FULL REVIEW
Full Review: Springs and Elastic Potential Energy — Foundational
Build a complete conceptual understanding of restoring force, elastic potential energy, turning points, and energy transformations in ideal spring systems.
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 Foundational 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: Foundational
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
Restoring-Force Direction
Identify the direction of the spring force from the displacement.
A spring is stretched to the right of equilibrium. Which way does the spring force point?
Reveal Answers
To the left, toward equilibrium.
Why it works: For an ideal spring, the restoring force always points opposite the displacement.
ACTIVITY 2
Stretch vs. Compression
Compare force magnitude and stored energy at equal-magnitude displacements.
The same spring is displaced by +0.10 m and then by −0.10 m. Compare the force magnitude and elastic potential energy.
Reveal Answers
The force magnitudes are equal, the force directions are opposite, and the elastic potential energies are equal.
Why it works: Force depends on the sign of x, while spring energy depends on x².
ACTIVITY 3
Turning Points
Connect maximum deformation with speed and energy.
A block attached to an ideal horizontal spring reaches maximum compression. What is true about its instantaneous speed?
Reveal Answers
Its speed is zero.
Why it works: Maximum compression is a turning point, so kinetic energy is zero while spring potential energy is maximum.
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 Equilibrium
An ideal spring exerts a restoring force proportional to displacement from equilibrium. The farther it is stretched or compressed, the larger the restoring-force magnitude.
Fs = −kx. At x = 0, Fs = 0.
Example: If the spring is stretched to the right, x is positive and the spring force points left.
Sensei Note: The negative sign describes direction; it does not mean the spring-force magnitude is negative.
KEY CONCEPT 2
Elastic Potential Energy
A stretched or compressed spring stores elastic potential energy. Equal-magnitude stretches and compressions store the same amount.
Us = ½kx².
Example: Doubling the deformation makes the stored spring energy four times larger.
Sensei Note: Spring energy is a scalar. Its value does not change sign when the displacement changes sign.
KEY CONCEPT 3
Energy Transformations and Turning Points
In an ideal spring system, spring potential energy and kinetic energy trade back and forth while total mechanical energy remains constant. Gravity can also participate when height changes.
K + Ug + Us = constant when only conservative forces do work.
Example: At maximum deformation, speed is zero. Near equilibrium, spring energy is minimum and speed can be maximum.
Sensei Note: Do not confuse zero spring force at equilibrium with zero speed.
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
Conceptual Spring Release
Track the direction of force and the energy change as a compressed spring returns toward equilibrium.
A block is released from rest from a compressed spring on a frictionless surface. As it moves toward equilibrium, what happens to spring energy and kinetic energy?
Reveal Answers
Spring potential energy decreases while kinetic energy increases.
Why it works: The spring does positive work on the block as stored spring energy is converted into kinetic energy.
PRACTICE 2
Energy Scaling
Use the x² dependence of elastic potential energy.
A spring is compressed twice as far as before. By what factor does its stored elastic potential energy change?
Reveal Answers
It becomes four times larger.
Why it works: Because Us = ½kx², doubling x multiplies the energy by 2² = 4.
PRACTICE 3
Spring and Gravity
Identify the energy forms in a vertical spring situation.
A spring launches an object upward. Which energy forms may need to be included between release and the highest point?
Reveal Answers
Elastic potential energy, kinetic energy, and gravitational potential energy.
Why it works: Choose initial and final states, then include only the energy forms present in those states.
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
Force or Energy?
Distinguish a directional force from scalar stored energy.
Which quantity changes sign when a spring moves from +x to −x: the spring-force component or the elastic potential energy?
Reveal Answers
The spring-force component changes sign; the elastic potential energy does not.
Why it works: Fs = −kx depends on x, while Us = ½kx² depends on x².
QUICK CHECK 2
At Equilibrium
Identify the force and energy conditions at x = 0.
A block passes through equilibrium in an ideal horizontal spring system. Is the spring force zero? Must the speed also be zero?
Reveal Answers
The spring force is zero, but the speed does not have to be zero; it can be maximum.
Why it works: At equilibrium x = 0, so Fs and Us are minimum, while kinetic energy can be maximum.
QUICK CHECK 3
At Maximum Extension
Identify turning-point conditions.
At maximum extension, what are the block’s speed and spring potential energy?
Reveal Answers
The speed is zero and the spring potential energy is maximum.
Why it works: Maximum extension is a turning point, so all available horizontal mechanical energy is stored in the spring.
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
Spring Force Restores Toward Equilibrium
The spring-force direction is opposite the displacement, and its magnitude grows with deformation.
KEY TAKEAWAY 2
Spring Energy Depends on Deformation Squared
Elastic potential energy is ½kx², so equal stretch and compression store equal energy.
KEY TAKEAWAY 3
Track Energy Between Turning Points and Equilibrium
Maximum deformation corresponds to zero speed; equilibrium corresponds to zero spring force and minimum spring energy.
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