = 0 and |a| is max; when x = 0 (passing equilibrium) |v| is max and a = 0. These relations (a = −ω^2 x, v_max = Aω, a_max = Aω^2) are exactly what the AP CED expects you to use for Topic 7.3. For a quick review, see the Topic 7.3 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/3-representing-and-analyzing-shm/study-guide/nHLSGDEgw3L81KKh) and try practice problems (https://library.fiveable.me/practice/ap-physics-c-mechanics).

In SHM (x = A cos(ωt + φ) or A sin(ωt + φ)), “maximum” or “minimum” means largest magnitude or most positive/negative value and tells you what the other quantities are: - Displacement at ±A (max/min): the mass is at an amplitude (farthest from equilibrium). Velocity is zero there (instantaneous turnaround). Acceleration is at its largest magnitude: a = −ω²x so |a| = Aω² and it points toward equilibrium (the largest restoring force). - Displacement = 0 (equilibrium): velocity is maximum in magnitude, vmax = Aω, and acceleration is zero (no restoring acceleration at that instant). - Velocity extrema: when |v| = Aω, x = 0. When v = 0, x = ±A. - Sign matters: a positive/negative extremum indicates direction (toward/away from origin). CED note: you should be able to identify these extrema and zeros from equations or graphs (7.3.A.1–3). For a quick refresher, check the Topic 7.3 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/3-representing-and-analyzing-shm/study-guide/nHLSGDEgw3L81KKh) and practice problems (https://library.fiveable.me/practice/ap-physics-c-mechanics).

The minus sign in a = −ω²x means acceleration always points opposite the displacement from equilibrium. Physically that’s a restoring acceleration: if x is positive (to the right), a is negative (to the left) and vice versa. Mathematically it comes from Newton’s 2nd law with a restoring force (e.g., Hooke’s law F = −kx), giving d²x/dt² = −(k/m)x so ω² = k/m. What that tells you about the motion (AP C ideas): the object oscillates about the equilibrium because acceleration always pulls it back toward x = 0. Acceleration is maximum at the amplitude (|a|max = Aω²) and zero at equilibrium, while velocity is zero at amplitude and max at equilibrium—a 90° phase difference between x and v, and a 180° difference between x and a. You don’t need to prove the differential equation on the exam, but you should be able to use x = A cos(ωt+φ) and a = −ω²x to describe these extrema (see the Topic 7.3 study guide on Fiveable: https://library.fiveable.me/ap-physics-c-mechanics/unit-7/3-representing-and-analyzing-shm/study-guide/nHLSGDEgw3L81KKh). For extra practice, check the AP Physics C practice problems on Fiveable (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Natural frequency is the frequency at which a system oscillates on its own when displaced and released (set by system parameters like m and k; e.g., ω = sqrt(k/m) or f = ω/2π). The external (driving) frequency is the frequency of a sinusoidal force you apply to the system. Resonance happens when the driving frequency matches the system’s natural frequency—then the oscillation amplitude grows large (CED 7.3.A.4.i–ii). Key extra points useful for the exam: resonance increases amplitude but doesn’t change the system’s natural period (7.3.A.4 & 7.3.A.5). If damping is present (not required to derive in AP C), the amplitude peak is reduced and the peak response can shift slightly from the undamped natural frequency. For more review on SHM and resonance, check the Topic 7.3 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/3-representing-and-analyzing-shm/study-guide/nHLSGDEgw3L81KKh) and practice problems (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Changing the amplitude A does not change the period T of simple harmonic motion. The position is x = A cos(ωt + φ), and T = 2π/ω, so T depends only on ω, not A (CED 7.3.A.1, 7.3.A.3). Physically this is because the restoring force is proportional to displacement (F ∝ −x), so the equation d²x/dt² = −ω²x is linear: increasing A just stretches the motion but leaves the curvature (ω) the same. What does change with A are the maximum speed and acceleration: vmax = Aω and amax = Aω² (CED 7.3.A.3.i–ii). For specific systems ω is set by system parameters (e.g., √(k/m) for a spring, √(g/L) for a small-angle pendulum), so only those parameters change T. You don’t need to prove the differential-equation solution on the AP exam, but you should know these forms and how amplitude affects extrema (see the Topic 7.3 study guide for examples: https://library.fiveable.me/ap-physics-c-mechanics/unit-7/3-representing-and-analyzing-shm/study-guide/nHLSGDEgw3L81KKh). For extra practice, try problems at https://library.fiveable.me/practice/ap-physics-c-mechanics.

Think of SHM as a sine wave for x(t): x = A cos(ωt + φ) (or A sin…). From that one equation you get velocity and acceleration by differentiation: - v = −Aω sin(ωt + φ) so v_max = Aω - a = −Aω^2 cos(ωt + φ) so a = −ω^2 x and a_max = Aω^2 Key quick rules to read graphs: - Position and acceleration are in phase opposition: when x is a maximum, a is a maximum in the opposite direction (x max → v = 0, a = −ω^2 x). - Velocity is 90° (quarter period) out of phase with position: when x = 0 (equilibrium) v is extremal; when x is extremal v = 0. - Zeros/extrema: zeros of x are peaks of v; zeros of v are extrema of x; extrema of a line up with extrema of x (but opposite sign). On the AP exam you should be able to identify these zeros/extrema and use a = −ω^2 x and v_max = Aω (you don’t have to prove the differential-solution). For worked examples and practice, check the Topic 7.3 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/3-representing-and-analyzing-shm/study-guide/nHLSGDEgw3L81KKh), the Unit 7 overview (https://library.fiveable.me/ap-physics-c-mechanics/unit-7), and tons of practice problems (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Resonance happens when a system is driven by a sinusoidal external force at (or very near) its natural frequency ω0. At that frequency the driving pushes in a way that continually adds energy to the oscillator at the same point in each cycle—the force and the oscillator’s motion have the right phase relation so each push increases the oscillator’s energy rather than canceling it. With little damping, the input energy per cycle accumulates, so the oscillation amplitude grows until energy lost to damping (friction, air resistance, etc.) equals the energy added each cycle. Mathematically, the steady-state amplitude for a driven damped oscillator peaks near ω = ω0; stronger damping lowers and broadens that peak. Important AP points from the CED: resonance occurs when drive = natural frequency and it increases amplitude (7.3.A.4.i–ii), and changing amplitude doesn’t change the period (7.3.A.5). For a compact review, see the Topic 7.3 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/3-representing-and-analyzing-shm/study-guide/nHLSGDEgw3L81KKh). For more practice, try problems at (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Solve it by looking for functions whose second derivative is −ω² times themselves. Try x(t)=e^{rt}. Plugging in gives r²e^{rt}=−ω²e^{rt} → r²+ω²=0, so r=±iω. That means the real general solution is x(t)=C cos(ωt)+D sin(ωt), or equivalently x(t)=A cos(ωt+φ), where A = √(C²+D²) and φ = arctan(−D/C) (choice of quadrant matters). Differentiate to get v(t)=dx/dt = −Aω sin(ωt+φ), a(t)=d²x/dt² = −Aω² cos(ωt+φ) = −ω² x(t), so vmax = Aω and amax = Aω². The AP CED expects you to know this solution and use x = A cos(ωt+φ) to get velocity/acceleration (7.3.A.1–3); you don’t need to prove the complex-root derivation on the exam. For a short topic review, see the Topic 7.3 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/3-representing-and-analyzing-shm/study-guide/nHLSGDEgw3L81KKh) and practice problems (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Look at the x(t) (displacement) graph and use two quick rules from the CED: v(t) = dx/dt (slope) and a(t) = −ω² x (so acceleration is proportional to −displacement). - Velocity zero: wherever the x(t) curve has a horizontal tangent (slope = 0)—those are the displacement maxima and minima. (From x = A cos(ωt+φ), v = −Aω sin(ωt+φ) so v = 0 at the cos peaks/troughs.) - Acceleration zero: wherever x = 0 (the equilibrium crossing). Because a = −ω² x, when x = 0 then a = 0. - Extras: v is largest (±Aω) when the mass passes equilibrium (steepest slope). Acceleration is largest in magnitude at the displacement extrema (|a|max = Aω²). These are exactly the features the exam expects you to recognize (7.3.A.1–3). For more examples and practice sketching, check the Topic 7.3 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/3-representing-and-analyzing-shm/study-guide/nHLSGDEgw3L81KKh) and try problems from the Unit 7 practice set (https://library.fiveable.me/practice/ap-physics-c-mechanics).

You use ω (angular frequency) instead of f (ordinary frequency) because ω naturally appears when you take time derivatives of sine/cosine and when you write the SHM differential equation. They’re related by ω = 2πf (so ω ≈ 6.28·f). That relation also gives T = 1/f = 2π/ω. Why prefer ω in equations: - The SHM DE from Newton’s 2nd law is d²x/dt² = −ω² x (CED 7.3.A.2). Writing x = A cos(ωt + φ) makes derivatives clean: v = −Aω sin(ωt+φ), a = −Aω² cos(ωt+φ), so a = −ω² x (CED 7.3.A.3, 7.3.A.3.i). - Max values use ω: vmax = Aω and amax = Aω² (CED 7.3.A.3.i). Using f would force extra 2π factors each time. On the AP exam you’re expected to know both forms (x = A cos(2π f t) and x = A cos(ωt+φ)) and how to move between them (CED 7.3.A.1 and 7.3.A.3). For more practice and a focused study guide on this topic see the Topic 7.3 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/3-representing-and-analyzing-shm/study-guide/nHLSGDEgw3L81KKh) and try problems at (https://library.fiveable.me/practice/ap-physics-c-mechanics).

φ (the phase constant) just sets where in its cycle the oscillator starts at t = 0. In x(t) = A cos(ωt + φ) it shifts the cosine left/right in time so the initial displacement and velocity match your conditions: x(0) = A cos φ and v(0) = −Aω sin φ. Given x(0) and v(0) you can solve for φ (tan φ = −v(0)/(ω x(0))), remembering φ is defined modulo 2π. Physically, different φ’s mean the same amplitude and frequency but different starting points (e.g., at an extreme, through equilibrium, or somewhere in between). Phase also lets you compare oscillators—a phase difference Δφ gives their time/position offset (important for phase difference problems on the AP). You don’t need to prove the SHM solution in AP C; just use the form x = A cos(ωt+φ) to get displacement, velocity, acceleration (CED 7.3.A.1–3). For a quick review, see the Topic 7.3 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/3-representing-and-analyzing-shm/study-guide/nHLSGDEgw3L81KKh) and more practice problems at (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Start with the standard SHM solution the CED gives: x(t) = A cos(ωt + φ) (you’re expected to know this form). Differentiate to get velocity and acceleration: - v(t) = dx/dt = −A ω sin(ωt + φ). The sine factor ranges between −1 and +1, so the largest magnitude of v is Aω. Thus v_max = A ω. - a(t) = d²x/dt² = −A ω² cos(ωt + φ). The cosine factor ranges between −1 and +1, so the largest magnitude of a is A ω². Thus a_max = A ω². (Also note a = −ω² x, so acceleration is always opposite displacement.) Quick numeric check: if A = 0.10 m and f = 2.0 Hz, then ω = 2πf = 4π rad/s. v_max = 0.10·4π ≈ 1.26 m/s. a_max = 0.10·(4π)² ≈ 15.8 m/s². These formulas (v_max = Aω, a_max = Aω²) are listed in the CED (7.3.A.3.i–ii). For more review and practice on Topic 7.3, see the study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/3-representing-and-analyzing-shm/study-guide/nHLSGDEgw3L81KKh) and thousands of practice problems (https://library.fiveable.me/practice/ap-physics-c-mechanics).