Simple harmonic motion (SHM) is a special kind of periodic motion where the net (restoring) force on an object is proportional to its displacement from equilibrium and points opposite that displacement: m a_x = −k Δx. That condition makes the motion sinusoidal (x(t) = A cos(ωt + φ)), with constant amplitude A, angular frequency ω = sqrt(k/m), period T = 2π/ω, and phase φ. Key ideas from the CED: a restoring force pulls the object back toward an equilibrium position (net force zero there), and SHM is the special case when that restoring force ∝ displacement (Hooke’s-law springs, small-angle pendulums). How it differs from “regular” motion: regular motion can have nonperiodic, non-repeating, or non-sinusoidal acceleration and forces that don’t follow −k x. Only when the force follows −k x (or its small-angle equivalent for a pendulum) do you get true SHM with fixed frequency independent of amplitude. For a quick topic review see the Topic 7.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/1-defining-simple-harmonic-motion-shm/study-guide/0XdktX7mCpAcsQF4). For more Unit 7 review and lots of practice problems, check the unit page (https://library.fiveable.me/ap-physics-c-mechanics/unit-7) and practice bank (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Because equilibrium is the spot where the net force is zero, any displacement away from that spot creates an imbalance of forces that pulls the object back. By definition (CED 7.1.A.2.i), a restoring force is a force exerted opposite the object’s displacement from equilibrium. For a mass on a spring (Hooke’s law) that imbalance is F = −kΔx: if Δx > 0 (to the right) the force is −kx (to the left); if Δx < 0 the force is +kx (to the right). Newton’s second law then gives max = −(k/m)Δx, so acceleration also points back toward equilibrium. That “opposite sign” relation is exactly what makes the motion oscillatory and, for linear proportionality, simple harmonic (CED 7.1.A.2). If you want a quick refresher and examples for AP-style questions, check the Topic 7.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/1-defining-simple-harmonic-motion-shm/study-guide/0XdktX7mCpAcsQF4) and the Unit 7 overview (https://library.fiveable.me/ap-physics-c-mechanics/unit-7). For practice, use the AP Physics C problem set (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Simple harmonic motion (SHM) is just a special kind of back-and-forth (periodic) motion where the restoring force always points toward equilibrium and is proportional to how far you’re displaced. In equations: m a_x = −k Δx. That minus sign means the force (and acceleration) is opposite the displacement—it “pulls” you back. Because of that linear restoring force, the motion is sinusoidal: x(t) = A cos(ωt + φ), with amplitude A, angular frequency ω = sqrt(k/m), period T = 2π/ω, and phase φ. Key ideas to remember from the CED: restoring force, equilibrium position, Hooke’s law form, and that SHM is a special case of periodic motion (Topic 7.1). You’ll see these concepts on MCQs and FRs in Unit 7. For a quick targeted review, check the Topic 7.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/1-defining-simple-harmonic-motion-shm/study-guide/0XdktX7mCpAcsQF4) and practice problems for extra work (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Periodic motion means any motion that repeats itself after a fixed time (period). SHM is a specific kind of periodic motion where the restoring force is proportional to the displacement from equilibrium and always points back toward equilibrium. Mathematically (CED 7.1.A.2): m ax = −k Δx. That proportionality gives sinusoidal motion (x(t) = A cos(ωt+φ)), a single natural frequency ω = sqrt(k/m), and predictable amplitude, period, and phase. So: every SHM is periodic, but not every periodic motion is SHM. For example, a square-wave oscillator is periodic but not simple harmonic; a mass-on-a-spring (small oscillations of a pendulum or Hooke’s-law spring) is SHM when the restoring force ≈ −k x (or small-angle approximation for pendulums). Topic 7.1 (Learning Objective 7.1.A) is what AP expects you to know—recognize the −k x law, equilibrium, and restoring force. For a quick review, see the Topic 7.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/1-defining-simple-harmonic-motion-shm/study-guide/0XdktX7mCpAcsQF4); for broader review and lots of practice, check the Unit 7 overview (https://library.fiveable.me/ap-physics-c-mechanics/unit-7) and AP Physics C practice problems (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Short answer: check whether the restoring force (or acceleration) is proportional to displacement from equilibrium with a negative sign. For SHM the CED condition is m a_x = −k Δx—meaning the net force pulls opposite the displacement and is linear in x. If F_net (or a) versus x is a straight line through the origin with negative slope, motion is SHM and x(t) is sinusoidal with constant ω = sqrt(k/m). How to tell in practice: draw a free-body diagram and identify the restoring force; measure or compute a for different displacements and test if a ∝ −x. For a pendulum, SHM holds only for small angles (small-angle approximation). If the motion is just back-and-forth but F_net is not proportional to x (or the a vs x plot is nonlinear), it’s periodic but not simple harmonic. For a quick review, see the Topic 7.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/1-defining-simple-harmonic-motion-shm/study-guide/0XdktX7mCpAcsQF4). More unit review and lots of practice problems are at (https://library.fiveable.me/ap-physics-c-mechanics/unit-7) and (https://library.fiveable.me/practice/ap-physics-c-mechanics).

F = −kx is Hooke’s law for a linear spring: the magnitude of the force the spring exerts is proportional to how far you displace it (k is the spring constant, in N/m) and the force always points back toward the equilibrium position. The negative sign shows that direction: if x is positive (mass pulled to the right), the spring force is negative (points left); if x is negative, F is positive. That “force back toward equilibrium” is exactly what the CED calls a restoring force and is the defining condition for SHM (ma_x = −kΔx). Because acceleration is proportional to −x, the motion is sinusoidal with angular frequency ω = sqrt(k/m) and period T = 2π/ω. For more AP-aligned review and practice on this idea, see the Topic 7.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/1-defining-simple-harmonic-motion-shm/study-guide/0XdktX7mCpAcsQF4) and extra problems (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Yes—lots of everyday motions are (or approximate) SHM because they have a restoring force proportional to displacement. Key idea: m a = −k Δx (restoring force toward equilibrium). Examples you’ll relate to: - Mass on a vertical spring (pull it down and release)—classic SHM; ω = √(k/m), T = 2π√(m/k). - Small-angle pendulum (swinging a watch, playground swing at small angles)—use small-angle approximation: T = 2π√(L/g). - Tuning fork or a struck bell (the tine’s tip moves roughly sinusoidally for small displacements). - A lightly plucked guitar string segment (local transverse motion near equilibrium approximates SHM). - Molecules in a solid: atoms vibrate about equilibrium positions like tiny springs (useful concept on exams). On the AP exam, be ready to identify the equilibrium, show the restoring force ∝ displacement, and apply equations for ω, T, amplitude, and phase. For a quick refresher, see the Topic 7.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/1-defining-simple-harmonic-motion-shm/study-guide/0XdktX7mCpAcsQF4) and try practice problems (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Because SHM is defined by a force that pulls an object back toward equilibrium with a strength proportional to how far it’s moved, we write ma_x = −kΔx. For a mass-on-a-spring that proportionality comes from Hooke’s law: the spring’s internal restoring force is F = −k x. More generally, any stable equilibrium lets you linearize forces: if F(x) is smooth, expand around the equilibrium x0 with a Taylor series. The constant term is zero at equilibrium (net force = 0), so the first nonzero term is F ≈ F′(x0)(x − x0)—a term proportional to displacement. That’s why small displacements often give SHM. (Large displacements can introduce nonlinear terms so motion isn’t simple harmonic.) This specific definition and the ma_x = −kΔx relation are in the CED for Topic 7.1 and show up on the exam—so study the derivation and when the small-angle/linear approximation applies (see the Topic 7.1 study guide: https://library.fiveable.me/ap-physics-c-mechanics/unit-7/1-defining-simple-harmonic-motion-shm/study-guide/0XdktX7mCpAcsQF4). For extra practice, try problems at https://library.fiveable.me/practice/ap-physics-c-mechanics.

Equilibrium position = the location where the net force on the object is zero (CED 7.1.A.2.ii). For SHM you measure displacement from that point: the restoring force is opposite the displacement and proportional to it (ma_x = −k Δx, CED 7.1.A.2). How to find it: write the sum of forces and set it equal to zero. Examples: - Horizontal mass–spring: ΣF = −k x = 0 → equilibrium at x = 0 (natural length). - Vertical mass–spring: ΣF = −k x + mg = 0 → equilibrium at x = mg/k (spring stretched). - Simple pendulum: equilibrium is the vertical (lowest) position where tangential force = 0. On AP free-response or MCQ, always state “net force = 0” and solve for the coordinate. For more worked examples and quick review, see the Topic 7.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/1-defining-simple-harmonic-motion-shm/study-guide/0XdktX7mCpAcsQF4) and practice problems (https://library.fiveable.me/practice/ap-physics-c-mechanics).

A mass–spring system is the canonical example of SHM because the spring provides a restoring force proportional to displacement from equilibrium—Hooke’s law: F = −kx. Plug that into Newton’s 2nd law (ma = F) and you get ma = −kx, or a = −(k/m) x. That’s the defining math of SHM: acceleration is proportional to, and opposite in direction to, displacement. Solutions are sinusoidal: x(t) = A cos(ωt + φ) with ω = sqrt(k/m). The period and frequency you’ll need on the exam are T = 2π sqrt(m/k) and f = 1/T. On AP C: Mechanics Topic 7.1 you should be able to identify the equilibrium, recognize the restoring force, and connect ma = −kx to sinusoidal motion. For a quick refresher, see the Topic 7.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/1-defining-simple-harmonic-motion-shm/study-guide/0XdktX7mCpAcsQF4). For extra practice, try problems at (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Draw the same kind of free-body diagram you always do, but focus on the force component that acts as the restoring force (the one proportional to displacement). Key steps: - Identify equilibrium and draw all forces (gravity, normal, tension, spring). For a horizontal mass-spring: draw tension from spring F_s = −k x pointing toward equilibrium; label mg and N vertically (they cancel). Use m a_x = −k x. - For a vertical mass-spring: include mg and spring force; draw net force along vertical and note equilibrium shift (stretch), but small oscillations still obey m a = −k Δx about equilibrium. - For a pendulum: draw mg downward and tension along the string; resolve mg into tangential component −mg sinθ (restoring). For small angles use −mg θ ≈ −(mg/L) s so motion is SHM. - Always show direction of displacement x, mark equilibrium, and draw the restoring force opposite x. On AP free-response you’ll often earn points for a correct FBD and writing m a = (restoring force) (see Topic 7.1 in the CED). For more examples and practice, check the Topic 7.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/1-defining-simple-harmonic-motion-shm/study-guide/0XdktX7mCpAcsQF4) and run practice problems (https://library.fiveable.me/practice/ap-physics-c-mechanics).

At maximum displacement (x = ±A) the restoring force is largest in magnitude and points back toward equilibrium. For a mass–spring SHM, Hooke’s law gives F = −kΔx, so at the amplitude |F|max = kA and the sign is opposite the displacement. Because F = ma, the acceleration is also maximal (and directed toward equilibrium) and the velocity is zero at that instant. Remember: equilibrium is where net force = 0; at the ends the net (restoring) force is maximum. This is exactly the SHM condition in the CED (7.1.A.2 and 7.1.A.2.i). For more on definitions and quick examples, check the Topic 7.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/1-defining-simple-harmonic-motion-shm/study-guide/0XdktX7mCpAcsQF4) and try practice problems at (https://library.fiveable.me/practice/ap-physics-c-mechanics).

You need SHM because it’s a compact, testable model that shows up in both multiple-choice and free-response problems and links lots of CED ideas (forces, energy, motion, small-angle approximation). SHM is defined by a restoring force proportional to displacement (m a_x = −k Δx), so knowing equilibrium, restoring force/torque, and sinusoidal solutions lets you: derive period/frequency for mass-spring and small-angle pendulum, relate acceleration to displacement, and use energy methods (KE ↔ PE)—all skills AP asks for (apply laws, derive relationships, translate reps). Unit 7 appears on the MC section ~10–15% and is assessed in FRs that ask for derivations and representations, so expect symbolic work, free-body diagrams, and justification. If you want a focused review and practice sets, check the topic study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/1-defining-simple-harmonic-motion-shm/study-guide/0XdktX7mCpAcsQF4), the Unit 7 overview (https://library.fiveable.me/ap-physics-c-mechanics/unit-7), and thousands of practice problems (https://library.fiveable.me/practice/ap-physics-c-mechanics).

Check the equilibrium point first: an equilibrium position is where the net force is zero. A force is a restoring force if (1) it points opposite the object’s displacement from that equilibrium and (2) for SHM the magnitude is proportional to that displacement so ma_x = −kΔx (Hooke’s-law form). Practically: draw the equilibrium, pick x>0 and x<0, and see whether the force points toward x=0 in both cases. If yes, it’s restoring. If its magnitude ∝ |x| (i.e., F = −k x), the motion can be SHM with angular frequency ω = √(k/m). Examples: a spring (F = −kx) is restoring and gives SHM; gravity on a pendulum is restoring only for small angles (small-angle approximation makes torque ∝ θ). If the force doesn’t reverse direction around equilibrium or isn’t proportional to displacement, it’s not a restoring force for SHM. More on this in the Topic 7.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/1-defining-simple-harmonic-motion-shm/study-guide/0XdktX7mCpAcsQF4)—and practice problems are at (https://library.fiveable.me/practice/ap-physics-c-mechanics).

For a mass on a spring the CED gives the restoring-law form m ax = −kΔx. That leads to simple harmonic motion with angular frequency ω = sqrt(k/m). So: - ω = √(k/m) (radians/s) - f = ω/(2π) = (1/2π)√(k/m) (Hz) - T = 2π√(m/k) (s) Interpretation: bigger k (stiffer spring) → larger ω and f, and a shorter period T—the system oscillates faster. Quantitatively, if you double k, ω increases by √2 and T decreases by 1/√2. This is exactly the Topic 7.1/7.2 connection in the CED: Hooke’s law gives the restoring force and k controls the oscillation rate. For a quick review see the Topic 7.1 study guide (https://library.fiveable.me/ap-physics-c-mechanics/unit-7/1-defining-simple-harmonic-motion-shm/study-guide/0XdktX7mCpAcsQF4) and try practice problems at (https://library.fiveable.me/practice/ap-physics-c-mechanics).