5.2ThermalCore

Circular Motion

An object moving in a circle at constant speed never stops accelerating — its velocity direction is changing every instant, even while its speed stays fixed. That single, slightly counter-intuitive idea underlies everything from a car cornering safely to a satellite staying in orbit.

40 min Video by ZPhysics Module 5.2 Circular motion

A Level Physics Revision: All of Circular MotionWatch the full walkthrough before the notes below.

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What you'll be able to do

Convert between degrees and radians, and use angular speed
Relate linear speed to angular speed, $v = ω r$
Calculate centripetal acceleration and centripetal force
Identify the real force providing the centripetal force in a given situation

Angular speed

Angles for circular motion are measured in radians; a full circle is $2 π$ radians. Angular speed, $ω$ , is the rate of change of angle.

ω = \frac{2 π}{T} = 2 π f

Angular speed from period or frequency.

1Use

ω = \frac{2 π}{T}

ω = \frac{2 π}{0.50}

Answer

ω \approx 12.6

rad s⁻¹

Centripetal acceleration and force

A point moving in a circle of radius $r$ at angular speed $ω$ has linear speed $v = ω r$ . Even at constant speed, the object accelerates, since its velocity direction is constantly changing — this centripetal acceleration always points toward the centre. By Newton’s second law, a resultant centripetal force toward the centre is required, provided by whatever real force applies (tension, friction, gravity, normal force).

a = \frac{v ^{2}}{r} = ω^{2} r F = \frac{m v ^{2}}{r} = m ω^{2} r

Centripetal acceleration and force.

1Use

F = \frac{m v ^{2}}{r}

F = \frac{1000 \times 1 8 ^{2}}{45} = \frac{1000 \times 324}{45}

Answer

F = 7200

Tip — Centripetal force is not a separate force — it is simply the name for the resultant force (tension, friction, gravity, etc.) that happens to be directed toward the centre.

Equation recap

ω = \frac{2 π}{T} = 2 π f

Angular speed.

v = ω r

Linear speed.

a = \frac{v ^{2}}{r} = ω^{2} r

Centripetal acceleration.

F = \frac{m v ^{2}}{r} = m ω^{2} r

Centripetal force.

Common mistakes to avoid

Thinking constant speed in a circle means zero acceleration.

Velocity direction constantly changes, so acceleration (centripetal) is nonzero even at constant speed.

Adding centripetal force as an extra force on a free-body diagram.

Centripetal force is the resultant of real forces already present (tension, friction, gravity) — not an additional force.

Drawing centripetal acceleration along the direction of motion.

Centripetal acceleration always points toward the CENTRE, perpendicular to instantaneous velocity.

Key takeaways

Angular speed ω = 2π/T = 2πf; linear speed v = ωr.
An object in circular motion at constant speed still accelerates, since velocity direction changes.
Centripetal acceleration a = v²/r = ω²r, always toward the centre.
Centripetal force F = mv²/r = mω²r is provided by a real force (tension, friction, gravity, etc.).

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Circular Motion

40 min Video by ZPhysics Module 5.2 Circular motion