3.6.1.1ThermalCore

Circular Motion

An object moving in a circle at constant speed is still $accelerating$ — its velocity is constantly changing direction, even if its speed never changes. That acceleration always points toward the centre of the circle, and understanding exactly why is the key to everything from a car cornering safely to a satellite staying in orbit.

40 min Video by DrPhysicsA 3.6.1 Periodic motion

Circular Motion — A Level PhysicsWatch 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 $ω$
Relate angular speed to linear speed, $v = ω r$
Explain why an object moving in a circle at constant speed is still accelerating
Calculate centripetal acceleration and centripetal force
Identify the real force (or combination of forces) providing the centripetal force in a given situation

Angles in radians and angular speed

Angles for circular motion are measured in $radians$ rather than degrees: one radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. A full circle is $2 π$ radians, equal to $360°$ .

$Angular speed$ , $ω$ , is the rate of change of angle, in radians per second. For an object completing one full revolution in a time period $T$ (or completing $f$ revolutions per second), the angular speed follows directly.

ω = \frac{θ}{t} = \frac{2 π}{T} = 2 π f

Angular speed from angle, period or frequency.

1Use

ω = \frac{2 π}{T}

ω = \frac{2 π}{0.40}

Answer

ω \approx 15.7

rad s⁻¹

Linear speed and centripetal acceleration

A point moving in a circle of radius $r$ at angular speed $ω$ has linear (tangential) speed $v = ω r$ — a larger radius at the same angular speed means a faster linear speed, which is why the outer edge of a spinning disc moves faster than a point near its centre.

Even at constant speed, an object moving in a circle is accelerating, because its $velocity$ (a vector) constantly changes direction. This acceleration, called $centripetal acceleration$ , always points directly toward the centre of the circle — never along the direction of motion — which is exactly why it changes direction without changing speed.

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

Linear speed and centripetal acceleration (two equivalent forms).

1Use

a = \frac{v ^{2}}{r}

a = \frac{1 5 ^{2}}{40} = \frac{225}{40}

Answer

a \approx 5.6

m s⁻²

Tip — Centripetal acceleration always points toward the centre — never draw it tangential to the circle (that would be a different quantity, the tangential acceleration, which is zero for constant-speed circular motion).

Centripetal force

By Newton’s second law, a centripetal acceleration requires a net force pointing toward the centre — the $centripetal force$ . Crucially, this is not some new, separate type of force: it is simply the name given to whatever resultant force is actually causing the circular motion in a given situation — tension in a string for a ball on a string, friction between tyres and road for a car cornering, gravity for a satellite in orbit, or the normal contact force for a rider on a curved track.

F = \frac{m v ^{2}}{r} = m ω^{2} r

Centripetal force.

1The tension provides the entire centripetal force here, so

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

T = \frac{0.20 \times 4. 0 ^{2}}{0.60} = \frac{3.2}{0.60}

Answer

T \approx 5.3

Tip — Never add "centripetal force" as an extra arrow on a free-body diagram alongside gravity, tension, friction etc. — it IS one of those forces (or their resultant), not an additional one.

Equation recap

ω = \frac{2 π}{T} = 2 π f

Angular speed.

v = ω r

Linear speed from angular speed and radius.

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 an object moving at constant speed in a circle has zero acceleration.

Acceleration depends on the rate of change of VELOCITY, a vector — direction is constantly changing in circular motion, so acceleration is nonzero even at constant speed.

Adding "centripetal force" as an extra force alongside tension, gravity, friction etc.

Centripetal force is just the name for the net (resultant) force toward the centre — it is provided BY one or more of the real forces already acting, not an additional one.

Drawing the centripetal acceleration or force pointing along the direction of travel.

Centripetal acceleration and force always point toward the centre of the circle, perpendicular to the 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 its velocity direction constantly changes.
Centripetal acceleration a = v²/r = ω²r, always directed toward the centre.
Centripetal force F = mv²/r = mω²r is provided by whichever real force (tension, friction, gravity, etc.) is acting — it is not a separate additional force.

Test yourself

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Circular Motion

An object moving in a circle at constant speed is still

accelerating

— its velocity is constantly changing direction, even if its speed never changes. That acceleration always points toward the centre of the circle, and understanding exactly why is the key to everything from a car cornering safely to a satellite staying in orbit.

40 min Video by DrPhysicsA 3.6.1 Periodic motion