Physics › Periodic motion › Circular motion
Circular motion
Going round a corner at a steady speed is accelerating, because velocity is a vector and its direction never stops changing. One angle-based toolkit, the radian and omega, turns every circular problem into three short formulas.
Builds on Newton's laws and the resultant force and Scalars and vectors.
IN THIS TOPIC
- Explain why constant-speed circular motion is accelerated motion requiring a centripetal force.
- Use radian measure and ω = v/r = 2πf.
- Apply a = v²/r = ω²r and F = mv²/r = mω²r, identifying what provides the force.
WHAT YOU PROBABLY THINK
You're flung outwards by centrifugal force.
Steady speed, changing velocity
An object circling at constant speed is accelerating, continuously, because velocity is a vector and its direction changes at every instant. The acceleration points to the centre of the circle, always at right angles to the velocity, which is exactly why it changes direction without changing speed.
By Newton's second law that centre-pointing acceleration requires a centre-pointing resultant force: the centripetal force. And here is the correction to the lie: nothing flings you outward. Take a corner in a car and your body tries to continue in a straight line, Newton's first law, while the car turns underneath you; the sensation of being thrown outward is the car door arriving to push you inward. Remove the centripetal force and the object does not fly radially away; it departs along the tangent.
The angular toolkit
Circles are easier in angle language. The radian is the natural unit: one radian is the angle whose arc length equals the radius, so θ = s/r and a full circle is 2π radians.
The angular speed ω is the angle swept per second, in rad s−1, and it links to everything else through
AQA notes that the direction of angular velocity is not considered: ω here is a magnitude. Keep the calculator in radian mode for this unit; degrees quietly poison every answer.
Acceleration and force
The centripetal acceleration comes in two equivalent forms, and the derivation is explicitly not examined:
Multiply by mass for the force:
The most-examined idea in the topic: centripetal force is never a new force. It is the job description filled by a real force, tension, gravity, friction, a normal reaction, or a combination. “What provides the centripetal force?” expects one of those by name, and “centripetal force” as the answer scores nothing.
THE EXAM BIT
- “Explain why the object accelerates at constant speed”: velocity is a vector; its direction changes; changing velocity is acceleration. Three short sentences, three marks.
- Name the provider: tension, gravity, friction or a normal reaction supplies the centripetal force. Writing “centripetal force” as the providing force is the classic zero.
- Radian mode. The formulas assume it, the data sheet assumes it, and a calculator in degrees fails every ω question by the same silent factor.
- If the centripetal force vanishes, the object leaves along the tangent, not radially outward. Sketch questions test exactly this.
- Choose the convenient form: v²/r when you know the speed, ω²r when you know the period or frequency (via ω = 2πf = 2π/T).
CHECK YOURSELF
A 900 kg car takes a bend of radius 50 m at a steady 15 m s−1. Find the centripetal force required, state what provides it, and explain what happens if the road cannot supply that much.
Show a hint
One formula, then think about which real force is doing the job on a flat road.
Show the answer
F = mv2/r = 900 × 152 / 50 = 4100 N (2 s.f.), directed toward the bend's centre.
On a flat road the provider is friction between the tyres and the surface.
If friction cannot reach 4100 N, on ice say, the car cannot follow that circle: it continues on a straighter path, drifting wide along a tangent, which is a skid described politely.
Circular motion is accelerated motion, toward the centre.
Something real must supply the force.
No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.