LF-04 · TEACHER LESSON PLAN

Pendulum period

How length sets a pendulum’s rhythm in the small-angle model

PhysicsGrades 8–122 sessions · 45 min totalHazard low; moving bobNGSS 3-PS2-2

01 //

Driving question

START THE LESSON WITH THISA 20 cm pendulum completes each swing sooner than an 80 cm pendulum, while a heavier bob on the same string barely changes the rhythm. What happened and why?

02 //

Learning objectives

Students predict how changing the pivot-to-bob length will affect the period while the bob and release angle stay the same.
Students calculate the mean time for ten cycles, period T, and T² for four tested string lengths.
Students graph and explain T² against length and use the near-linear pattern to describe how length affects a pendulum's timing.

03 //

Standards

Dimension	Alignment
Performance expectation	3-PS2-2Make observations and/or measurements of an object's motion to provide evidence that a pattern can be used to predict future motion. (Partial alignment: this is a grade 3 performance expectation, and its assessment boundary excludes the terms period and frequency, which this lesson uses. It supports the pattern idea (a swing repeats, so future motion can be predicted from it).)
Practices	SEP 3Plan and carry out investigations (control variables, provide evidence). SEP 4Analyze and interpret data.
Core idea	3-PS2 core ideaThe patterns of an object's motion in various situations can be observed and measured; when that past motion exhibits a regular pattern, future motion can be predicted from it.
Crosscutting	PatternsPatterns of change can be used to make predictions.

Standards text paraphrased from nextgenscience.org. Alignment is partial; this hands-on activity does not cover every part of the performance expectation.

03A //

State standards crosswalk

Use the row as a planning crosswalk, then confirm local district pacing and grade placement. “Partial” labels describe a skill connection, not full mastery of the complete standard.

State	Verified standard	How this lesson aligns	Official source
California	CA NGSS MS-PS2-2	Partial connection: students plan a controlled motion investigation, but the changed variable is pendulum length, not the net force named in this expectation.	Open standard ↗
Texas	Science.8.7(B)	Limited connection: students investigate a moving system, but do not analyze all three Newton laws as the expectation requires.	Open standard ↗
Florida	SC.6.P.12.1	Partial connection: students measure motion over time, but the pendulum does not move at constant speed as this benchmark specifies.	Open standard ↗
New York	NYSSLS MS-PS2-2	Partial connection: students plan a controlled investigation, but vary pendulum length rather than the force acting on an object.	Open standard ↗
Pennsylvania	STEELS 3.2.3.A	Topic match for observing and measuring a repeatable motion pattern to support prediction; this lesson extends the pattern work to period and T-squared.	Open standard ↗

04 //

Materials and prep timeline

PER GROUP

1 secure clamp stand, 1 light bob and 1 string adjustable to 20, 40, 60 and 80 cm
1 ruler marked in centimeters, measured from pivot to bob center
1 stopwatch, recording sheet and calculator
Safety glasses if required by local classroom rules

TEACHER ONLY

Check each clamp and knot before class; mark a clear swing zone
One printed student sheet per student

CLASS-SIZE KIT PLANNER

One per-group kit contains every item in the team materials list above. Groups are balanced around the listed 3 students target.

Students	Groups	Per-group kits	Student sheets	Suggested group split
20	6	6 complete kits	20 copies	2 groups of 4 and 4 groups of 3
25	8	8 complete kits	25 copies	1 group of 4 and 7 groups of 3
30	10	10 complete kits	30 copies	10 groups of 3

SESSION 1 · 20 MINSet up and time the cycles

Secure the support, predict the pattern, then take three ten-cycle timing trials at each listed length with the same bob and small release angle.

SESSION 2 · 25 MINCalculate and interpret

Average each length's trials, find T and T², graph T² against length, and explain the pattern and limits of the ideal model.

04A //

Lab figures and data visuals

Reviewed article images support the lesson flow. Use the tables and student sheet to record actual observations; pictured patterns are not target results.

A pendulum bob swings along a luminous arc with its string length marked from the pivot to the center of the bob. — **A longer string swings more slowly.**In the ideal small-angle model, period depends on pendulum length and gravity; bob mass cancels from the equation.

A student times complete pendulum swings with a stopwatch while measuring the string from the pivot to the center of the bob. — **Measure length and time full swings.**Measure from pivot to bob center, release from a small angle, time ten complete swings and divide by ten to estimate one period.

Two labeled graphs compare the curved T-versus-L relationship with the straight T-squared-versus-L relationship and show that the latter slope is 4π² divided by g. — **T² versus length should form a straight line.**The ideal small-angle relationship predicts a T²-versus-L slope of 4π²/g; use your measured slope to estimate local gravitational acceleration.

04B //

Printable shopping list

TEACHER PREP · Pendulum period

Shopping and staging checklist

Multiply the per-group kit contents by the class-size planner. The teacher section includes teacher-provided materials and setup or safety notes.

PER-GROUP KITS

1 secure clamp stand, 1 light bob and 1 string adjustable to 20, 40, 60 and 80 cm
1 ruler marked in centimeters, measured from pivot to bob center
1 stopwatch, recording sheet and calculator
Safety glasses if required by local classroom rules

TEACHER-PROVIDED MATERIALS + SETUP RULES

Check each clamp and knot before class; mark a clear swing zone
One printed student sheet per student

QUANTITIES BY CLASS SIZE

Material	Per group kit	20 students	25 students	30 students
Secure clamp stand	1	6	8	10
Light bob, string, ruler, stopwatch and calculator	1 of each	6 of each	8 of each	10 of each
Student recording sheets	1 per student	20	25	30
Safety glasses, only if local rules require them	1 pair per student	20 pairs	25 pairs	30 pairs

Prepare 6 kits for 20 · 8 kits for 25 · 10 kits for 30 students.

05 //

Lesson flow

Minutes	What happens
S1 · 0–4	Engage Show the short-versus-long pendulum and ask why a heavier bob on the same length barely changes its rhythm. Students predict a pattern before measuring.
S1 · 4–8	Explore Secure the stand. Demonstrate measuring from pivot to the bob's center and choosing a release angle under 15°; clear the swing path.
S1 · 8–20	Explore Use one bob, release without a push, time ten full cycles three times at each of 20, 40, 60 and 80 cm, and record all trials.
S2 · 0–9	Explain Average the three total times at each length. Divide each mean by ten for T, then square T; keep seconds and seconds squared distinct.
S2 · 9–17	Plot T² (s²) against length (cm), label both axes and draw a best-fit trend rather than joining points in order.
S2 · 17–22	Elaborate Compare the curved T-versus-length idea with the straighter T²-versus-length pattern; discuss how pivot friction or a large angle can alter results.
S2 · 22–25	Evaluate Write a claim with a measured mean and explain what the classroom setup can and cannot establish about mass and length.

06 //

Discussion questions

1. Which length should have the longest period, and what result would support that claim?

SAMPLE ANSWERThe 80 cm setting should take longer per cycle than 20 cm; the evidence is a larger mean ten-cycle time divided by ten, not simply that it looks slower.

2. Why time ten cycles three times instead of timing one cycle once?

SAMPLE ANSWERA start-stop reaction error is a smaller fraction of a ten-cycle total. Three repeats let us compare means and notice timing spread at each length.

3. Why graph T² against length rather than expecting T to double when length doubles?

SAMPLE ANSWERThe model says T depends on the square root of L. Squaring T makes the tested relationship closer to a straight trend; doubling length does not double T.

4. What does this apparatus leave out of the ideal pendulum model?

SAMPLE ANSWERThe stand pivots have friction and air resistance gradually reduces the swing. A large release angle, a moving clamp, or timing differences also limit the comparison; this setup does not prove every pendulum behaves identically.

07 //

Common misconceptions

STUDENTS OFTEN SAY“A heavier bob must swing slower because it weighs more.”

BETTER IDEAWith the same string length and a small release angle, bob mass has little effect on the ideal period. Test mass separately while holding the pivot-to-center length fixed.

STUDENTS OFTEN SAY“Twice the string length means twice the period.”

BETTER IDEAThe model predicts period grows with the square root of length. Compare T² with L to examine the more nearly linear relationship.

STUDENTS OFTEN SAY“The visible string length is the pendulum length.”

BETTER IDEAMeasure from the pivot point to the bob's center of mass. Measuring only the exposed string changes the length used in the model.

08 //

Differentiation

SUPPORT

Provide a four-row table for 20, 40, 60 and 80 cm and model one mean from three times.
Give a calculator-ready sequence: mean ten-cycle time ÷ 10 = T; T × T = T².
Use a partner counter and a marked center crossing so cycle starts and finishes at the same point and direction.

EXTEND

Test a second light bob mass at one fixed length and compare the mean T values.
Compare two small release angles at the same length; keep the bob and support unchanged.
Convert centimeters to meters and compare the slope of T² against L with 4π²/g, explaining timing uncertainty.

09 //

Assessment rubric

Criterion	Beginning	Developing	Secure
Claim	Does not identify how length relates to period.	Names a direction but does not distinguish period from ten-cycle time.	States how period changes with length and limits the claim to the tested setup.
Evidence	Gives no timing result.	Uses one trial or omits its unit.	Uses a mean supported by repeated ten-cycle times with seconds and length units.
Reasoning	Says only that a long string is slower.	Mentions T² or length without connecting them.	Explains why T² plotted against L supports the small-angle model's pattern.
Data care	Changes bob, angle, or timing method across lengths.	Controls most conditions but leaves a measurement or repeat unclear.	Uses the same bob, small angle, pivot, cycle definition and three recorded trials at each length.

10 //

Vocabulary

Period – Time for one complete back-and-forth cycle.
Pivot – The fixed point from which the pendulum hangs.
Bob – The mass attached to the lower end of the string.

Cycle – One repeat of the motion, returning to the same side in the same direction.
Mean – The sum of repeated measurements divided by the number of trials.
Amplitude – The greatest displacement from the resting position.

11 //

Safety, cleanup and troubleshooting

BEFORE STUDENTS TOUCH ANYTHING

Clamp or hold the support securely and use a light bob. Keep faces, fingers, and other people outside the swing path; do not wrap the string around a hand or neck.
Have an adult help with any elevated support, and stop if the stand shifts or the bob becomes loose.

Troubleshooting: If repeated times scatter, agree on the same center crossing and direction for starting and stopping, then repeat the three trials. If the string slips at the clamp or the support shifts, stop and reset the length before collecting more data.