What makes an oscillation simple harmonic?
You will be able to: Identify SHM from a net restoring force proportional to displacement from equilibrium.
Why does a displaced spring block come back toward the middle?
Pull a block 0.10 m to the right on an ideal horizontal spring and release it. The spring pulls left. Pull it twice as far and the force doubles. This proportional pull toward the middle is what makes the motion simple harmonic in the ideal model.
A useful starting point: Choosing a system →
Words and symbols before equations
- Equilibrium position
- For this one-dimensional system, the position where net force is zero.
- Displacement x
- Signed position measured from equilibrium, in m; right is positive here.
- Restoring force
- A force directed toward equilibrium, opposite the displacement.
- Spring constant k
- Spring stiffness, in N/m; a larger k gives more force at the same extension.
- Simple harmonic motion (SHM)
- Motion resulting from a net restoring force proportional to displacement: F_net=−kx, with constant positive k.
What this picture assumes
Horizontal ideal spring and 2 kg block. Friction and spring mass neglected. The vertical normal force and weight balance. Right is positive; the arrow shows net horizontal force. Each setting is a state, not elapsed time.
Connect the picture to the physics
The minus sign in F_net=−kx describes direction. At x>0 the net force is left; at x<0 it is right. At x=0 it is zero. Hooke’s law describes an ideal spring within its elastic range; the whole net force must have this restoring form for this mass–spring system to execute SHM.
Use Newton’s second law: a=F_net/m=−(k/m)x, where m is block mass in kg. Acceleration is largest in magnitude farthest from equilibrium. A restoring force opposes displacement, not necessarily velocity: while the block moves toward the middle, force and velocity point the same way and the block speeds up.
Periodic motion repeats, but not every repeating motion is SHM. Look for the linear restoring relationship, not merely back-and-forth movement. With negligible resistance the block passes through equilibrium because it has velocity there; zero net force at one instant does not require zero velocity.
A worked example, step by step
A 2 kg block on a horizontal ideal spring with k=20 N/m is at x=+0.20 m. Find the net force and acceleration. Compare with x=−0.10 m.
- At +0.20 m, F_net=−(20)(0.20)=−4 N: left.
- a=F/m=−4/2=−2 m/s²: left.
- At −0.10 m, F_net=−(20)(−0.10)=+2 N and a=+1 m/s²: right.
- In both states force and acceleration point toward x=0; halving the displacement magnitude halves both magnitudes.
A restoring force opposes displacement, not always motion. Zero acceleration does not mean zero speed.
A block is right of equilibrium and moving left. Does the restoring force slow it?
Compare with an explanation
No. The force is left, in the same direction as its velocity, so it speeds up as it approaches equilibrium.
Predict. Change one thing. Explain.
Move the block to positive and negative x, then double k at fixed x. Predict force direction and magnitude before each change. The position slider selects a snapshot, not a time trajectory.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
x=0.2 m; F_net=-4 N; a=-2 m/s². Force and acceleration point toward x=0.
Horizontal ideal spring and 2 kg block. Friction and spring mass neglected. The vertical normal force and weight balance. Right is positive; the arrow shows net horizontal force. Each setting is a state, not elapsed time.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant force, motion or energy relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA 0.5 kg block has F_net=−10x, with force in N and x in m. (a) Identify k. (b) Find force at x=+0.15 m. (c) Find acceleration there. (d) Explain whether the block speeds up or slows down if it is moving left at that position.
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Compare with the answer and four-point rubric
- 1 point: k=10 N/m.
- 1 point: F=−1.5 N, left.
- 1 point: a=−1.5/0.5=−3 m/s², left.
- 1 point: It speeds up because velocity and acceleration are both leftward.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What is special about the restoring force in SHM?
It is proportional to displacement and directed toward equilibrium.
RECALL 2Where is net force zero in this horizontal spring model?
At x=0, the equilibrium position.
RECALL 3Does F_net=0 at one instant require v=0?
No; a moving block can pass through equilibrium.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What makes an oscillation simple harmonic?
- Ideal SHM: F_net=−kx; a=−(k/m)x.
- x is measured from equilibrium; k>0 and mass is constant.
Remember: A restoring force opposes displacement, not always motion. Zero acceleration does not mean zero speed.
Conditions: Horizontal ideal spring and 2 kg block. Friction and spring mass neglected. The vertical normal force and weight balance. Right is positive; the arrow shows net horizontal force. Each setting is a state, not elapsed time.
Refresh Kid · Unit 7 · Objectives 7.1.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 7.1, objectives 7.1.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. Fall-2026 corrections also checked. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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