Lesson 8: Springs & Hooke's Law | PHY 2.4

Lesson at a Glance | He Tirohanga Whakamua

Do NowWhy stretching a spring twice as far takes four times as much energy10 min

Hooke's Law ( $F = k x$ )Restoring force & spring constant

k

15 min

Elastic Potential EnergyArea under Force-Extension graph

E_{e p} = \frac{1}{2} k x^{2}

15 min

Portfolio EntryWrite Section 8: Hooke's Law & Elastic Energy Transformations10 min

Exit DrillCalculate launch speed of a 0.1kg pinball launched by a compressed spring5 min

Ngā Whāinga Ako | Learning Intentions

Students will know

Hooke's Law ( $F = k x$ ): The restoring force exerted by an elastic spring is directly proportional to extension or compression $x$ up to the elastic limit.
Elastic Potential Energy ( $E_{e p} = \frac{1}{2} k x^{2}$ ): Energy stored in a distorted spring equals the triangular area under a Force-Extension ( $F$ vs $x$ ) graph.
How spring potential energy converts into Kinetic ( $E_{e p} \to E_{k}$ ) and Gravitational Potential Energy ( $E_{e p} \to E_{p}$ ).

Students will demonstrate

By calculating spring constant $k$ , stored $E_{e p}$ , and projectile launch velocity from a spring launcher.
By completing Section 8 of their Level 2 Physics Mechanics Mastery Portfolio.

Do Now | Tīmatanga Whakaaro (10 min)

Spring Energy Scaling Prompt:

"If compressing a spring by 0.1 metres stores 5 Joules of energy, how much energy is stored if you compress the exact same spring by 0.2 metres?"

Unpack: 20 Joules! Because Elastic Potential Energy is proportional to extension SQUARED ( $E_{e p} = \frac{1}{2} k x^{2}$ ). Doubling extension ( $x \times 2$ ) increases stored energy by $2^{2} = 4$ times!

Hooke's Law & Force-Extension Graphs (15 min)

1. Hooke's Law Equation ( $F = k x$ )

$F$ : Force ( $N$ )
$k$ : Spring constant ( ${N m}^{- 1}$ , gradient of $F$ vs $x$ graph)
$x$ : Extension or compression ( $m$ ).

2. Energy Stored ( $E_{e p} = \frac{1}{2} k x^{2}$ )

Area under Force-Extension triangle: $Area = \frac{1}{2} \times base \times height = \frac{1}{2} x F = \frac{1}{2} k x^{2}$ .

Elastic Potential Energy (15 min)

1. The area under the graph is the energy

For a spring obeying Hooke's law, the force-extension graph is a straight line through the origin. The work done stretching it is the area under that line: a triangle of base $x$ and height $F$ , so $E_{p} = \frac{1}{2} F x$ .

2. Substituting Hooke's law

Since $F = k x$ , the same area gives $E_{p} = \frac{1}{2} k x^{2}$ . The square matters: stretch a spring twice as far and it stores four times the energy. Use the area method when the graph is NOT a straight line, because the formula only holds while the spring is still elastic.

Do this now: a spring with $k = 250$ N m⁻¹ is stretched 0.08 m. Find the stored energy two ways, from the area and from the formula, and check they agree.

📁 Physics Mechanics Portfolio — Section 8: Hooke's Law & Spring Transformations

Students open their Level 2 Physics Portfolio and complete Section 8:

Section 8 Requirements:

1. Force-Extension Graph: Plot Force vs Extension for a spring ( $k = 400 {N m}^{- 1}$ ) up to $x = 0.15 m$ , shading the area representing stored $E_{e p}$ .

2. Spring Launcher Calculation: Solve $E_{e p} = \frac{1}{2} k x^{2} \to E_{k} = \frac{1}{2} m v^{2}$ to find the launch speed of a 0.05kg dart launched from a spring compressed by 0.12m.

3. Excellence Vertical Spring Equilibrium Rationale: 1-paragraph explanation of vertical mass-spring equilibrium where hanging weight $m g$ equals restoring force $k x$ ( $m g = k x$ ).

Exit Verification | Ka Mutu Hoki (5 min)

Exit Check:

"My Section 8 sets 1/2kx^2 = 1/2mv^2 to prove launch speed v = 10.7 m s⁻¹ for the spring launcher."

Teacher Planning & NCEA Alignment

NCEA Level 2 Physics Alignment (6 Credits External):

Hooke's Law & Springs: Demonstrate understanding of spring constant, restoring force ( $F = k x$ ), elastic potential energy ( $E_{e p} = \frac{1}{2} k x^{2}$ ), and energy conservation.
Graph Analysis: Calculate spring constant from gradient and stored energy from area under $F$ vs $x$ graph.

Vocabulary: Hooke's Law, spring constant ( $k$ ), extension ( $x$ ), Elastic Potential Energy ( $E_{e p}$ ), elastic limit, restoring force.

Same concept, shorter route: the Guided Viewing & Problem Practice version of this unit covers it in Hooke's Law and Simple Harmonic Motion. That route is a compact viewing-and-practice sequence; this one builds the concept over ten sections with a formative portfolio.

Lesson 8: Hooke's Law & Elastic Potential Energy in Springs