Simple harmonic motion sits at the centre of the AP Physics 1 unit on oscillations, and almost every FRQ that touches it asks the candidate to do one thing first: represent the motion. Not analyse it, not solve a numerical answer, but write the equations, sketch the graphs, and label the relationships between position, velocity, acceleration, period, frequency, and angular frequency. This article is a tutor's working read of how to handle that representation step, what the rubric actually checks, and the most common ways candidates lose the points before the algebra even begins.
The SHM representation question family and where it lives on the exam
The AP Physics 1 exam asks SHM questions in two distinct formats. The first is a multiple-choice item, usually worth a single point, that presents a mass on a spring or a pendulum in words and asks for the period, the frequency, the maximum acceleration, or the phase of the motion. The second is a free-response item, typically a sub-part of a larger oscillation problem, that awards partial credit for an equation of motion, a labelled graph, or a verbal justification of why two graphs are out of phase by a quarter period.
Within the FRQ, representation work is almost always weighted. A 12-point FRQ on a mass-spring system will typically assign 1 point for writing the position equation in the form x(t) = A cos(ωt + φ), 1 point for the velocity equation or graph, 1 point for the acceleration equation or graph, and 1 point for naming the period or angular frequency. The remaining points go to energy conservation, force analysis, or a follow-on quantitative sub-part. In other words, four points of a 12-point question can be lost before a candidate ever substitutes a number.
The reason the College Board weights representation so heavily is that SHM is a calculus-ready topic, but the AP Physics 1 exam does not require calculus. A correctly written x(t) = A cos(ωt + φ) is a way of saying 'I understand the motion' without writing a derivative. Likewise, a graph that shows displacement peaking when velocity crosses zero is a way of saying 'I understand the phase relationship' without invoking differentiation explicitly. The rubric reads these representations as proof of conceptual understanding, which is the heart of the course.
The three core quantities: period, frequency, and angular frequency
Candidates who read the question carefully will notice that SHM questions use three quantities interchangeably on the surface, but the rubric treats them as three different rows. The period T is the time for one full cycle, measured in seconds. The frequency f is the number of cycles per second, measured in hertz. The angular frequency ω is the rate at which the phase advances, measured in radians per second. They are linked by two equations that must be memorised:
- ω = 2π / T
- f = 1 / T
- ω = 2π f
A common FRQ pattern reads: 'A 0.40 kg mass on a spring with k = 80 N/m oscillates with amplitude 0.05 m. Calculate the period, the frequency, and the angular frequency, and write an equation for x(t) assuming the mass is released from rest at maximum positive displacement at t = 0.' The rubric row for period awards the point for any equivalent expression: T = 2π√(m/k), the substituted value 2π√(0.40/80), the numerical answer 2π√(0.005) seconds, or a simplified decimal in the right units. The row for ω requires the conversion to radians per second; writing 0.05 Hz and calling it ω costs the row.
For a pendulum, the same three quantities emerge from T = 2π√(L/g). The rubric is stricter here: a candidate who uses T = 2π√(m/k) for a pendulum loses both the period row and the angular frequency row, because the FRQ scorer will not retroactively award credit for the wrong underlying model. This is one of the more common representation errors and worth flagging during preparation: read the system description, decide whether it is a spring or a pendulum, and write the period equation that matches the system before any substitution.
The position equation x(t) = A cos(ωt + φ): what the rubric reads
The first representational task on most SHM FRQs is to write an equation of motion in the form x(t) = A cos(ωt + φ) or, less commonly, x(t) = A sin(ωt + φ). The rubric does not care which trigonometric function is used, as long as two conditions hold: the phase angle φ matches the initial condition, and the amplitude A matches the maximum displacement. A typical error is to write x(t) = A cos(ωt) when the mass is released from rest at x = 0 at t = 0, which is a sine function, not a cosine function. The rubric's phase row reads 'φ = 0 if cos, φ = π/2 if sin' for that initial condition, and a candidate who writes x(t) = A cos(ωt) for an initial condition that demands sine loses the row.
The amplitude row is the simplest and most often earned. A = 0.05 m in the example above. Candidates who try to solve for A from a graph rather than reading it lose a minute, and a minute is significant when the FRQ budget is roughly 25 minutes per question. Train yourself to read the amplitude directly from the problem statement, write it down, and move on.
The phase row is where the rubric does most of its work. Consider an FRQ that says 'at t = 0 the mass is at x = A/2 and moving in the negative direction.' The correct phase for a cosine representation is φ = +π/3, because cos(π/3) = 1/2 and the derivative -Aω sin(ωt + φ) is negative at t = 0. A candidate who writes φ = -π/3 has the right cosine value but the wrong sign on the velocity, and loses the row. The shorthand I use with my own students: pick the initial position, pick the initial velocity, and test both with the chosen function. If a cosine test gives the wrong sign for velocity, switch to a negative cosine or to a sine with the appropriate phase.
Velocity and acceleration representations: phase, sign, and the 90-second check
Once the position equation is in hand, the velocity equation is its time derivative: v(t) = -Aω sin(ωt + φ). The acceleration equation is the second derivative: a(t) = -Aω² cos(ωt + φ) = -ω² x(t). These are the two equations that connect SHM to Newton's second law, and the rubric is alert to candidates who write them as separate facts rather than as consequences of one position equation.
A useful 90-second check on the FRQ: at t = 0 with the mass at maximum positive displacement, v(0) should be zero and a(0) should be at its maximum negative value. This matches the physical picture: at the turning point, the mass is momentarily at rest and the spring is fully compressed or stretched, exerting maximum restoring force. A candidate who writes v(0) = Aω and a(0) = 0 at that turning point has the phase wrong by π/2, and the rubric's sign row will mark it down.
The acceleration equation can also be written in the form a = -ω² x, which is a direct statement of the defining property of SHM: the acceleration is proportional to the displacement and points back toward equilibrium. This form is the one that earns the SHM definition point on FRQs that ask candidates to justify why a motion is or is not simple harmonic. A candidate who writes F = -kx and then divides by m to get a = -(k/m)x is implicitly stating ω² = k/m, which is the correct relation for a mass-spring system. A candidate who writes F = -kx and stops has not yet linked to the SHM representation, and the row may go unearned depending on the rubric's exact wording.
Graph families: x(t), v(t), a(t), and the energy plot
The graphical representation of SHM is a separate rubric row from the equation, and the two are not interchangeable. A candidate can earn the equation row by writing x(t) = A cos(ωt) and lose the graph row by drawing a sine curve. This is a recurring trap. The four standard graphs for an undamped mass-spring system released from rest at maximum positive displacement are:
- x(t): cosine, peaks at +A, troughs at -A, zero at t = T/4 and t = 3T/4.
- v(t): negative sine, zero at t = 0, t = T/2, t = T, and extremum at ±Aω at t = T/4 and t = 3T/4.
- a(t): negative cosine, trough at t = 0 at -Aω², peak at t = T/2 at +Aω².
- E(t): constant total energy, with kinetic and potential bars summing to the same height at every instant.
Notice the phase relationships: x and a are in phase, both being cosine-like functions (or both negative-cosine-like in the case of a), while v is a quarter cycle out of phase with both. This is the 'x and a together, v by itself' rule that the rubric enforces on graph questions. A candidate who draws a(t) in phase with v(t) has the phase wrong by a quarter period and loses the row, even if the period itself is correct.
Common pitfalls and how to avoid them
The first pitfall is mixing up angular frequency with ordinary frequency when sketching a graph. The horizontal axis on a sketch is usually labelled t in seconds, and the period T is the interval between two successive peaks. A candidate who labels the period as 2π/ω on the axis has it right; a candidate who labels the period as 1/ω has used f instead of T and loses any row that depends on the period value.
The second pitfall is drawing the velocity graph with the wrong sign convention. If positive x is defined as 'to the right,' then positive v is also to the right. A mass moving from +A toward 0 has negative velocity. A mass moving from 0 toward -A also has negative velocity. A candidate who draws a positive sine for v(t) in this situation has the sign wrong.
The third pitfall is treating the energy bar chart as if it changes shape. In an undamped oscillator, total mechanical energy is constant. The kinetic energy bar grows as the potential energy bar shrinks, but their sum is a flat line. A candidate who draws a sinusoidal total energy line has confused the SHM energy plot with a non-conservative system, and loses the conservation row on any FRQ that includes it.
The phase constant φ: reading initial conditions without calculus
The phase constant is the most under-represented quantity on AP Physics 1 SHM questions, and the one most likely to cost a candidate the full-credit row on a 5-target attempt. The rubric awards the phase point only when the written phase matches both the initial position and the initial velocity, with the correct sign. A typical four-option initial condition is 'released from rest at x = +A.' For this condition, using a cosine representation, φ = 0. Using a sine representation, φ = -π/2 (or equivalently +3π/2). Both are correct; the rubric does not penalise the choice of trigonometric function.
A trickier initial condition is 'passing through equilibrium moving in the positive x-direction.' For this, the cosine representation requires φ = -π/2 (so that the derivative -Aω sin(ωt + φ) is positive at t = 0) and the sine representation requires φ = 0. The cleanest way to set φ is to use the identity at t = 0: x(0) = A cos(φ) and v(0) = -Aω sin(φ). The two equations together pin φ down. A common error is to set φ using only x(0) and ignore v(0), which leaves a sign ambiguity that the rubric catches.
A third initial condition appears occasionally: 'the motion can be described by x(t) = A cos(ωt) for some constant phase reference.' The question may then ask the candidate to rewrite the equation so that the new reference is t = 0 when the mass is at x = 0 moving in the positive direction. The answer is x(t) = A sin(ωt), a phase shift of π/2. This is a clean way for the College Board to test phase manipulation without forcing calculus on the candidate, and it is worth rehearsing.
Energy representation: which bar is which, and the conservation argument
Energy is a representational system in its own right, and the FRQ often asks for an energy bar chart at two instants. For a horizontal mass-spring oscillator, the relevant energies are translational kinetic energy (½mv²), spring potential energy (½kx²), and total mechanical energy (½kA²). The gravitational potential energy is constant and can be ignored if the reference is set at the equilibrium height; the rubric penalises candidates who include gravitational potential energy in the bar chart without changing the reference.
The first row of the energy representation is the conservation statement: E_total = ½kA², constant for all time. The second row is the energy partition: E_total = ½mv² + ½kx² at every instant. The third row is the maximum/minimum identification: at x = ±A, all energy is potential; at x = 0, all energy is kinetic. A candidate who draws the kinetic bar larger than the potential bar at x = A has the partition wrong, and loses the conservation row on a strict rubric.
The SHM energy equation can also be written as ½mv² + ½kx² = ½kA². This is a useful bridge between the kinematic representation (x, v, a) and the energy representation, and the FRQ will sometimes ask candidates to solve for v in terms of x, yielding v = ±ω√(A² - x²). This equation is what the rubric reads when an energy sub-part asks for the speed at a particular displacement. A candidate who writes v = √(k/m) · x has confused SHM with constant-velocity motion and loses the row.
Tying representations together: the chain that earns full credit on a 5-target attempt
The full-credit representation on a typical SHM FRQ is a chain of five rows: the period equation, the angular frequency, the position equation with the correct phase, the velocity and acceleration equations or graphs, and the energy representation. The chain is most reliable when built in this order, because each row uses the previous one. Candidates who try to write all four equations at once and then check phase at the end often find that they have to rewrite everything, costing two or three minutes.
Worked example: a mass-spring system released from x = 0 with positive initial velocity
A 0.50 kg mass on a spring with k = 200 N/m is at x = 0 at t = 0 and is given an initial velocity of +1.5 m/s. (a) Write an equation for x(t). (b) Sketch x(t), v(t), and a(t) over one period. (c) Determine the period and the amplitude. (d) Determine the maximum acceleration.
Step one, the period: T = 2π√(m/k) = 2π√(0.50/200) = 2π√(0.0025) = 2π(0.05) = 0.10π seconds, approximately 0.314 s. The rubric's period row accepts any of these equivalent forms.
Step two, the angular frequency: ω = 2π / T = 2π / 0.10π = 20 rad/s. Equivalently, ω = √(k/m) = √(200/0.50) = √400 = 20 rad/s. The two methods must agree; if they do not, a candidate has a sign error or a misread value.
Step three, the amplitude. The energy at t = 0 is ½mv² = ½(0.50)(1.5)² = 0.5625 J. Setting this equal to ½kA² gives A² = 2(0.5625)/200 = 0.005625, so A = 0.075 m, or 7.5 cm. The amplitude row accepts this calculation or a direct energy equation with the substitution.
Step four, the position equation. The mass is at x = 0 at t = 0 and moving in the positive direction. The cosine form requires φ = -π/2 (so that the velocity -Aω sin(ωt + φ) is positive at t = 0). The sine form requires φ = 0. Either is acceptable; a clean write-up is x(t) = 0.075 sin(20t) metres.
Step five, the velocity and acceleration equations. v(t) = Aω cos(20t) = 1.5 cos(20t) m/s. a(t) = -Aω² sin(20t) = -30 sin(20t) m/s². The graphs: x(t) is a sine, v(t) is a cosine starting at its maximum, a(t) is a negative sine. Maximum acceleration magnitude is Aω² = 0.075(400) = 30 m/s², which is the answer to part (d).
This worked example covers all four rows: period, angular frequency, amplitude, phase, and acceleration maximum. A candidate who writes the equations in this order and checks each phase row will pick up the full four points on a typical FRQ.
Comparison of representation systems and the rows they earn
The table below summarises the four representation systems a candidate can use on an AP Physics 1 SHM question, the rubric row each system primarily satisfies, and the most common error that costs the row.
| Representation system | Primary rubric row | Most common error |
|---|---|---|
| Equation x(t) = A cos(ωt + φ) | Position and phase row | Using the wrong phase for the initial condition |
| Graph of x(t), v(t), a(t) | Graph and phase relationship row | Drawing v(t) in phase with a(t) instead of a quarter period ahead |
| Period / frequency / angular frequency | Period row | Labelling ω as f or using the wrong formula for the system |
| Energy bar chart ½mv² + ½kx² = ½kA² | Conservation and partition row | Drawing a non-constant total energy line on an undamped system |
Preparation strategy: how to rehearse the representation step
For candidates aiming for a 5, the representation step is best rehearsed in isolation, separate from numerical problem-solving. A useful drill: take a single SHM scenario, write the four equations (x, v, a, plus period) and the energy partition, and then sketch all three graphs on the same time axis. Repeat the drill for the four standard initial conditions: released from rest at +A, released from rest at -A, passing through equilibrium moving positive, and passing through equilibrium moving negative. Twelve drills, twenty minutes total, and the representation rows become automatic.
A second preparation habit is to write the phase constant in degrees for visual checks, then convert to radians for the final answer. The rubric accepts either, but a candidate who has miscomputed the phase by π/2 will see the error more clearly if the degrees are 90° than if the radians are π/2. A 90° error in degrees is a clean mistake to catch; a π/2 error in radians is the kind of thing that gets written down and carried through to the final answer.
A third preparation habit is to read the FRQ for the word 'justify' or 'explain.' When the rubric asks for a justification of a phase relationship, it is asking for a verbal statement of which graph is zero, which graph is at its maximum, and which graph is at its minimum at t = 0. A one-sentence answer ('at t = 0 the mass is at x = +A, so v = 0 and a is at its most negative value') earns the row. A candidate who writes a paragraph that does not include the t = 0 snapshot has not addressed the question and loses the row even if the equation is correct.
Conclusion and next steps
Representing SHM is a skill that pays off across the rest of the AP Physics 1 oscillations unit and on every FRQ that touches a mass-spring or pendulum system. The four rows to focus on are the period row, the position equation row, the phase row, and the energy partition row. The rest of the FRQ analysis energy, force, numerical answer follows from these four, and a candidate who has them nailed can complete the remaining sub-parts under the time pressure of the exam.
AP Courses' one-to-one AP Physics 1 programme works through a student's position-equation and phase rows on the SHM FRQ, scores them against the rubric, and builds a representation drill set around the four standard initial conditions so that the 5-target becomes a concrete preparation plan.