SCIENCE · INVESTIGATE AND EXPLAIN

Reject a circular path that needs a pulling surface

At the top of a convex circular support, inward points downward toward the circle center. A 2 kg point has gravity downward and a surface normal upward. The term centripetal force names the inward resultant needed for circular motion; it is not a third interaction to add to gravity and normal.

Learning outcome: Compare speeds 4,6,8 m/s at R=4 m. Draw the two actual interaction forces, then calculate the inward resultant. Keep failed required normal separate from actual force and state what is not predicted after separation.

Starting recap

Draw recipient-specific interactions, resolve signed components and identify the domain of each force/contact model before substituting.

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Try

At R=4 m, the zero-normal speed is√40≈6.32456 m/s. A prescribed6 m/s state remains feasible, whereas8 m/s does not follow the circular-support model.

Open the experiment

Predict

A negative required normal at a convex crest means:

Your prediction and reason

Experiment

Compare speeds 4,6,8 m/s at R=4 m. Draw the two actual interaction forces, then calculate the inward resultant. Keep failed required normal separate from actual force and state what is not predicted after separation.

Test whether a circular crest can supply its required force

Worked teaching model · synthetic values · A 2 kg point is at the top of a convex circular support, gravity 10 m/s². Normal force is upward, desired inward direction downward. Contact cannot pull inward. Failure means circular constraint invalid; no road-speed or ride-safety recommendation.

Crest radius: 4 m Instantaneous horizontal speed: 6 m/s

Reset keeps your prediction, captures and practice history.

Required circular-contact normal

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: conditional equation, negative means failure (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Required circular-contact normal

conditional equation, negative means failure

speed (m/s)	required upward normal (N)
0	20
0.2	19.98
0.4	19.92
0.6	19.82
0.8	19.68
1	19.5
1.2	19.28
1.4	19.02
1.6	18.72
1.8	18.38
2	18
2.2	17.58
2.4	17.12
2.6	16.62
2.8	16.08
3	15.5
3.2	14.88
3.4	14.22
3.6	13.52
3.8	12.78
4	12
4.2	11.18
4.4	10.32
4.6	9.42
4.8	8.48
5	7.5
5.2	6.48
5.4	5.42
5.6	4.32
5.8	3.18
6	2
6.2	0.78
6.4	-0.48
6.6	-1.78
6.8	-3.12
7	-4.5
7.2	-5.92
7.4	-7.38
7.6	-8.88
7.8	-10.42
8	-12

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Required circular inward acceleration	9	m/s²
Required upward normal	2	N
Actual contact normal	2	N
Actual instantaneous downward acceleration	9	m/s²
Circular contact branch feasible	1	flag
Critical speed for zero normal	6.32456	m/s

Gravity minus the support force supplies the inward resultant at the crest.

At the top, mg−N=mv²/R. Centripetal force names that inward resultant, not a third interaction to add to gravity and normal. Equality N=0 is a marginal instantaneous contact boundary; beyond it actual free-flight acceleration is g, not v²/R constrained to the surface. No full post-separation trajectory is computed.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Gravity minus the support force supplies the inward resultant at the crest.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Define the recipient and model

Build the algebraic relationship

If contact maintains the circular path, mg−N=mv²/R. Rearranging gives N=m(g−v²/R). Higher speed requires a larger inward resultant and therefore a smaller upward normal at the crest. This may initially seem opposite to intuition developed for the bottom of a valley, where the radial direction is different.

Check the changed state

Contact is feasible only while the required normal is nonnegative. At v=√(gR), N=0 is the marginal instantaneous boundary. Above it, the surface would need to pull downward; instead the point loses contact and follows a new free-flight trajectory. Its immediate acceleration is then gravity, not the imposed circular acceleration v²/R.

Interpret evidence and limits

With R=4 m and v=6 m/s, inward demand is 9 m/s² and N=2 N. At v=8 m/s, the circular equation demands N=−12 N, which is rejected. The graph intentionally shows that conditional negative requirement to reveal the failure; the ledger separately gives actual normal zero. No road, ride or athletic speed limit is certified by this virtual model.

Worked original packet

At R=4 m, the zero-normal speed is√40≈6.32456 m/s. A prescribed6 m/s state remains feasible, whereas8 m/s does not follow the circular-support model.

Bounded investigation

Changed-case numerical practice

Try the changed case before feedback

For m=3 kg, R=5 m, v=5 m/s and g=10 m/s² at a convex crest, find required upward normal in newtons.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original prose, data and diagrams checked 30 October 2026 against OpenStax College Physics 2e. This is an algebra-based virtual evidence exercise, not a physical experiment, equipment certification, transport recommendation or independent mastery result. Save assumptions and a competing explanation with the course evidence; supervised practical requirements remain external.

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Public practice checks a specific claim. It does not establish independent mastery.

Why does changing from crest to valley change the signs in the radial balance?

Your explanation and limitations

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