Uniform Circular Motion

A particle moving in a circle at constant speed still accelerates because its direction of velocity is always changing. The acceleration points toward the center — centripetal acceleration.

Concept

Setup

A particle moves on a circle of radius $r$ with constant speed $v$.

• Angular velocity: $\omega = v/r$ (radians/second).
• Period: $T = 2\pi r / v = 2\pi/\omega$.
• Frequency: $f = 1/T$.

Centripetal Acceleration

Although speed is constant, the velocity vector rotates. The instantaneous acceleration always points to the center, with magnitude:

$a_c = \frac{v^2}{r} = \omega^2 r = v\omega.$

Derivation (sketch)

Velocity at time $t$ rotates by angle $\Delta\theta = \omega\,\Delta t$ in time $\Delta t$. The change $\Delta\vec{v}$ has magnitude $|\vec{v}|\Delta\theta = v\omega\,\Delta t$, directed toward the center. So: $a = \lim_{\Delta t \to 0}\frac{|\Delta\vec{v}|}{\Delta t} = v\omega = v^2/r.$

Centripetal Force

The net force required for UCM is: $F_c = m a_c = \frac{mv^2}{r} = m\omega^2 r.$ It points toward the center.

Examples of what provides this force:

• Tension (ball on string).
• Friction (car turning).
• Gravitational attraction (planetary orbits).
• Normal force (banked road; loop-the-loop).
• Electrostatic force (electron in Bohr model).

Vector Picture

If the position is $\vec{r}(t) = r(\cos\omega t\,\hat{i} + \sin\omega t\,\hat{j})$:

• Velocity: $\vec{v} = r\omega(-\sin\omega t\,\hat{i} + \cos\omega t\,\hat{j})$, tangential.
• Acceleration: $\vec{a} = -\omega^2 \vec{r}$, radially inward.

Always: $\vec{v}\perp\vec{r}$ and $\vec{a}\perp\vec{v}$.

Worked Example

Q1: A car turns on a circular bend of radius $50$ m at $20$ m/s. Find centripetal acceleration. If the car has mass 1000 kg, what frictional force is needed?

Solution: $a_c = v^2/r = 400/50 = 8$ m/s².

$F_c = ma_c = 1000(8) = 8000$ N, directed toward center of curve.

Q2: A geostationary satellite orbits Earth at radius $r = 4.2 \times 10^7$ m with period $T = 24$ h. Find its centripetal acceleration.

$\omega = 2\pi/T = 2\pi/(86400) \approx 7.27 \times 10^{-5}$ rad/s.

$a_c = \omega^2 r = (7.27 \times 10^{-5})^2 \times 4.2 \times 10^7 \approx 0.22$ m/s².

(This equals $g$ at that altitude — the satellite is in free fall.)

Common Confusions

• "There is also a tangential acceleration." — Only if speed changes (non-uniform circular motion). In UCM, tangential $a$ = 0.
• "Centripetal force is a new fundamental force." — No, it's the role played by some real force (tension, gravity, friction) directed inward.
• "Centrifugal force pulls outward." — Only in a rotating (non-inertial) frame; in inertial frames it doesn't exist.
• "Speed and velocity are constant in UCM." — Speed is constant; velocity (direction) constantly changes.

Key Takeaways

• UCM: constant speed but changing velocity direction.
• $a_c = v^2/r = \omega^2 r$, directed toward center.
• Net force $F_c = mv^2/r$ must be supplied by some real force.
• $\vec{a}\perp\vec{v}$ in UCM; both are perpendicular to $\vec{r}$ from center.
• Tangential acceleration is zero in UCM (non-zero in non-uniform).