FULL REVIEW
Full Review: Springs and Elastic Potential Energy — Algebra-Based
Review the complete quantitative toolkit for Hooke’s law, spring work, elastic potential energy, and conservation-of-energy problems.
TIME
45–60 minutes
BEST FOR
A complete topic review
FINISH WITH
A readiness check
After this full review, you’ll be able to...
analyze springs and elastic potential energy using the Algebra-Based treatment with confidence.
Choose how you want to review
Course Alignment
This Physics Sensei Unit Review reinforces the unit below. Use it to review core ideas, prepare for homework, or refresh before a quiz or exam.
RESOURCE: Physics Sensei Unit Review
UNIT: Springs and Elastic Potential Energy
TREATMENT: Algebra-Based
COURSE LEVEL: Introductory college physics
BEST USED
✓ After learning the unit
✓ Before starting homework
✓ Before a quiz or exam
Physics Sensei is an independent educational resource organized around physics concepts, problem-solving strategies, and guided review.
Your Review Plan
Complete these six stages in order. Each stage builds on the previous one and prepares you for the final readiness check.
6 Stages • Approximately 45–60 minutes.
Warm-Up Check
Before you begin, take a moment to see what you already remember. Do not worry about getting everything right. This is only a starting point.
ACTIVITY 1
Hooke’s Law
Calculate spring force from stiffness and displacement.
A 250 N/m spring is stretched 0.080 m. What is the spring-force magnitude?
Reveal Answers
20 N, directed toward equilibrium.
Why it works: |Fs| = k|x| = (250)(0.080) = 20 N.
ACTIVITY 2
Elastic Energy
Calculate stored spring energy.
A 180 N/m spring is compressed 0.12 m. How much elastic potential energy is stored?
Reveal Answers
1.30 J.
Why it works: Us = ½kx² = ½(180)(0.12)² = 1.296 J ≈ 1.30 J.
ACTIVITY 3
Turning Point
Connect maximum deformation with v = 0.
At maximum extension of a frictionless horizontal spring-block system, what is the block’s speed?
Reveal Answers
0 m/s.
Why it works: Maximum extension is a turning point, so kinetic energy is momentarily zero.
Ready to strengthen your understanding?
You've refreshed what you already know. Next, you'll reinforce the essential concepts that will help you solve problems with confidence. Need to see the learning path again?
Core Concepts
Let's rebuild the key ideas one step at a time. Focus on understanding the relationships before worrying about solving problems.
KEY CONCEPT 1
Hooke’s Law and Force Graphs
For an ideal spring, the force is linear in displacement. The slope of an F-versus-x graph is −k.
Fs = −kx; |Fs| = k|x|.
Example: A graph slope of −400 N/m corresponds to k = 400 N/m.
Sensei Note: The negative sign tracks restoring direction; use magnitude form only when direction is handled separately.
KEY CONCEPT 2
Elastic Potential Energy and Spring Work
A deformed spring stores elastic potential energy. Spring work is the negative change in that potential energy.
Us = ½kx²; Ws = ½kxi² − ½kxf² = −ΔUs.
Example: If a spring moves toward equilibrium, Us decreases and the spring does positive work.
Sensei Note: Do not use W = Fd with one endpoint force; spring force changes with position.
KEY CONCEPT 3
Mechanical Energy with Springs
For conservative motion, combine kinetic, gravitational, and spring potential energy between carefully chosen initial and final states.
Ki + Ug,i + Us,i = Kf + Ug,f + Us,f.
Example: For a horizontal spring released from rest at amplitude A: ½kA² = ½mv² + ½kx².
Sensei Note: Set zero terms only after identifying the physical state.
Ready to apply these ideas?
You've reinforced the essential concepts. Now it's time to put them into practice by working through guided examples and building your problem-solving confidence. Need a quick reminder?
Guided Practice
Now it's time to apply what you've reviewed.
Work through each activity in order. The examples become gradually more challenging, and each one prepares you for the final readiness check.
PRACTICE 1
Spring Energy
Calculate elastic potential energy directly.
A 160 N/m spring is compressed 0.20 m. Find Us.
Reveal Answers
3.2 J.
Why it works: Us = ½(160)(0.20)² = 3.2 J.
PRACTICE 2
Speed at Equilibrium
Convert spring potential energy into kinetic energy.
A 0.80 kg block is released from rest after compressing a 320 N/m spring by 0.10 m. Find its speed at equilibrium.
Reveal Answers
2.0 m/s.
Why it works: ½kx² = ½mv², so v = x√(k/m) = 0.10√(320/0.80) = 2.0 m/s.
PRACTICE 3
Maximum Compression
Solve for deformation from incoming kinetic energy.
A 1.0 kg block moving at 3.0 m/s compresses a 450 N/m spring on a frictionless surface. Find the maximum compression.
Reveal Answers
0.141 m, about 0.14 m.
Why it works: At maximum compression v = 0, so ½mv² = ½kx² and x = v√(m/k).
Ready to check your understanding?
You've practiced the essential skills with guidance. Now it's time to solve a few short problems on your own and confirm you're ready to move forward. Need a quick reminder?
Confidence Check
You've rebuilt the key ideas and practiced them with guidance. Now try these short questions on your own to check your understanding before moving on.
QUICK CHECK 1
Energy Scaling
Use proportional reasoning before calculating.
If the spring displacement doubles while k is unchanged, how does Us change?
Reveal Answers
It becomes four times larger.
Why it works: Us is proportional to x².
QUICK CHECK 2
Spring Work
Determine the sign of spring work from an energy change.
A spring moves from x = 0.15 m to x = 0.05 m. Is the work done by the spring positive or negative?
Reveal Answers
Positive.
Why it works: The spring moves toward equilibrium, so Us decreases and Ws = −ΔUs is positive.
QUICK CHECK 3
Conservation Setup
Choose the correct energy terms.
A vertical spring launches a mass upward. Which additional potential-energy term must be included if height changes?
Reveal Answers
Gravitational potential energy, Ug = mgy.
Why it works: Mechanical-energy accounting must include every conservative potential that changes between the chosen states.
How did it go?
You've checked your understanding. Take one final look at the essential ideas before deciding what to do next. Need a quick reminder?
Summary
Before moving on, take one final look at the most important ideas from this review.
KEY TAKEAWAY 1
Use Hooke’s Law with Direction
Fs = −kx; the slope of an F-x graph is −k.
KEY TAKEAWAY 2
Connect Work and Spring Energy
Us = ½kx² and Ws = −ΔUs.
KEY TAKEAWAY 3
Solve Spring Problems by States
Choose initial and final states, identify energy forms, set legitimate zero terms, then solve.
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