Uniform Circular Motion

Period, frequency, angular velocity, and linear speed

Uniform circular motion — key relationships between period $T$, angular velocity $\omega$, linear speed $v$ and radius $r$.

A satellite dish tracks a geostationary satellite: it rotates once every 24 hours, sweeping through $2\pi$ radians while the satellite completes one full orbit 35,900 km above Earth. Both the dish motor and the satellite are performing circular motion — the dish slowly, the satellite at 3.07 km/s. Describing this motion requires quantities beyond the linear speed you already know:

• Period ($T$): The time for one complete revolution. Measured in seconds (s).
• Frequency ($f$): The number of revolutions per second. $f = 1/T$, measured in hertz (Hz).
• Angular velocity ($\omega$): The rate at which the angle changes. $\omega = 2\pi/T = 2\pi f$, measured in rad/s.
• Linear (tangential) speed ($v$): The distance travelled per unit time along the circular path. $v = 2\pi r/T = \omega r$.

The relationship $v = \omega r$ is particularly important. It tells us that for a given angular velocity, points farther from the centre travel faster. This is why the outer edge of a spinning DVD moves faster than the inner part.

For circular motion: $f = 1/T$ (Hz); $\omega = 2\pi/T = 2\pi f$ (rad/s); $v = 2\pi r/T = \omega r$ (m/s). At fixed $\omega$, larger $r$ means faster linear speed — outer edge of a spinning disc moves faster than the centre.

Pause — copy the highlighted circular motion relationships into your book before moving on.

Why constant speed does not mean zero acceleration

We just saw how to describe circular motion using $f$, $\omega$ and $v$. That raises a question: if speed is constant, how can there be acceleration? This card answers it → velocity is a vector; continuous change of direction means $a_c = v^2/r$ directed toward the centre.

In uniform circular motion, the speed is constant but the velocity is not. Velocity is a vector — it has both magnitude (speed) and direction. As the object moves around the circle, its direction continuously changes. A change in velocity means there is acceleration.

Consider the velocity vectors at two nearby points on the circle. The velocity is always tangent to the circle. When we subtract these vectors to find $\Delta \vec{v}$, the result points toward the centre of the circle. Taking the limit as the time interval approaches zero gives the instantaneous acceleration.

This acceleration is called centripetal acceleration ($a_c$), and it is always directed toward the centre of the circle.

The two forms are equivalent since $v = \omega r$:

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

Centripetal acceleration $a_c = v^2/r = \omega^2 r$ (m s⁻²), always directed toward the centre of the circle. Increasing $v$ or decreasing $r$ both increase $a_c$; constant speed does NOT mean zero acceleration.

Add the highlighted centripetal acceleration formula and direction rule to your notes before the check below.

The net force required to maintain circular motion

We just saw that centripetal acceleration is always directed toward the centre. That raises a question: what force causes this inward acceleration? This card answers it → by Newton's 2nd Law, $F_c = mv^2/r$; it is not a new force type — tension, friction or gravity can each provide it.

By Newton's Second Law, any acceleration requires a net force. Since centripetal acceleration is directed toward the centre, there must be a net force directed toward the centre. This is called centripetal force ($F_c$).

Critical understanding: Centripetal force is not a new type of force. It is simply the name we give to the net force directed toward the centre that causes circular motion. This net force can be provided by:

• Tension — a ball on a string being whirled in a horizontal circle
• Friction — a car turning on a flat road
• Gravity — a satellite in orbit, or the Moon around Earth
• Normal force — a car on a banked curve
• Any combination of these forces

If the net inward force is removed, the object travels in a straight line at constant velocity (Newton's First Law) — it does not fly radially outward.

Centripetal force $F_c = mv^2/r = m\omega^2 r$ (N), directed toward the centre. It is the net inward force — provided by tension, friction, gravity or normal force (not a new type). Remove it and the object moves in a straight line (Newton's 1st Law).

Pause — copy the highlighted centripetal force formula and the "not a new force" rule into your book before moving on.

Syllabus 5.21 — Why the centripetal force does no work

We just saw that centripetal force acts inward at every point. That raises a question: if a force is always acting, does it change the object's energy? This card answers it → $F_c$ is always perpendicular to displacement, so $W = Fd\cos90° = 0$ and $\Delta KE = 0$.

In uniform circular motion, speed is constant. Because $KE = \tfrac{1}{2}mv^2$, the kinetic energy never changes:

Why is the work zero? At every instant, the centripetal force points toward the centre, while the displacement is tangent to the circle. These two directions are perpendicular ($\theta = 90°$), so $\cos 90° = 0$ and no work is done.

Key insight: The centripetal force continuously redirects the velocity vector, but it never speeds up or slows down the object. No energy is transferred into or out of the system by the centripetal force. This is why satellites in circular orbits maintain constant speed without any engine thrust.

In uniform circular motion $\Delta KE = 0$ because $F_c \perp$ displacement ($\theta = 90°$, so $W = Fd\cos90° = 0$). Centripetal force changes the direction of velocity but never its magnitude — satellites orbit at constant speed with zero thrust.

Add the highlighted work-energy result and the zero-work reason to your notes before moving on.

Misconceptions — final check

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