Contact at the top of a loop
You will be able to: Write radial force equations at the top and bottom of an inside circular track.
When can a cart just maintain contact at the top?
A small cart rides inside a vertical loop. At the top, both gravity and the track’s push can point toward the center, which is below the cart. The direction “inward” changes as the cart goes around.
A useful starting point: Circular motion →
Words and symbols before equations
- Radial direction
- Along the radius; choose inward positive at each point.
- Contact condition
- The track can push but cannot pull the cart toward itself.
- Minimum top speed
- Limiting speed for inside contact, where the top normal force reaches zero.
What this picture assumes
Snapshot at the top of a radius 2 m inside loop, mass 1 kg, g=10 m/s². This checks local contact only; it does not simulate a full loop or conserve orbital speed.
Connect the picture to the physics
At the top of an inside loop, mg and N both point inward: mg+N=mv_top²/r. Contact requires N≥0. The limiting case N=0 gives v_top=√(gr).
At the bottom, inward is upward: N−mg=mv_bottom²/r. Do not reuse the top signs at the bottom. The bottom normal force must exceed mg if the speed there is nonzero.
This gives a local contact condition at the top. It does not determine the launch height or prove a particular initial speed will carry the cart around the entire loop; that needs the motion or energy analysis. A rigid rod that can pull and push has different constraints from a loose string or one-sided track.
A worked example, step by step
A 1 kg cart is at the top of an inside loop with radius 2 m and speed 6 m/s. Use g=10 m/s². Find normal force and minimum top speed.
- Inward requirement is mv²/r = 1×36/2 = 18 N downward.
- At top, mg+N=18, so N=18−10=8 N downward.
- Limiting contact: v_min=√(10×2)=√20≈4.47 m/s.
- Below this limiting speed, a negative computed N flags lost contact; the inside track cannot supply that pull.
The cart can have zero normal force at the limit while still having downward acceleration from gravity.
At the minimum top speed is net force zero?
Compare with an explanation
No. Gravity alone supplies mv²/r, so net force is mg inward.
Predict. Change one thing. Explain.
Hold r=2 m. Change the top speed across √20≈4.47 m/s. Read whether circular contact is feasible. A negative required normal force is shown as an invalid-contact condition, not a physical pull.
Required top normal force 8 N. Circular contact is feasible at this instant. Minimum top speed 4.47 m/s. Force arrows use 6 drawing units/N.
Snapshot at the top of a radius 2 m inside loop, mass 1 kg, g=10 m/s². This checks local contact only; it does not simulate a full loop or conserve orbital speed.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use a force or motion 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 2 kg cart is at the bottom of a 2 m-radius inside loop at 5 m/s. (a) Name inward direction, (b) write the radial equation, (c) calculate N, and (d) explain why N exceeds weight.
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Compare with the answer and four-point rubric
- 1 point: Upward.
- 1 point: N−mg=mv²/r.
- 1 point: N=20+2×25/2=45 N.
- 1 point: The upward normal force must both offset weight and leave a net inward force.
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 inward at the top?
Downward.
RECALL 2What does negative required N mean here?
The assumed one-sided contact is impossible.
RECALL 3Does minimum top speed specify the starting height?
No; that requires additional analysis.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Contact at the top of a loop
- Top inside loop: mg+N=mv_top²/r.
- Bottom: N−mg=mv_bottom²/r.
- Minimum top contact speed √(gr), for the stated one-sided contact model.
Remember: The cart can have zero normal force at the limit while still having downward acceleration from gravity.
Conditions: Snapshot at the top of a radius 2 m inside loop, mass 1 kg, g=10 m/s². This checks local contact only; it does not simulate a full loop or conserve orbital speed.
Refresh Kid · Unit 2 · Objectives 2.9.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.9, objectives 2.9.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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