Circular Motion
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Circular Motion
- Radians measure angles using arc length ÷ radius — they're the "natural" unit for anything going in circles.
- Angular displacement (Δθ) is how far round the circle something has swept, in radians.
- Angular velocity (ω) is how fast that angle changes — the rotational equivalent of linear velocity.
- Even at constant , an object going in a circle is always accelerating, because velocity is a vector and direction is constantly changing.
- This acceleration is called centripetal acceleration — always pointing toward the centre of the circle.
- Newton's Second Law says acceleration needs a force — that resultant force is the centripetal force, also always pointing toward the centre.
- Centripetal force isn't a new type of force — it's whatever existing force (tension, friction, gravity...) happens to be doing the job of keeping the object curving.
- In vertical circular motion, tension/normal force varies around the loop because gravity adds to it at the top and subtracts from it at the bottom (or vice versa depending on setup).
What actually is a radian?
You're used to measuring angles in degrees, where a full circle is 360°. That number is completely arbitrary — it comes from ancient Babylonian counting systems, not from anything physical about circles. Radians, on the other hand, are defined , which is exactly why physicists love them for circular motion.
Here's the definition: one radian is the angle you get at the centre of a circle when the arc (the curved distance around the edge) is exactly as long as the radius. Imagine taking a piece of string the same length as the radius, laying it along the curved edge of the circle, and then drawing two lines from its ends back to the centre. The angle between those two lines is 1 radian — roughly 57.3°.
Because the full circumference of a circle is 2πr, and each "radius-length" of
arc corresponds to 1 radian, a complete circle (360°) is exactly 2π radians (≈ 6.28 rad). That's
not a coincidence you need to memorise separately — it falls straight out of the definition.
- Δθ = angular displacement (radians)
- Δs = arc length / distance travelled around the circle (m)
- r = radius of the circle (m)
θ° × (π/180) = θ rad.
To go the other way: θ rad × (180/π) = θ°. Common ones worth knowing cold:
90° = π/2, 180° = π, 270° = 3π/2, 360° = 2π.
An angle is given as θ = π/3 radians. What is this in degrees?
A toy car drives around a circular track of radius 2.5 m. It travels an arc length of 4.0 m. What angle (in radians) has it swept through?
How fast is it spinning?
Angular velocity, ω (the Greek letter omega), is the rotational cousin of linear velocity. Where linear velocity tells you metres covered per second, angular velocity tells you . It's defined as the rate of change of angular displacement.
- ω = angular velocity (rad s⁻¹)
- Δθ = change in angular displacement (rad)
- Δt = time interval (s)
Angular velocity connects to the linear speed you're already comfortable with through a really useful equation. Think about it intuitively: if two people are on a spinning merry-go-round, one near the centre and one on the outer edge, they both sweep the in the same time (same ω) — but the person on the edge is physically moving much faster because they're covering more distance. That's why linear speed depends on how far out you are (the radius).
- v = linear speed (m s⁻¹)
- ω = angular speed (rad s⁻¹)
- r = radius of the circular path (m)
There's also a neat relationship between ω and how long a full revolution takes. If one entire lap is 2π radians, and it takes time T (the period), then:
- T = time period — time for one full revolution (s)
- f = frequency — revolutions per second (Hz)
A bird flies in a horizontal circle of radius 650 m with angular speed 5.25 rad s⁻¹. (a) Find its linear speed. (b) Find its frequency of rotation.
Why "constant speed" doesn't mean "no acceleration"
This is the idea that trips almost everyone up the first time they meet it. In everyday language, "accelerating" means "speeding up." But in physics, acceleration is the rate of change of velocity — and velocity is a vector. A vector has both size (speed) and direction. If the direction changes, even while the speed stays exactly the same, the velocity has still changed — which means there's still an acceleration.
Picture a ball on a string being swung in a horizontal circle at a steady speed. At every instant, its velocity arrow points in a new direction — tangent to the circle. Since that arrow is constantly rotating, there's a constant acceleration happening, even though the length of the arrow (the speed) never changes. This acceleration is called centripetal acceleration, and — as the derivation in your notes shows using vector diagrams and the small-angle approximation (sin θ ≈ θ for tiny θ) — it always points of the circle.
"Centripetal" literally means "centre-seeking." It's a slightly misleading name in one sense: it doesn't make the object move toward the centre and crash into it — it just continuously bends the object's path so that it curves around the centre rather than flying off in a straight line.
- a = centripetal acceleration (m s⁻²)
- v = linear speed (m s⁻¹)
- r = radius of the circular path (m)
- ω = angular speed (rad s⁻¹)
A ball on a string moves in a horizontal circle, radius 1.5 m, angular speed 3.5 rad s⁻¹. If both the radius and the angular speed are doubled, by what factor does the centripetal acceleration change? Then calculate the new value.
What actually keeps something moving in a circle?
Newton's First Law says an object keeps moving in a straight line unless a resultant force acts on it. An object going around in a circle is clearly moving in a straight line — its direction is constantly being bent inward. That bending requires a resultant force, and by Newton's Second Law (F = ma), that force must point in the exact same direction as the acceleration it causes — toward the centre. This resultant force is the centripetal force.
| Situation | What provides the centripetal force? |
|---|---|
| Car going round a roundabout | Friction between tyres and road |
| Ball on a string swung in a circle | Tension in the string |
| Earth orbiting the Sun | Gravitational force |
| Charged particle in a magnetic field | Magnetic force (always perpendicular to velocity) |
Calculating the force, and what happens when gravity gets involved
Since F = ma, and we already have three equivalent expressions for centripetal acceleration, we get three equivalent expressions for centripetal force just by multiplying by mass:
- F = centripetal force (N)
- m = mass of the object (kg)
- v = linear speed (m s⁻¹)
- ω = angular speed (rad s⁻¹)
- r = radius of the circular path (m)
Things get more interesting when the circle is vertical instead of horizontal — think of a ball on a string being swung in a vertical loop, or a rollercoaster doing a loop-the-loop. Now there are two forces acting on the object at all times: the tension (or normal force) pointing toward the centre, and the object's own weight (mg), which always points straight down no matter where the object is in the loop.
Because weight never changes direction but the required centripetal force always points toward the centre, the relationship between tension and weight — and therefore the value of the tension itself — changes continuously as the object goes around.
A bucket of mass 8.0 kg filled with water is attached to a string of length 0.5 m and swung in a vertical circle. What is the minimum speed the bucket must have at the top of the circle so that no water spills out?
A 0.20 kg ball on a 0.80 m string is swung in a vertical circle, passing through the bottom point at 4.0 m s⁻¹. Calculate the tension in the string at that instant. (g = 9.81 m s⁻²)
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