DSE Physics | Hooke's Law and Springs Lesson-Prep Guide (for tutors) | what k means, energy from area, four common errors

DSE Physics | Hooke’s Law and Springs Lesson-Prep Guide (for tutors) | what k means, energy from area, four common errors

2026-11-05

Tutor lesson-prep guides · Secondary Physics

Students can write F = kx yet drop 15 cm straight in and get an answer a hundred times too big. This lesson fixes units, proportionality and graph slope together, then uses the area under the line to explain why elastic energy carries a half.

Objectives — Students can: (1) convert an extension from cm to m before using F = kx, and state that k is in N/m, meaning newtons needed per metre; (2) take k from the slope of a load-extension graph and find elastic potential energy as the triangle area, half k x squared (k = 200 N/m, extension 15 cm gives 30 N and 2.25 J); (3) work out how series and parallel springs change the effective k and explain with T = 2 pi root(m/k) why a stiffer spring oscillates faster.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Unit conversion	How many metres is 15 cm?	0.15 m
Equilibrium of forces	A 3 kg mass hangs still on a spring. What is the tension?	About 29.4 N, taking g as 9.8
Straight-line graphs	What does the gradient of a straight line represent?	Change in y divided by change in x

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-6 min	Prior check	Three quick items, especially cm to m	Answer orally
6-20 min	Read k from the graph	Draw the F-x line and take two points	Students pick their own two points
20-32 min	Unit and ratio traps	Show that 15 cm must be written 0.15 m	Students correct three wrong calculations
32-44 min	Elastic energy from area	Triangle area is 2.25 J, and doubling x quadruples it	Students compute 30 cm and compare
44-56 min	Series, parallel and period	Parallel 400 N/m, series 100 N/m, period 0.314 s	Students predict and then verify the period
56-60 min	Exit check	Collect three items, mark at once, tag the act	Work independently

03 Teaching script (read it out)

Scene 1: Act 1: k means newtons per metre, not per centimetre

Open with two lines on the board: F = kx and the units N = (N/m)(m). Then ask: if k is 200 N/m, how much force stretches the spring by 15 cm? Most students write 200 x 15 = 3000 N because they treat centimetres as metres.
The correct line is 0.15 m times 200, which gives 30 N. Say it twice: the unit is part of the equation, and an answer without units is not an answer. Then unpack k: a larger k means a stiffer spring, so the same force stretches it less. It is the spring identity card, fixed for a given spring inside the elastic limit.
Give a feel for the size: 200 N/m means a load of about 3.1 kg stretches it 15 cm, so everyday springs sit in the hundreds of N/m.

Students repeat: metres first, then multiply by k, and k is in N/m.

Scene 2: Act 2: the slope is k and the area is energy

Act two works from the graph. With extension in metres on the x axis and force in newtons on the y axis, Hooke law is a straight line through the origin and its gradient is k. Taking the point (0.15, 30) gives 30 / 0.15 = 200 N/m, matching the equation.
Then the energy: the force rises linearly from zero to F, so the average force is F/2 and the work done is half F x, which is half k x squared, here 2.25 J. Students must see that the half comes from the force not being at full value from the start, and that the area under the line is a triangle rather than a rectangle. That picture matters more than the formula.
Then compare: extending from 15 cm to 30 cm doubles the force to 60 N but takes the energy from 2.25 J to 9 J, four times as much, because extension is squared. This comparison appears almost every year.

Students repeat: slope gives k, area gives energy, and doubling x gives four times the energy.

Scene 3: Act 3: series is soft, parallel is stiff, and the period follows

Act three covers combinations and oscillation. Two identical springs in parallel share the load, so the effective k is 2k = 400 N/m and the pair is stiffer; in series the same force stretches each section, so the effective k is k/2 = 100 N/m and the pair is softer. The slogan is parallel stiffens, series softens, the opposite of resistors, so flag the contrast explicitly.
Hanging 0.5 kg on 200 N/m gives T = 2 pi root(0.5/200) = 0.314 s; switching to the parallel pair (400 N/m) gives 0.222 s, about thirty per cent shorter. Students should state the cause: a larger k means a larger restoring force, faster motion and a shorter period, while more mass does the reverse. Add that the period is independent of amplitude in the small-amplitude approximation, which is the point most often confused.

Students repeat: series softens, parallel stiffens, and a larger k shortens the period.

04 Question types and worked steps

Type	Example	Steps	Answer
Force to extension	k = 200 N/m, load 60 N: how far does it stretch	1 convert units 2 x = F/k 3 give the answer in m or cm and say which	30 cm
k from a graph	The F-x line passes (0,0) and (0.15, 30)	1 take two points 2 gradient = change in F over change in x 3 unit N/m	200 N/m
Elastic potential energy	Energy stored at 15 cm extension with k = 200 N/m	1 half k x squared 2 x in metres 3 unit J	2.25 J
Series, parallel, period	Period of 0.5 kg on two parallel 200 N/m springs	1 effective k 2 T = 2 pi root(m/k) 3 unit s	0.222 s

Sanity check (teach this): After each answer ask whether it is plausible: 200 N/m holding 30 N (about 3.1 kg) stretches 15 cm, which is visible by eye, so a stretch of metres or millimetres means a unit slip. Energy is always positive and scales with the square of extension. A series combination must be softer than either spring and a parallel one stiffer, so a series result above 200 N/m means the formula was inverted. A period should land between 0.1 and 2 s for classroom apparatus.

05 Independent practice (3 questions)

Question	Answer	What to watch
A 200 N/m spring is stretched 15 cm. Find the force and the elastic energy stored.	30 N; 2.25 J	Did they convert cm to m and use half k x squared
The same spring is stretched 30 cm. How many times the earlier energy is stored?	Four times, 9.0 J	Do they say the energy scales with the square of extension
Two parallel 200 N/m springs carry 0.5 kg. Find the period.	0.222 s	Is the effective k taken as 2k, and is T = 2 pi root(m/k) used

06 Common mistakes and what to say

Mistake	Student writes	Say this
Dropping cm into the formula	200 x 15 = 3000 N	Convert first: 15 cm is 0.15 m, not 15.
Using cm inside the energy formula	Half x 30 x 15 = 225 J	Energy needs SI units, so x in metres, and then the answer is in joules.
Swapping series and parallel	Treating a series pair as 400 N/m	Parallel springs push together so the pair is stiff; series springs stretch in turn so it is soft.
Believing the period depends on amplitude	Saying a bigger pull gives a longer period	At small amplitude the period depends only on mass and k, not on how far you pull.

07 Homework (paper, section and question numbers)

Redo this chapter — Redo practice items 1 to 3, starting each with the unit conversion
Physics chapter index — Open the circular motion chapter and note how period is used there
Draw an F-x graph — Plot a 200 N/m line on graph paper and mark (0.1, 20) plus the gradient

08 Exit check (last 5 minutes)

Question	Answer	Decision
What is the unit of k	N/m, newtons per metre	Answering N/cm means replay Act 1
What does the area under an F-x graph mean	Elastic potential energy stored	Answering k means replay Act 2
Does a series pair have a larger or smaller effective k	Smaller, k/2	Answering larger means replay Act 3

09 Tutor FAQ

When does Hooke law fail?

Beyond the elastic limit, where the spring deforms permanently and the graph curves instead of staying straight.

Why is elastic energy half k x squared rather than k x squared?

The force rises linearly from zero to F, so the average force is F/2, and the work is average force times displacement, the triangle area.

Should the mass of the spring be included?

DSE questions normally ignore it; use it only when the question supplies it.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.