Circular-Motion Force Models

Circular-motion dynamics is Newton’s second law applied along a changing radial direction. The reliable method is to model the interactions first and introduce mv²/r only as the radial ma term.

1. The Five-Step Method

Step 1: Isolate the object

Choose the object whose motion you are analysing. Draw only forces exerted on that object by other bodies: weight, tension, normal reaction, friction, or another stated interaction.

Step 2: Mark the centre and radial direction

At the instant shown, draw an axis pointing towards the centre and define it as positive. The inward direction can be horizontal, upward, downward or inclined depending on the object’s position.

Step 3: Resolve the real forces

Take the component of each real force along the radial axis. Forces perpendicular to that axis do not enter the radial equation. If the speed changes, use a separate tangential equation for the tangential component.

Step 4: Write Newton’s second law symbolically

With inward positive,

sum Fᵢₙwₐᵣd = maᵣ = mv²/r = mrω².

The left side contains named forces or components. The right side contains the required radial acceleration.

Step 5: Solve and check the model

Substitute only after the symbolic equation is correct. Then check:

• units are consistent;
• the radius belongs to the path of the chosen object;
• the answer respects physical constraints such as tension ≥ 0;
• increasing speed at fixed radius demands a larger inward resultant proportional to v².

Scroll diagram horizontally to read all labels.

2. Choosing the Radial Equation

These are not separate formulae to memorise. Each row follows from the same free-body-diagram method.

3. Worked Examples

Example 1: Level bend with a given friction limitCore

A 900 kg car travels around a level bend of radius 45 m. The maximum available horizontal frictional force is 6.5 kN. Find the greatest speed for circular motion without skidding.

Show Answer

Friction is the only horizontal inward force. At the limiting speed,

6500 = 900v²/45.

Hence

v = sqrt(6500)(45)/900 = 18.0 m s⁻¹.

Weight and normal reaction balance vertically and are absent from the horizontal radial equation.

Example 2: Rebuild the radial equation at each positionCore

A 0.30 kg ball on a light string moves in a vertical circle of radius 0.80 m. Its speed is 5.0 m s⁻¹ at the top and 7.0 m s⁻¹ at the bottom. Find the tension at each point. Take g = 9.81 m s⁻².

Show Answer

At the top, inward is downward:

Tₜₒₚ + mg = mvₜₒₚ²/r,

Tₜₒₚ = 0.30(5.0)²/0.80-0.30(9.81) = 6.43 N.

At the bottom, inward is upward:

Tbₒₜₜₒₘ-mg = mvbₒₜₜₒₘ²/r,

Tbₒₜₜₒₘ = 0.30(7.0)²/0.80 + 0.30(9.81) = 21.3 N.

The signs differ because “towards the centre” reverses between the two positions.

Example 3: Banked bend from force componentsCore

On a frictionless banked bend, the normal reaction is at angle θ to the vertical. Derive the design-speed relation.

Show Answer

There is no vertical acceleration, while the horizontal component is inward:

Ncosθ = mg,

Nsinθ = mv²/r.

Dividing the radial equation by the vertical equation eliminates N and m:

tanθ = v²/rg.

This is a consequence of the force model, not a starting formula.

4. Common Mistakes

• Drawing mv²/r as a force arrow instead of putting it on the ma side.
• Assuming tension, friction or weight always equals mv²/r without checking other radial forces.
• Using the same signed equation at the top and bottom of a vertical circle.
• Resolving forces relative to the page rather than relative to the radial axis.
• Using the diameter instead of the radius of the object’s path.

5. Transfer Check

Mind stretcher 1: Diagnose an impossible modelExtension

At the top of a vertical circle, a student’s calculation gives a string tension of -2.0 N. What does the result mean physically?

Show Answer

A string cannot push, so negative tension is impossible. The assumed taut circular path cannot be maintained at that speed. The string goes slack and the subsequent motion is no longer the same circular-motion model.

After this method lesson, use the Circular Motion quiz for retrieval and the structured set for multi-step transfer.

8. Practice, Quiz and Next Step

Close your notes and use Circular-Motion Force Models in the supplied context below. This requires a constructed explanation or working, not recognition of an option.

Fresh context: An unfamiliar data set or physical system requires you to apply Circular-Motion Force Models while stating the model, regime and assumptions.

1. Retrieve: define circular-motion force models in your own words, including units, sign or conditions where relevant.
2. Represent: Choose and label an appropriate diagram, graph, table or symbolic model; derive or justify the relationship used.
3. Apply: Reach a conclusion, then evaluate it using units, uncertainty, a limiting case and one practical or modelling limitation.

Check the response before looking back

• The model, regime, coordinates and assumptions are explicit.
• The derivation or multi-step reasoning is visible rather than implied.
• The conclusion is tested against units, data quality and a limiting case.
• A practical control, uncertainty or model limitation is evaluated where applicable.

If one check fails, name that exact gap, revisit the matching explanation or worked example, and redo the task with different values or a different situation. Then use theA-Level Physics course hub orpractice browser for an independent re-test.