Representing and Analyzing SHM
Once a system has been identified as SHM, its displacement can be written x(t)=A cos(ωt+φ) or an equivalent sine form. Differentiation gives v(t)=-Aω sin(ωt+φ) and a(t)=-Aω² cos(ωt+φ)=-ω²x. The extrema and zero crossings of x, v, and a are therefore locked by phase: speed is greatest at equilibrium, acceleration magnitude is greatest at the turning points, and changing amplitude does not change the period of an ideal SHM system. This topic also includes natural frequency and resonance under a sinusoidal drive.
Topic sequence · 3 of 5
Learning objectives
Objective 1
Translate among the SHM differential equation, sinusoidal functions, graphs, and physical motion.
Objective 2
Use calculus to derive velocity, acceleration, phase relationships, and the maximum speed/acceleration from x(t).
Objective 3
Analyze natural frequency and resonance qualitatively while keeping amplitude, phase, and period conceptually distinct.
- Topic 7.1 defining SHM and the relation a=-ω²x.
- Topic 7.2 period, frequency, and angular frequency.
- Topic 1.2 derivatives and Topic 1.3 motion graphs.
Definitions first
Representations you must read and create
Representation 1
Aligned x(t), v(t), and a(t) graphs with the same time axis: v is shifted by one-quarter cycle from x, while a is opposite in sign to x.
Representation 2
Phase-circle representation: projection of uniform circular phase can organize sine/cosine values without claiming the physical oscillator itself moves in a circle.
Representation 3
State table at x=+A, 0, -A showing signs and extrema of x, v, and a.
Representation 4
Driven-response sketch identifying the natural frequency and a resonance peak qualitatively without importing a full damping theory.
Core relationships and calculus
Variables: x,A m; ω rad/s; t s; φ rad
Use: one valid phase convention for ideal SHM
Variables: v m/s
Use: time derivative of displacement
Variables: a m/s²
Use: second derivative and SHM restoring relation
Variables: v_max m/s
Use: maximum speed at equilibrium
Variables: a_max m/s²
Use: maximum acceleration magnitude at |x|=A
Variables: T s
Use: one complete 2π phase advance
Differentiate x(t)=A cos(ωt+φ): v=dx/dt=-Aω sin(ωt+φ). Differentiate again: a=dv/dt=-Aω² cos(ωt+φ)=-ω²x. Because |sin|≤1 and |cos|≤1, |v|≤Aω and |a|≤Aω². The calculus automatically produces the quarter-cycle phase shift of velocity and the half-cycle sign reversal between acceleration and displacement.
The SHM equation x¨+ω²x=0 has sinusoidal solutions. The course expects you to know and use the solution, not prove the full differential-equation solution theorem. To determine A and φ from initial conditions, use x(0)=A cosφ and v(0)=-Aω sinφ. Squaring x(0)/A and v(0)/(Aω), then adding, gives A²=x₀²+(v₀/ω)²; the signs of x₀ and v₀ then select the correct phase quadrant.
The course expects students to know the sinusoidal solution of the SHM differential equation and use it, but not to mathematically prove that solution. Resonance is treated as a sinusoidal external force applied at the natural frequency and an increased response amplitude. A full damped/forced-oscillator differential-equation solution, quality factor, and resonance linewidth are beyond this lesson’s required scope.
Physical examples
Example 1
Released from rest at x=+A at t=0: choose φ=0 in the cosine convention, so x=A cosωt and v=-Aω sinωt.
Example 2
At equilibrium x=0, acceleration is zero and speed magnitude is maximum; the oscillator is not in equilibrium statically because it is moving through the equilibrium position.
Example 3
At x=+A, velocity is zero while acceleration is -Aω², directed back toward equilibrium.
Example 4
Doubling A doubles v_max and a_max but leaves ω and T unchanged for the same ideal linear oscillator.
Example 5
A periodic driving force near the natural frequency can increase oscillation amplitude through resonance.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Given x(t), differentiate rather than memorizing disconnected velocity/acceleration formulas; then evaluate signs at the requested time.
Reasoning pattern 2
Use a=-ω²x as a fast consistency check between a graph and the displacement.
Reasoning pattern 3
At x=0: |v| is maximum and a=0. At x=±A: v=0 and |a| is maximum.
Reasoning pattern 4
Extract A, T, ω, and φ from a graph before substituting values into equations.
Reasoning pattern 5
For resonance, identify the natural frequency and explain why a sinusoidal drive at that frequency increases the response amplitude.
Deeper calculus and limiting-case check
Initial conditions provide a compact amplitude relation: A=√(x₀²+(v₀/ω)²). This follows from sin²φ+cos²φ=1 and is a strong cross-check when the initial state is neither an endpoint nor equilibrium. The phase still requires sign/quadrant information; an inverse tangent without quadrant awareness can return the wrong physical state.
Model-selection workflow: decide what the problem is before choosing an equation
For Representing and Analyzing SHM, the fastest reliable route is not equation hunting. Start by identifying the equilibrium point and restoring interaction, encode it with the phase/energy representation, choose the oscillator differential equation, and then test the small-displacement or ideal-oscillator assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat amplitude A and phase θ=ωt+φ as physical quantities first and symbols second.
Choose a representation
Build Aligned x(t), v(t), and a(t) graphs with the same time axis: v is shifted by one-quarter cycle from x, while a is opposite in sign to x.. If the prompt supplies Phase-circle representation: projection of uniform circular phase can organize sine/cosine values without claiming the physical oscillator itself moves in a circle., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is x(t) = A cos(ωt+φ). State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
The course expects students to know the sinusoidal solution of the SHM differential equation and use it, but not to mathematically prove that solution. Resonance is treated as a sinusoidal external force applied at the natural frequency and an increased response amplitude. A full damped/forced-oscillator differential-equation solution, quality factor, and resonance linewidth are beyond this lesson’s required scope. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | Aligned x(t), v(t), and a(t) graphs with the same time axis: v is shifted by one-quarter cycle from x, while a is opposite in sign to x. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | x(t) = A cos(ωt+φ) | one valid phase convention for ideal SHM. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | v(t) = -Aω sin(ωt+φ) | time derivative of displacement. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | Differentiate x(t)=A cos(ωt+φ): v=dx/dt=-Aω sin(ωt+φ). Differentiate again: a=dv/dt=-Aω² cos(ωt+φ)=-ω²x. Because |sin|≤1 and |cos|≤1, |v|≤Aω and |a|≤Aω². The calculus automatically produces the quarter-cycle phase shift of velocity and the half-cycle sign reversal between acceleration and displacement. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
An oscillator has x(t)=0.080 cos(5.0t+π/3) m. The amplitude is 0.080 m and angular frequency is 5.0 rad/s, so T=2π/5.0=1.257 s. At t=0, x=0.080 cos(π/3)=0.040 m. Its velocity is v=-Aω sin(π/3)=-(0.080)(5.0)(√3/2)=-0.346 m/s, so it is moving toward decreasing x. Acceleration is a=-ω²x=-(25)(0.040)=-1.00 m/s². Independently, v_max=Aω=0.400 m/s and a_max=Aω²=2.00 m/s². The signs and magnitudes agree with the phase: the oscillator begins right of equilibrium and moving left. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 7.3
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship x(t) = A cos(ωt+φ) should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Aligned x(t), v(t), and a(t) graphs with the same time axis: v is shifted by one-quarter cycle from x, while a is opposite in sign to x. and Phase-circle representation: projection of uniform circular phase can organize sine/cosine values without claiming the physical oscillator itself moves in a circle.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Use a motion sensor to record a low-friction spring oscillator. Fit x(t) to a sinusoid, then numerically differentiate or use the fitted function to generate v(t) and a(t). Check that a versus x is linear with slope -ω², that the measured time between repeated states equals T, and that v peaks near x=0. If a periodic driver is available, sweep its frequency slowly and identify the frequency at which the response amplitude is largest.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Using φ as a time shift with units of seconds; φ is a phase angle in radians. | Return to the phase/energy representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Saying velocity is maximum at a turning point. At x=±A, v=0; speed is maximum at x=0. | Return to the oscillator differential equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Confusing maximum acceleration with acceleration at equilibrium. a=0 at x=0 and |a| is maximum at |x|=A. | Return to the phase/energy representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Changing amplitude and then claiming the ideal SHM period changes. For the same linear oscillator, A changes energy and extrema, not ω or T. | Return to the oscillator differential equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Sketching x, v, and a with independent periods or inconsistent phase shifts. | Return to the phase/energy representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Assuming every driven oscillator at every frequency is in resonance; resonance specifically refers to driving at the natural frequency in the course model. | Return to the oscillator differential equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 1.2 turns a position function into velocity and acceleration by differentiation. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 7.2 converts the fitted angular frequency into period and frequency. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 7.4 uses the same A and x to describe the kinetic/potential energy exchange. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write x(t) = A cos(ωt+φ) and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 7.3 mastery ladder
Interactive SHM phase-state check
For x=A cos(ωt+φ), enter amplitude, angular frequency, phase constant, and time. The tool returns x, v, a and the extrema.
Experimental connection
Use a motion sensor to record a low-friction spring oscillator. Fit x(t) to a sinusoid, then numerically differentiate or use the fitted function to generate v(t) and a(t). Check that a versus x is linear with slope -ω², that the measured time between repeated states equals T, and that v peaks near x=0. If a periodic driver is available, sweep its frequency slowly and identify the frequency at which the response amplitude is largest.
Connections across Physics C
Connection 1
Topic 1.2 turns a position function into velocity and acceleration by differentiation.
Connection 2
Topic 7.2 converts the fitted angular frequency into period and frequency.
Connection 3
Topic 7.4 uses the same A and x to describe the kinetic/potential energy exchange.
Connection 4
E&M Topic 13.6 later uses the same second-order SHM structure for an ideal LC circuit.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Using φ as a time shift with units of seconds; φ is a phase angle in radians.
Avoid this
Saying velocity is maximum at a turning point. At x=±A, v=0; speed is maximum at x=0.
Avoid this
Confusing maximum acceleration with acceleration at equilibrium. a=0 at x=0 and |a| is maximum at |x|=A.
Avoid this
Changing amplitude and then claiming the ideal SHM period changes. For the same linear oscillator, A changes energy and extrema, not ω or T.
Avoid this
Sketching x, v, and a with independent periods or inconsistent phase shifts.
Avoid this
Assuming every driven oscillator at every frequency is in resonance; resonance specifically refers to driving at the natural frequency in the course model.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
Why can SHM use either sine or cosine?
They are the same family of solutions with different phase constants. The initial conditions determine the convenient phase.
Where is an SHM object fastest?
At equilibrium, x=0, where the restoring acceleration is zero and speed magnitude reaches Aω in the ideal model.
Does resonance mean the oscillator’s natural frequency changes?
No. Resonance is the enhanced response when the driving frequency matches the natural frequency; the natural frequency is a property of the undriven system model.
Does a larger amplitude change the period?
Not for ideal linear SHM. It increases the extrema and the total energy, while ω and T stay fixed for the same oscillator parameters.