Topic 7.1: Defining Simple Harmonic Motion (SHM)

What Topic 7.1 requires

One learning objective, one sentence of definition, and three clauses that pin down its words. LO 7.1.A asks you to describe simple harmonic motion. That is the entire required content of the topic.

EK 7.1.A.1 places SHM inside a larger family: simple harmonic motion is a special case of periodic motion. EK 7.1.A.2 supplies the definition itself, that SHM results when the magnitude of the restoring force exerted on an object is proportional to that object's displacement from its equilibrium position, and the CED prints one derived equation beneath it:

Three sub-statements do the defining work. EK 7.1.A.2.i says a restoring force is a force exerted in a direction opposite to the object's displacement from an equilibrium position. EK 7.1.A.2.ii says an equilibrium position is a location at which the net force exerted on an object or system is zero. EK 7.1.A.2.iii is the pendulum clause: the motion of a pendulum with a small angular displacement can be modeled as simple harmonic motion because the restoring torque is proportional to the angular displacement.

None of the four topics in Unit 7 carries a boundary statement, so nothing in this topic is fenced off by the CED. The unit is weighted at 5 to 8 percent of the multiple-choice section, tied with Unit 6 for the smallest share of the eight units, and the CED budgets roughly 5 to 10 class periods for it.

One oddity is worth knowing before you plan around the skills. The Topic 7.1 page of the CED lists suggested skills 1.C, 2.A, 2.B and 3.B in its sidebar, while the Unit 7 at-a-glance table lists 1.C, 2.B, 3.B and 3.C for the same topic. The two lists agree on 1.C, 2.B and 3.B and part company over 2.A and 3.C. Treat all five as fair game rather than picking a side.

The definition is about the force, not the shape

Most study material introduces SHM by drawing a sine wave. The CED does not, and the order matters.

EK 7.1.A.2 defines SHM by a condition on the force: the magnitude of the restoring force is proportional to the displacement from equilibrium. Nothing in that sentence mentions sines, cosines, amplitude, or period. Those arrive later, in Topic 7.3, where the CED states that the displacement of an object exhibiting SHM can be represented by $x = A\cos(2\pi f t)$ or $x = A\sin(2\pi f t)$.

So the logic runs one way only. Restoring force proportional to displacement is the hypothesis. Sinusoidal position against time is the conclusion. Getting this backwards costs points on justification questions, because a graph that happens to look wavy is not evidence that a system is in SHM, while a force law of the form $F = -kx$ is.

The practical version of the test is short. Ask two questions about the net force along the direction of motion. Does it point back toward a single equilibrium position at every point of the motion? Does its size grow in direct proportion to the distance from that position? Two yeses and the motion is simple harmonic. One no and it is not, whatever the graph looks like.

Restoring force and equilibrium, in the CED's own terms

Both supporting definitions are worth reading slowly, because each is narrower than the everyday word suggests.

Restoring force (EK 7.1.A.2.i) is defined purely by direction: it is exerted opposite to the object's displacement from an equilibrium position. That is a claim about where the force points, not about how big it is. A restoring force alone does not make SHM. A ball bouncing on the floor feels a restoring force in this sense, and its motion is periodic, but the size of that force has nothing to do with how high the ball was. The proportionality in EK 7.1.A.2 is the extra condition that narrows restoring force down to simple harmonic motion.

Equilibrium position (EK 7.1.A.2.ii) is defined as a location where the net force on the object or system is zero. Net force, not spring force. That single word decides the most common setup in the unit. A block hanging on a vertical spring sits at rest where the spring force and the gravitational force cancel, which is a distance $mg/k$ below the spring's relaxed length. That loaded position, not the relaxed length, is the equilibrium position for the oscillation, and displacement in the SHM definition is measured from it. Topic 2.8 works through the spring force itself and the two reference points it can be measured from.

Once you measure from the loaded equilibrium, gravity drops out of the restoring force entirely. Pull the hanging block down an extra $A$ and the net force on it is $kA$ upward, the same size as on a frictionless horizontal track with the same spring and displacement. That is why the two cases behave identically here while looking so different on a diagram.

Every SHM is periodic, but not every periodic motion is SHM

EK 7.1.A.1 is one sentence and it is easy to skim past: simple harmonic motion is a special case of periodic motion. The word special is doing real work.

Periodic motion is any motion that repeats itself at regular intervals. Simple harmonic motion is the subset of periodic motion in which the restoring force obeys the proportionality condition. Every question that offers you a repeating motion and asks whether it is SHM is really asking whether that condition holds.

Three repeating motions that are not simple harmonic, and why:

• A ball bouncing elastically on the floor. Between bounces the only force is gravity, which is constant in size and always points the same way. It does not grow with distance from any equilibrium position, so the motion is periodic but not simple harmonic.
• A pendulum released from a large angle. The restoring torque goes as $\sin\theta$, not as $\theta$, and the two part company as the angle grows. The first worked example below puts numbers on how fast.
• A spring stretched past the range where its force stays proportional to its extension. The ideal-spring model that Topic 2.8 sets up is an assumption about the spring, and when a data set stops being proportional the SHM conclusion goes with it. The third worked example shows what that looks like in a table of measurements.

Uniform circular motion is worth naming too, because it is periodic and it does produce a sinusoid: the shadow of an object moving round a circle at constant speed, projected on any one axis, moves in simple harmonic motion. The circular motion itself is not SHM, since the net force keeps a constant magnitude and points at the center rather than growing with displacement along a line. Topic 2.9 covers that force.

What the derived equation actually says

The CED prints one equation under EK 7.1.A.2 and labels it a derived equation:

It is Newton's second law with the spring force substituted in. The left side is $ma_x$, from $\vec{F}_{\text{net}} = m\vec{a}$ on the equation sheet. The right side is Hooke's law, $\vec{F}_s = -k\Delta\vec{x}$, also on the sheet. Setting them equal assumes the spring force is the net force along the axis of the motion, which is what a frictionless horizontal track buys you, and what the loaded-equilibrium argument above buys you in the vertical case.

Divide by the mass and the statement becomes a claim about acceleration:

Read it as three separate facts. The acceleration is proportional to the displacement, so doubling the displacement doubles the acceleration. The acceleration is opposite in sign to the displacement, so it always points back at equilibrium. And the constant of proportionality is $k/m$, in units of one over seconds squared, which is the only place the properties of the system enter. That last quantity is what Topic 7.2 turns into a period.

Two consequences fall straight out, and both are heavily tested. Acceleration is largest where displacement is largest, at the two turning points, and zero at equilibrium. Speed does the opposite: zero at the turning points, largest at equilibrium. Expecting the two to peak together gets the whole of Topic 7.3 wrong in one stroke.

The pendulum qualifies only at small angles

EK 7.1.A.2.iii is careful, and the care is the point: the motion of a pendulum with a small angular displacement can be modeled as simple harmonic motion because the restoring torque is proportional to the angular displacement. Both hedges are in the CED's own sentence. A pendulum is not simple harmonic motion; it is a system that a small-angle model treats as simple harmonic motion.

Notice also that the CED reaches for torque rather than force here. That is the Unit 5 vocabulary applied to a swinging bob: the gravitational force acts at a distance $\ell$ from the pivot, and the restoring torque about that pivot has magnitude $mg\ell\sin\theta$. SHM needs the torque to be proportional to the angular displacement $\theta$, and $\sin\theta$ is not proportional to $\theta$. It is only close to it when $\theta$, measured in radians, is small.

That is the entire location of the approximation. It is not in the definition of SHM, and it is not in the spring case at all. It lives in one step, replacing $\sin\theta$ with $\theta$, and everything downstream inherits it, including the pendulum period equation in Topic 7.2.

The CED never names a cutoff angle, so do not quote one as if it did. Show the size of the error instead, which is what the first worked example does. A common classroom rule of thumb keeps the amplitude under about 15 degrees, and those numbers show why that is a reasonable line to draw rather than a law of physics.

One further warning about units. The small-angle statement is true for angles in radians and false for angles in degrees: $\sin(10^\circ) = 0.1736$, nowhere near 10. Every calculation that compares $\sin\theta$ with $\theta$ has to have $\theta$ in radians first.

What the equation sheet hands you, line by line

Students lose time in Unit 7 by re-deriving things that are printed and by hunting for things that are not. Here is the actual inventory on the AP Physics 1 equation sheet, checked item by item.

In the group headed Mechanics and Fluids (rotational, oscillations, and fluids), five lines belong to oscillations:

Both sinusoidal position functions are printed. That is worth saying plainly, because plenty of material implies you have to produce them yourself.

Two more lines that Unit 7 leans on sit in the translational group instead, which is why they are easy to miss when you are scanning the oscillations block: $\vec{F}_s = -k\Delta\vec{x}$ and $U_s = \frac{1}{2}k(\Delta x)^2$.

What is not printed matters just as much. The derived equation $ma_x = -k\Delta x$ from EK 7.1.A.2 is not on the sheet as its own line; you assemble it from $\vec{F}_{\text{net}} = m\vec{a}$ and $\vec{F}_s = -k\Delta\vec{x}$, which is precisely the derivation skill the topic is testing. Neither is $E_{\text{total}} = \frac{1}{2}kA^2$, which the CED gives as a relevant equation under EK 7.4.A.4.ii but which the sheet leaves out. And there is no $\omega = 2\pi f$ on the AP Physics 1 sheet: the Unit 7 statements are written in terms of the frequency $f$ throughout, so angular frequency is a tool you may use but never one you can cite.

How Topic 7.1 shows up on the exam

The skills attached to this topic tell you the shape of the questions before you see one.

1.C, sketch graphs. Science Practice 1 is not assessed in the multiple-choice section at all, so every 1.C task reaches you through the free-response section, where Science Practice 1 carries 20 to 35 percent of the points.

2.A and 2.B, derive and calculate. Deriving a symbolic expression is 15 to 20 percent of the multiple-choice section and calculating an unknown quantity is 20 to 25 percent. Here that usually means building $a_x = -(k/m)\Delta x$ and reading a number off it.

3.B and 3.C, apply a model and justify a claim. Applying a law, definition, theoretical relationship, or model to make a claim is 20 to 25 percent of the multiple-choice section, and 35 to 45 percent of the free-response section falls to Science Practice 3. A definition-based topic maps onto 3.B almost exactly: name the model, then show that its conditions are met.

The CED is unusually specific about where Unit 7 lands in the free-response section. Its Unit 7 overview says the second free-response question, Translation Between Representations, requires students to create graphical and verbal models and compare them to mathematical representations, and it gives the example directly: a student might be asked to sketch free-body diagrams of a block oscillating on a spring at maximum displacement and at equilibrium, then create energy bar charts for the block and spring system at those same two positions, then explain how the two representations are consistent with each other. It adds that while Unit 7 content is especially good practice for that question, content from any unit may appear in it.

Those two free-body diagrams are Topic 7.1 in one task: largest spring force and largest acceleration at maximum displacement, and no net force arrow at equilibrium by EK 7.1.A.2.ii, even though the object is moving at its fastest there. A four-function, scientific, or graphing calculator is allowed on both sections of the exam, so arithmetic is never what makes these hard.

Errors that turn a right idea into a wrong answer

• Calling a motion SHM because the graph looks sinusoidal. The sinusoid is the conclusion, not the premise. Justify with the force law.
• Drawing a net force arrow at equilibrium. EK 7.1.A.2.ii defines equilibrium as zero net force. The object is moving fast there, but velocity is not force, and a free-body diagram with an arrow at that instant contradicts the definition it was drawn from.
• Measuring displacement from the spring's relaxed length in a hanging system. The definition says displacement from the equilibrium position, and for a loaded vertical spring those are two different places, separated by $mg/k$.
• Treating the minus sign as part of a magnitude. In $ma_x = -k\Delta x$ the sign encodes the word restoring. The size of the force is $k|\Delta x|$, a positive number of newtons.
• Assuming speed and acceleration peak together. Acceleration peaks at the turning points, speed peaks at equilibrium. They are never large in the same place.
• Quoting a maximum angle as if the CED gave one. EK 7.1.A.2.iii says small angular displacement and stops. Show the size of the error instead, and remember that comparing $\sin\theta$ with $\theta$ means nothing until $\theta$ is in radians.