Frequency and Period of SHM
The period T is the time for one complete oscillation, frequency f is cycles per second, and angular frequency ω is the phase rate in radians per second: f=1/T and ω=2πf=2π/T. For an ideal mass-spring oscillator, T=2π√(m/k_eff). For a simple pendulum at small angle, T=2π√(L/g). These periods come from the system’s restoring-force-to-inertia ratio, not from amplitude.
Topic sequence · 2 of 5
Learning objectives
Objective 1
Relate period, frequency, and angular frequency with correct units.
Objective 2
Derive the mass-spring and small-angle simple-pendulum periods from Newton’s second law.
Objective 3
Use functional dependence to predict how T changes with mass, stiffness, length, and gravitational field strength.
- Topic 7.1 defining SHM.
- Topic 2.8 spring forces.
- Topic 2.6 gravitational force near a planet surface.
- Basic small-angle approximation in radians for pendulum reasoning.
Definitions first
Representations you must read and create
Representation 1
T versus √m is linear for fixed k, while T² versus m is linear with slope 4π²/k.
Representation 2
T² versus L for a small-angle pendulum is linear with slope 4π²/g.
Representation 3
Log-log scaling: T∝m^1/2, T∝k^-1/2, T∝L^1/2, T∝g^-1/2.
Representation 4
Cycle diagram connecting one period to 2π radians of phase.
Core relationships and calculus
Variables: f Hz; T s
Use: any periodic motion
Variables: ω rad/s
Use: phase relation for SHM
Variables: m kg; k_eff N/m
Use: ideal mass-spring SHM
Variables: ω rad/s
Use: same model
Variables: L m; g m/s²
Use: simple pendulum, small-angle approximation
Variables: ω rad/s
Use: same small-angle model
For a spring, m x¨=-kx gives x¨+(k/m)x=0. Comparing with x¨+ω²x=0 gives ω=√(k/m), then T=2π/ω. For a pendulum, tangential motion obeys mL θ¨=-mg sinθ. At small angle, sinθ≈θ, so θ¨+(g/L)θ=0; hence ω=√(g/L) and T=2π√(L/g). The derivation makes the small-angle condition explicit.
Both canonical formulas are instances of the same pattern: period is set by inertia divided by restoring stiffness. The spring has translational inertia m and stiffness k; the simple pendulum’s small-angle angular equation has effective ratio L/g. Dimensional analysis supports the square-root forms but cannot supply the factor 2π; that comes from one full sinusoidal phase cycle.
The simple-pendulum formula here assumes a point-like bob, massless inextensible string, negligible damping, and small angular displacement. Topic 7.5 develops pendulum models in more detail. The ideal spring formula assumes a linear restoring force and an effective oscillating mass represented by m.
Physical examples
Example 1
Quadrupling a spring-mass oscillator’s mass doubles its period.
Example 2
Quadrupling spring stiffness halves the period.
Example 3
Doubling pendulum length multiplies period by √2.
Example 4
Changing pendulum bob mass does not change the small-angle period.
Example 5
Changing spring amplitude does not change period in the ideal linear SHM model.
Example 6
A vertical ideal spring has the same small-oscillation period 2π√(m/k) as the same horizontal spring; gravity shifts the equilibrium position but does not appear in the period when x is measured from equilibrium.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Convert cleanly among T, f, and ω; the factor 2π separates cycles from radians.
Reasoning pattern 2
Use functional dependence before arithmetic: T∝√m/√k for springs and T∝√L/√g for a small-angle simple pendulum.
Reasoning pattern 3
For a vertical ideal spring, separate equilibrium shift from oscillation about equilibrium; g does not enter T=2π√(m/k).
Reasoning pattern 4
For experimental linearization, use T² versus m or T² versus L so the slope has a direct physical meaning.
Reasoning pattern 5
State the small-angle approximation for the simple-pendulum formula instead of applying it automatically to a large-amplitude swing.
Deeper calculus and limiting-case check
For an unfamiliar oscillator, identify an effective restoring coefficient κ and effective inertia μ so that q¨+(κ/μ)q=0. Then ω=√(κ/μ). This unifies springs, pendulums, torsional systems, and later LC oscillations.
Model-selection workflow: decide what the problem is before choosing an equation
For Frequency and Period of 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 period T and frequency f as physical quantities first and symbols second.
Choose a representation
Build T versus √m is linear for fixed k, while T² versus m is linear with slope 4π²/k.. If the prompt supplies T² versus L for a small-angle pendulum is linear with slope 4π²/g., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is f = 1/T. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
The simple-pendulum formula here assumes a point-like bob, massless inextensible string, negligible damping, and small angular displacement. Topic 7.5 develops pendulum models in more detail. The ideal spring formula assumes a linear restoring force and an effective oscillating mass represented by m. 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 | T versus √m is linear for fixed k, while T² versus m is linear with slope 4π²/k. | 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 | f = 1/T | any periodic motion. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | ω = 2πf = 2π/T | phase relation for SHM. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | For a spring, m x¨=-kx gives x¨+(k/m)x=0. Comparing with x¨+ω²x=0 gives ω=√(k/m), then T=2π/ω. For a pendulum, tangential motion obeys mL θ¨=-mg sinθ. At small angle, sinθ≈θ, so θ¨+(g/L)θ=0; hence ω=√(g/L) and T=2π√(L/g). The derivation makes the small-angle condition explicit. | 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
A 0.50 kg mass oscillates on an ideal spring with k=32 N/m. The angular frequency is ω=√(k/m)=√64=8.0 rad/s, so T=2π/8=0.785 s and f=1/T=1.27 Hz. If the mass is quadrupled to 2.0 kg, ω is halved and T doubles to 1.57 s. The same period applies if the original spring is mounted vertically: gravity changes the equilibrium extension, but once x is measured from that equilibrium the oscillation equation remains m x¨=-kx. This is a model-selection result, not a memorized exception. 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 7.2
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 f = 1/T should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between T versus √m is linear for fixed k, while T² versus m is linear with slope 4π²/k. and T² versus L for a small-angle pendulum is linear with slope 4π²/g.. 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
For a spring, measure T over many cycles for several masses and plot T² versus m; use slope=4π²/k to estimate k. For a pendulum, vary L at small amplitude and plot T² versus L; slope=4π²/g estimates g. Timing many cycles reduces fractional reaction-time uncertainty.
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 |
|---|---|---|
| Confusing frequency f in hertz with angular frequency ω in rad/s; they differ by 2π. | 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? |
| Using T=2π√(L/g) for a pendulum at large amplitude without acknowledging the small-angle approximation. | 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? |
| Claiming the ideal SHM period depends on amplitude. | 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? |
| Using total spring length or arbitrary geometry instead of the effective spring constant actually seen by the mass. | 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? |
| Timing one cycle when experimental uncertainty could be reduced by timing many cycles and dividing. | 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? |
| Putting g into the ideal spring-mass period because the spring is vertical. Gravity shifts equilibrium, but after measuring displacement from equilibrium the linearized equation is m x¨=-kx and the period is unchanged. | 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? |
| Treating 'small angle' as a universal fixed numerical cutoff. The required approximation is sinθ≈θ in radians; validity depends on the accuracy demanded by the problem or experiment. | 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? |
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 7.1 supplies the restoring-force criterion and SHM differential equation. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 2.8 determines k_eff for spring arrangements before the period formula is used. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 7.5 will distinguish simple and physical pendulum periods and their validity conditions. 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 f = 1/T 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 7.2 mastery ladder
Interactive SHM period comparison
Compare an ideal mass-spring oscillator with a small-angle simple pendulum.
Experimental connection
For a spring, measure T over many cycles for several masses and plot T² versus m; use slope=4π²/k to estimate k. For a pendulum, vary L at small amplitude and plot T² versus L; slope=4π²/g estimates g. Timing many cycles reduces fractional reaction-time uncertainty.
Connections across Physics C
Connection 1
Topic 7.1 supplies the restoring-force criterion and SHM differential equation.
Connection 2
Topic 2.8 determines k_eff for spring arrangements before the period formula is used.
Connection 3
Topic 7.5 will distinguish simple and physical pendulum periods and their validity conditions.
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
Confusing frequency f in hertz with angular frequency ω in rad/s; they differ by 2π.
Avoid this
Using T=2π√(L/g) for a pendulum at large amplitude without acknowledging the small-angle approximation.
Avoid this
Claiming the ideal SHM period depends on amplitude.
Avoid this
Using total spring length or arbitrary geometry instead of the effective spring constant actually seen by the mass.
Avoid this
Timing one cycle when experimental uncertainty could be reduced by timing many cycles and dividing.
Avoid this
Putting g into the ideal spring-mass period because the spring is vertical. Gravity shifts equilibrium, but after measuring displacement from equilibrium the linearized equation is m x¨=-kx and the period is unchanged.
Avoid this
Treating 'small angle' as a universal fixed numerical cutoff. The required approximation is sinθ≈θ in radians; validity depends on the accuracy demanded by the problem or experiment.
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
What is the difference between f and ω?
f counts cycles per second; ω measures phase advance in radians per second, so ω=2πf.
Does amplitude affect the period of ideal SHM?
No. In the ideal linear SHM model, period is amplitude-independent. Nonlinear systems can lose that property.
Why does a heavier spring mass oscillate more slowly?
Greater inertia reduces angular frequency: ω=√(k/m), so the period increases as √m.
Why is pendulum period independent of bob mass?
Both the tangential gravitational force and inertia are proportional to mass, so mass cancels from the equation of motion.