Lesson 3: Circular Motion | PHY 2.4

Lesson at a Glance | He Tirohanga Whakamua

Do NowWhy a car passenger slides outward when taking a sharp turn10 min

Tangential Speed vs VelocityContinuous directional change = continuous acceleration15 min

Centripetal FormulasSolving

v = \frac{2 π r}{T}

a_{c} = \frac{v^{2}}{r}

, and

F_{c} = \frac{m v^{2}}{r}

15 min

Portfolio EntryWrite Section 3: Centripetal Force & Circular Dynamics10 min

Exit VerificationDraw velocity and force vectors for a ball whirled on a string5 min

Ngā Whāinga Ako | Learning Intentions

Students will know

That an object in uniform circular motion travels at constant speed, but its velocity direction is constantly changing, creating a centripetal acceleration ( $a_{c}$ ) directed towards the centre.
The core formulas for circular motion:
• Tangential Speed: $v = \frac{2 π r}{T}$
• Centripetal Acceleration: $a_{c} = \frac{v^{2}}{r}$
• Centripetal Force: $F_{c} = m a_{c} = \frac{m v^{2}}{r}$
That Centripetal Force ( $F_{c}$ ) is not a new magic force, but the net result of real physical forces (e.g. friction on tires, tension in a rope, gravity).

Students will demonstrate

By calculating centripetal force and required tire friction for a car turning a corner of radius 25m.
By completing Section 3 of their Level 2 Physics Mechanics Mastery Portfolio.

Do Now | Tīmatanga Whakaaro (10 min)

Centripetal Inertia Prompt:

"When a car turns sharply to the left, passengers feel pushed to the right against the car door. Is an outward force pushing them, or are they experiencing Newton's 1st Law?"

Unpack: There is NO outward force (so-called 'centrifugal force' is an illusion!). Due to inertia (Newton's 1st Law), the passenger's body wants to continue travelling straight ahead. The car door turns left into them, exerting a net inward Centripetal Force ( $F_{c}$ ) to pull them around the turn.

Vector Relationships in Circular Motion (15 min)

1. Velocity Vector ( $v$ )

Directed tangent to the circular path at any given instant. If the string breaks, the object flies off in this straight line.

2. Acceleration & Force Vectors ( $a_{c}, F_{c}$ )

Directed perpendicular to velocity, pointing straight towards the centre of the circle at all times.

Centripetal Equations (15 min)

1. Speed from the period

An object completing one circle of radius $r$ in period $T$ travels $2 π r$ , so $v = \frac{2 π r}{T}$ . Use this whenever a question gives you revolutions per minute or a time per lap rather than a speed.

2. Acceleration and force toward the centre

$a_{c} = \frac{v^{2}}{r}$ and $F_{c} = \frac{m v^{2}}{r}$ . Both point at the centre, not along the motion. Note the square: doubling the speed quadruples the force needed, which is why a corner taken at twice the speed is far more than twice as demanding on the tyres.

Do this now: a 0.4 kg ball on a 0.8 m string completes one circle in 0.5 s. Find $v$ , then $F_{c}$ . Name the physical thing actually providing that force.

Whakawhiti | Take it further — the physics of poi

Poi are taonga of kapa haka, and were used to build the wrist strength and coordination needed to handle rākau. A poi is also, exactly, the situation you have just solved: a mass on a cord in uniform circular motion, with the cord supplying $F_{c}$ .

Open Physics of Traditional Māori Games and use its poi scenario table — those figures are illustrative values chosen for calculation practice, not measurements of real practice, and the sheet says so. Predict first, then calculate: fire poi carry two and a half times the mass of traditional raupō poi but spin more slowly. Which needs the greater centripetal force — and by what factor? Work out $T$ , $v$ and $F_{c}$ for both before you commit to an answer.

Then say where that force is felt. By Newton’s third law the cord pulls the performer’s wrist inward just as hard as it pulls the poi — which is why the conditioning is real, and why a kaihaka can tell you the answer before the maths does.

📁 Physics Mechanics Portfolio — Section 3: Centripetal Force & Circular Dynamics

Students open their Level 2 Physics Portfolio and complete Section 3:

Section 3 Requirements:

1. Circular Motion Vector Diagram: Draw a circle showing tangential velocity $v$ and inward $F_{c} / a_{c}$ vectors at 4 cardinal positions.

2. Centripetal Calculation: Calculate $v$ , $a_{c}$ , and $F_{c}$ for a 1200kg car navigating a 30m radius bend in 8 seconds.

3. Excellence Speed Doubling Analysis: 1-paragraph explanation of why doubling a car's cornering speed quadruples the required centripetal friction force ( $F_{c} \propto v^{2}$ ), creating extreme skidding hazards on wet roads.

Exit Verification | Ka Mutu Hoki (5 min)

Exit Check:

"My Section 3 diagram shows tangential velocity perpendicular to inward centripetal acceleration, and proves Fc quadruples when speed doubles."

Teacher Planning & NCEA Alignment

NCEA Level 2 Physics Alignment (6 Credits External):

Circular Motion: Demonstrate understanding of period, tangential speed, centripetal acceleration, and centripetal force.
Force Relationships: Relate centripetal force to friction, tension, gravity, and normal forces.

Vocabulary: Uniform circular motion, period ( $T$ ), radius ( $r$ ), tangential velocity ( $v$ ), centripetal acceleration ( $a_{c}$ ), centripetal force ( $F_{c}$ ), inertia.

Same concept, shorter route: the Guided Viewing & Problem Practice version of this unit covers it in Circular Motion and Centripetal Force. That route is a compact viewing-and-practice sequence; this one builds the concept over ten sections with a formative portfolio.

Lesson 3: Circular Motion & Centripetal Acceleration