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4 Jan 2026

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8 pages

Understanding Circular Motion in Further Mechanics

Jessie M

@studywithjessie

Ever wondered why you feel pressed against the car door... Show more

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$Circular Measure Sept 2020 $\theta = s/r$ in radians. 1 rad = 360/2$\pi$ = 57.3 * The term angular displacement is used to describe$

Circular Measure and Angular Motion

When objects move in circles, we measure their movement differently than straight-line motion. Instead of just looking at distance, we use angular displacement - how much something has rotated around a fixed point.

Angular velocity (ω) tells us how fast something is spinning, measured in radians per second. Think of it like the speedometer for rotation. The formula ω = 2π/T shows us that faster spinning means a shorter time period (T) to complete one full rotation.

Here's where it gets interesting: there's a direct connection between how fast the edge of a spinning object moves (v) and its angular velocity. The relationship v = ωr means that points further from the centre move faster than points closer to the centre - just like the outside of a record player moves faster than the inside.

Quick Check: A bicycle wheel spinning at the same rate will have different speeds at the hub compared to the rim!

$Circular Measure Sept 2020 $\theta = s/r$ in radians. 1 rad = 360/2$\pi$ = 57.3 * The term angular displacement is used to describe$

Centripetal Acceleration

Here's something that might surprise you: even when an object moves at constant speed in a circle, it's still accelerating! This happens because velocity includes both speed and direction, and circular motion constantly changes direction.

Centripetal acceleration always points towards the centre of the circle. You can calculate it using a = v²/r or a = ω²r, depending on what information you have. The key thing to remember is that this acceleration doesn't speed up or slow down the object - it just keeps changing its direction.

This concept explains why satellites don't fly off into space and why your phone slides across the car seat when you go round corners. The acceleration is always there, pulling things towards the centre of the circular path.

Real-World Example: The Earth constantly accelerates towards the Sun due to changing direction in its orbit, even though its orbital speed stays roughly the same!

$Circular Measure Sept 2020 $\theta = s/r$ in radians. 1 rad = 360/2$\pi$ = 57.3 * The term angular displacement is used to describe$

Why Earth Always Accelerates

The Earth's orbital motion perfectly demonstrates circular motion principles. Even though Earth travels at roughly the same speed around the Sun, it's constantly accelerating because its direction keeps changing.

Since velocity is a vector quantity (meaning direction matters), any change in direction means the velocity is changing. When velocity changes, acceleration must be occurring - that's just basic physics using a = Δvelocity/time.

This centripetal acceleration keeps Earth in its orbital path rather than flying off in a straight line. Without this constant acceleration towards the Sun, we'd be heading off into deep space instead of enjoying our yearly trip around our star.

Mind-Bender: Every object in circular motion, from electrons around atoms to planets around stars, is constantly accelerating even at constant speed!

$Circular Measure Sept 2020 $\theta = s/r$ in radians. 1 rad = 360/2$\pi$ = 57.3 * The term angular displacement is used to describe$

Understanding Centripetal Force

Now we get to the force that makes circular motion possible. Using Newton's second law $F = ma$ , we know that centripetal force creates centripetal acceleration and always acts towards the centre of the circle.

Imagine a train going round a bend with a ball hanging from the ceiling and another ball on the floor. The floor ball will slide towards the outer wall because it wants to continue in a straight line (Newton's first law). The hanging ball behaves differently because the string tension pulls it around the corner.

The hanging ball experiences two forces: its weight pulling down and tension pulling at an angle. These forces don't balance out - their vector sum creates an unbalanced force pointing towards the centre of the circular path. This unbalanced force is what we call centripetal force.

Key Insight: The ball on the string shows us that circular motion needs a force pulling towards the centre - objects naturally want to travel in straight lines!

$Circular Measure Sept 2020 $\theta = s/r$ in radians. 1 rad = 360/2$\pi$ = 57.3 * The term angular displacement is used to describe$

What Creates Centripetal Forces?

Here's a crucial point that many students get wrong: circular motion doesn't cause force - force causes circular motion. You need an inward force first, then you get circular motion as a result.

The centripetal force can come from various sources depending on the situation. For a ball on a string, it's tension. For planetary orbits, it's gravitational attraction. For electrons around atoms, it's electrostatic force. Cars going round corners rely on friction between tyres and road.

The formula F = mv²/r tells us that heavier objects or faster speeds need stronger centripetal forces to maintain the same circular path. This explains why cars need to slow down for tight corners and why racing drivers need special tyres for grip.

Memory Tip: Think of centripetal force as the "string" that keeps objects from flying away - without it, everything would travel in straight lines!

$Circular Measure Sept 2020 $\theta = s/r$ in radians. 1 rad = 360/2$\pi$ = 57.3 * The term angular displacement is used to describe$

When Objects Leave Circular Paths

Sometimes the required centripetal force becomes too large, and objects can't maintain their circular path. This happens with cars going over hills too fast - they can actually become airborne!

At the top of a hill, a car experiences its weight (mg) pulling down and the normal force (R) from the road also pushing down. Both forces contribute to centripetal force: F = mg + R = mv²/r.

When speed increases enough, the required centripetal force equals the car's weight alone. At this critical speed, R = 0, meaning no contact force between car and road. The maximum speed before losing contact is v_max = √(gr), which depends only on the hill's curvature and gravity.

Thrill Factor: This principle explains why you feel weightless going over hills quickly, and why roller coaster designers carefully calculate speeds for different sections!

$Circular Measure Sept 2020 $\theta = s/r$ in radians. 1 rad = 360/2$\pi$ = 57.3 * The term angular displacement is used to describe$

Forces at the Bottom of Circular Paths

The situation completely reverses when you're at the bottom of a circular path, like in a valley or at the bottom of a roller coaster loop. Now gravity and the normal force work in opposite directions.

Your weight still pulls you down (mg), but the normal force (R) pushes upward from your seat. The net upward force provides the centripetal acceleration: R - mg = mv²/r, which gives us R = mg + mv²/r.

This explains why you feel heavier when going through dips or at the bottom of roller coaster loops. You're experiencing not just your normal weight, but an additional force of mv²/r pushing you into your seat. The faster you're going, the heavier you feel!

Experience Check: Next time you go over a hill or through a dip while driving, notice how your stomach responds to these changing forces!

$Circular Measure Sept 2020 $\theta = s/r$ in radians. 1 rad = 360/2$\pi$ = 57.3 * The term angular displacement is used to describe$

Essential Circular Motion Equations

Master these key equations and you'll be able to solve most circular motion problems. v = rω connects linear and angular velocity - essential for understanding how different parts of rotating objects move at different speeds.

For acceleration calculations, use a = v²/r when you know speed and radius, or a = ω²r when working with angular velocity. Both give the same result, so choose whichever fits your given information better.

The force equations F = mv²/r and F = mω²r are your go-to formulas for centripetal force problems. Remember that this force always points towards the centre and can come from tension, gravity, friction, or other sources depending on the situation.

Exam Success: These equations work together - if you know any three variables, you can always find the fourth. Practice switching between them fluently!

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537

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4 Jan 2026

•

8 pages

Understanding Circular Motion in Further Mechanics

Jessie M

@studywithjessie

Here's a crucial point that many students get wrong: circular motion doesn't cause force - force causes circular motion. You need an inward force first, then you get circular motion as a result.

Memory Tip: Think of centripetal force as the "string" that keeps objects from flying away - without it, everything would travel in straight lines!

$Circular Measure Sept 2020 $\theta = s/r$ in radians. 1 rad = 360/2$\pi$ = 57.3 * The term angular displacement is used to describe$

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When Objects Leave Circular Paths

Sometimes the required centripetal force becomes too large, and objects can't maintain their circular path. This happens with cars going over hills too fast - they can actually become airborne!

At the top of a hill, a car experiences its weight (mg) pulling down and the normal force (R) from the road also pushing down. Both forces contribute to centripetal force: F = mg + R = mv²/r.

Thrill Factor: This principle explains why you feel weightless going over hills quickly, and why roller coaster designers carefully calculate speeds for different sections!

$Circular Measure Sept 2020 $\theta = s/r$ in radians. 1 rad = 360/2$\pi$ = 57.3 * The term angular displacement is used to describe$

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Forces at the Bottom of Circular Paths

Experience Check: Next time you go over a hill or through a dip while driving, notice how your stomach responds to these changing forces!

$Circular Measure Sept 2020 $\theta = s/r$ in radians. 1 rad = 360/2$\pi$ = 57.3 * The term angular displacement is used to describe$

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Essential Circular Motion Equations

Exam Success: These equations work together - if you know any three variables, you can always find the fourth. Practice switching between them fluently!