H2 Physics Circular Motion Notes | A-Level 9478

Q: What does A-Level Physics: 7) Circular Motion Guide cover?
A: From angular displacement to centripetal force, this guide unpacks Topic 7 of the 2026 H2 Physics syllabus for IP students and parents.

TL;DR
Circular motion looks deceptively “plug-and-chug”, but it underpins vertical-circle forces, banked tracks, and conical pendulums - and it feeds directly into gravitation, satellites and SHM later. Nail radians, $v = r \omega$, $a = r \omega^2$ and $F = \dfrac{mv^2}{r}$ early to free up mental bandwidth for the harder integrations later in the IP track.

Concrete example: how to use this page

For a car turning a corner, do not invent an outward force. Draw the real force that points inward, then set it equal to $mv^2/r$. The same habit works for strings, tracks, and satellites.

Cross-reference our free H2 Physics notes for adjoining gravitation/SHM chapters and downloadable drills so your revision plan follows the full Topic 7→Topic 9 arc. For the full topic map and paper weightings, see our H2 Physics Syllabus 2026-27 overview.

1 Where this fits in the syllabus

The SEAB 2026 H2 Physics syllabus positions Circular Motion as Topic 7 in Section I - Mechanics. IP schools typically teach it right after Dynamics so that normal-reaction questions in vertical circles feel like natural extensions.

1.1 Why parents should care

Sign slips in the radial direction (and mixing up centripetal vs “centrifugal” language) are common sources of lost method marks early on; fixing them early makes later gravitation/satellite/SHM work much smoother.

2 Angular displacement in radians

Radians measure “arc length per radius”, making every calculus derivative in later topics cleaner.

• Quick check: Half a revolution = $\pi$ rad ($= 180^{\circ}$).
• WA tip: Always label the axis on your graph in radians (not degrees) to avoid losing easy accuracy marks.

3 Angular velocity $\omega$

Define: $$\omega = \dfrac{\Delta \theta}{\Delta t}$$ with $\theta$ in radians and $\omega$ in $\pu{s-1}$

$$v = r \omega.$$

This falls straight out of “distance = rate × time” when the arc length $s = r \theta$ is divided by $t$.

4 Centripetal acceleration $a_c$

Uniform circular motion means the speed is constant but the velocity changes direction, producing an inward acceleration

$$a_c = r \omega^2 = \dfrac{v^2}{r}.$$

Direction: Always toward the geometric centre, perpendicular to $v$.

Intuitive cue: Rotate the velocity vector by 90° then scale it by $\dfrac{v}{r}$.

5 Centripetal force $F_c$

Newton's Second Law packages the previous result into

$$F_c = m a_c = m r \omega^2 = \dfrac{m v^2}{r}.$$

The sharper the bend (small $r$) or the faster the motion (large $v$), the larger the required inward force.

5.1 Radial force setup routine

Use this routine before substituting into $F = \dfrac{mv^2}{r}$. It keeps "centripetal force" as a resultant, not a new force to add to the diagram.

Worked force map:

Trap check: if your free-body diagram contains a separate arrow labelled "centripetal force", redraw it. The inward resultant is made from real forces already on the object.

5.2 Common exam archetypes

6 Mini-drill (3 min)

1. Express 720° in radians.
   Answer: $4 \pi \text{ rad}$.
2. Solve: A $0.40 \pu{kg}$ mass whirls at $5.0 \pu{m.s-1}$ in a horizontal circle on a $0.60 \pu{m}$

7 Bridging to Paper 4 practical

To verify $F_c = \dfrac{m v^2}{r}$ experimentally, keep one variable fixed and plot a straight-line graph:

• fixed $r$: plot $F_c$ against $v^2$ → gradient $= m/r$

Always quote $\pm$ one standard error from the LINEST output.

8 Three WA timing rules

1. Use syllabus pacing as a guide: Paper 2/3 average ~1.6 min/mark; Paper 4 ~3 min/mark.
2. Write the radial direction beside your FBD before summing forces.
3. Keep units visible; missing the “per-second-squared” costs one accuracy mark.

Need structured practice on Circular Motion? Our H2 Physics tuition programme covers this topic with weekly problem sets and Paper 4 practical drills.

Comprehensive revision pack

9478 Section II, Topic 7 Syllabus outcomes

Candidates should be able to:

• (a) express angular displacement in radians.
• (b) show an understanding of and use the concept of angular velocity.
• (c) recall and use $v = r\omega$ to solve problems.
• (d) show an understanding of centripetal acceleration in the case of uniform motion in a circle, and qualitatively describe motion in a curved path (arc) as due to a resultant force that is both perpendicular to the motion and centripetal in direction.
• (e) recall and use centripetal acceleration $a = r\omega^2$, and $a = \dfrac{v^2}{r}$

Concept map (in words)

Convert to radians, compute angular quantities, then translate back to linear variables when forces are required. Always identify the radial direction because the net inward force equals $\dfrac{m v^2}{r}$. Vertical situations demand additional energy/force checks at top and bottom. Banked motion and conical pendula link circular motion to equilibrium concepts from Topic 2.

Key relations

Derivations & reasoning to master

1. Centripetal acceleration: derive $a_c = \dfrac{v^2}{r}$ using vector subtraction or calculus (limit of chord angles).
2. Banked-track formula: resolve normal reaction components and solve for frictionless optimum.
3. Vertical circle tension: write Newton's 2nd law at top and bottom, showing how minimum speed arises from $T \ge 0$

Vertical-loop contact checkpoint

For vertical-circle questions, decide first whether the object is held by a string, pressed by a track, or just maintaining contact with a surface. The radial equation changes because the real inward forces change at each position.

Worked check: if a cart just maintains contact at the top of a loop of radius $\pu{1.8 m}$, then

$$mg = \dfrac{mv^2}{r} \Rightarrow v = \sqrt{gr} = \sqrt{9.81 \times 1.8} \approx \pu{4.2 m.s-1}.$$

Misconception check: the normal reaction becomes zero at the limiting top speed, but weight is still acting and still supplies the required inward resultant.

Worked example 1 - vertical loop

A $\pu{0.50 kg}$ cart enters a vertical loop of radius $1.8 \pu{m}$ at $12 \pu{m.s-1}$ at the bottom. Find (a) its speed at the top (neglect friction), (b) the normal reaction at the top and bottom.

Sketch solution: use energy to determine the speed at the top $v^2 = u^2 - 4 g R$. Substitute into $\dfrac{m v^2}{r}$

Taking $g = \pu{9.81 m.s-2}$,

$$v = \sqrt{12^2 - 4g(1.8)} \approx \pu{8.56 m.s-1}.$$

Top: $N = \dfrac{0.50\times v^2}{1.8} - 0.50g \approx \pu{15.5 N}$.

Worked example 2 - conical pendulum

A $\pu{0.25 kg}$ bob moves in a horizontal circle of radius $0.65 \pu{m}$ with period $1.7 \pu{s}$. Determine the string tension and the angle it makes with the vertical.

Approach: compute angular speed $\omega = \dfrac{2 \pi}{T}$, then $v = r \omega$. Apply the radial equation $T \sin\theta = \dfrac{m v^2}{r}$

$$\omega = \dfrac{2\pi}{1.7} \approx \pu{3.70 s-1},\qquad v = 0.65\,\omega \approx \pu{2.40 m.s-1}.$$

$$\tan\theta = \dfrac{v^2}{rg} \approx 0.904\;\Rightarrow\; \theta \approx \pu{42^{\circ}},\qquad T = \dfrac{mg}{\cos\theta} \approx \pu{3.31 N}.$$

Practical & data tasks

• Perform a whirling bung experiment; plot $v^2$ vs radius and extract $g$ from the gradient.
• Use accelerometer data on a rotating turntable to compare measured radial acceleration with $\dfrac{v^2}{r}$

Common misconceptions & exam traps

• Treating “centripetal force” as a new force; it is just the resultant toward the centre.
• Using $mg = \dfrac{m v^2}{r}$ blindly without checking position in the vertical circle.
• Mixing tangential and radial components when acceleration is not uniform.
• Forgetting to convert revolutions per minute to radians per second.

Quick self-check quiz

1. Express 3 revolutions per second in $\pu{rad.s-1}$. - $\pu{ 6 \pi rad.s-1}$.
2. What provides the centripetal force for a satellite in circular orbit? -

Revision workflow

1. Redo three past-paper problems covering vertical circles, conical pendula and banked tracks.
2. Re-derive $a_c = \dfrac{v^2}{r}$ weekly to keep the concept fresh.
3. Create a table comparing radial forces in different contexts (tension, weight, normal, electrostatic).
4. Practise sketching velocity and acceleration vectors for points around a circle without referencing notes.

Practice Quiz

Test yourself on the key concepts from this guide.

9 Further reading

• SEAB H2 Physics 9478 syllabus (2026)
• Lumen Learning - Centripetal Acceleration
• Wikipedia - Angular Velocity
• IOP Spark - Proof of F = mv^2/r

10 Call-to-action

Parents: Book a 60-min Circular Motion clinic before WA 2 to bullet-proof free-body diagrams. Students: Paste $a_c = \dfrac{v^2}{r}$ on your water-bottle and test it on tomorrow's vertical-circle worksheet.

Last updated 14 Jul 2025. Next review when SEAB issues the 2027 draft syllabus.