Period vs Radius
Introduction
Uniform circular motion occurs when an object travels along a circular path at a constant tangential speed. The period T is the time required to complete one full revolution, and the frequency f is the number of revolutions per second. Both quantities depend on just two parameters: the radius of the circle and the tangential speed of the object. Holding speed fixed and changing the radius is the cleanest way to isolate how geometry alone governs timing in circular motion.
This relationship appears across an enormous range of physical systems. Satellite engineers specify orbital radii to achieve target communication windows. Mechanical designers choose gear and pulley radii to set shaft rotation rates. Particle accelerators tune the radius of the beam path to control the revolution frequency of protons and ions. In every case the underlying arithmetic is the same: the circumference of the orbit divided by the speed gives the period.
A common first guess is that a larger orbit must mean a faster period because the object has more distance to cover. The simulator shows the opposite: with speed fixed at v = 5 m/s, increasing the radius from r = 2 m to r = 4 m doubles the Period readout from 2.51 s to 5.03 s — a larger orbit takes longer, not shorter, to complete.
The Physics Explained
At constant tangential speed, the object traces out the full circumference 2π·r in every complete revolution. The time to cover that distance is simply the circumference divided by the speed: T = 2π·r / v. This is not an approximation — it is exact for perfectly circular motion at constant speed. The frequency f = 1 / T = v / (2π·r) is the reciprocal, telling how many complete orbits fit inside one second. The simulator's Period and Frequency readouts update continuously as the radius slider moves, and they track T = 2π·r / v and f = v / (2π·r) to the displayed precision.
The linear dependence of T on r is the central result. Doubling the radius doubles the circumference and therefore doubles the time for one lap at the same speed. With v = 5 m/s and r = 1 m, the Period readout shows T = 1.26 s and the Frequency readout shows f = 0.796 Hz. Increasing r to 2 m gives T = 2.51 s and f = 0.398 Hz — both values shift by exactly a factor of two. This proportionality holds regardless of the speed chosen; only the overall scale of T changes when v is adjusted.
Angular velocity ω connects the period to the geometry through the relation ω = 2π / T = v / r. It measures how many radians of arc the object sweeps per second. A small orbit at the same tangential speed produces a high angular velocity because the radius that converts arc length into angle is short. With v = 5 m/s and r = 1 m the angular velocity is ω = 5 rad/s; at r = 2 m it drops to ω = 2.5 rad/s, halving as the radius doubles. This is why gears of different radii rotating at different angular velocities can still have matching tangential speeds at their contact point — the product r·ω is the same for both.
Centripetal acceleration a = v² / r also changes with radius. At fixed speed, a larger radius means lower centripetal acceleration because the path curves less sharply. With v = 5 m/s and r = 2 m, centripetal acceleration is 12.5 m/s²; at r = 4 m it drops to 6.25 m/s². The period, however, grows linearly with r, not as the inverse square — these two dependencies coexist because period measures timing while centripetal acceleration measures the inward force requirement.
Key Equations
T is the time for one complete revolution, r is the orbital radius, and v is the constant tangential speed. With r = 3 m and v = 6 m/s: T = 2π·3 / 6 = π ≈ 3.14 s. The simulator's Period readout confirms this value when the radius slider is set to 3 m and the speed slider to 6 m/s.
f is the number of complete orbits per second (Hz), the reciprocal of T. With r = 3 m and v = 6 m/s: f = 6 / (2π·3) = 1/π ≈ 0.318 Hz. The simulator's Frequency readout shows 0.318 Hz for these settings, matching the formula exactly.
ω is measured in rad/s. With v = 6 m/s and r = 3 m: ω = 6 / 3 = 2 rad/s. Alternatively, ω = 2π / 3.14 ≈ 2 rad/s — both routes agree. Halving the radius to r = 1.5 m at the same speed doubles ω to 4 rad/s, which the Period readout confirms indirectly: T drops from 3.14 s to 1.57 s and ω = 2π / 1.57 ≈ 4 rad/s.
The period formula is simply C / v. With r = 3 m the circumference is 2π·3 ≈ 18.85 m. Dividing by v = 6 m/s returns T ≈ 3.14 s. This decomposition makes clear why T scales linearly with r at fixed v: every extra metre of radius adds 2π metres of circumference, each of which costs 2π / v seconds of travel time.
Key Variables
| Symbol | Name | Unit | Meaning |
|---|---|---|---|
| r | Orbital radius | m | Distance from the centre of the circle to the object |
| v | Tangential speed | m/s | Constant speed of the object along the circular path |
| T | Period | s | Time for one complete revolution; T = 2π·r / v |
| f | Frequency | Hz | Revolutions per second; f = 1 / T |
| ω | Angular velocity | rad/s | Rate of angle swept per second; ω = v / r |
| C | Circumference | m | Total path length per orbit; C = 2π·r |
Real World Examples
Why do the outer planets of the solar system take longer to complete one orbit?
Each planet orbits the Sun at a speed set by gravitational dynamics, but the key observation for uniform-circular-motion purposes is that the orbital circumference grows linearly with radius while the orbital speed decreases as radius increases. The combined effect means that period grows faster than radius alone would suggest — for Newtonian gravity it scales as r^(3/2) — but even in the simplified case of fixed orbital speed, doubling the radius doubles the circumference and therefore doubles the period.
Saturn orbits at roughly 9.5 times Earth's orbital radius. If both moved at the same tangential speed, Saturn's period would be 9.5 times longer than Earth's — 9.5 years instead of 1. The actual period is about 29.5 years because Saturn also moves more slowly, compounding the circumference effect.
The simulator demonstrates the circumference-period link directly. With speed held constant at v = 5 m/s, increasing the radius from r = 2 m to r = 4 m doubles the Period readout from 2.51 s to 5.03 s, matching the prediction T = 2π·r / v exactly. The outer-planet pattern is this same proportionality, extended across the full span of the solar system.
How do engineers choose the radius of a geostationary satellite orbit?
A geostationary satellite must complete exactly one orbit in 24 hours so that it appears fixed over a point on the equator. The required period is fixed by Earth's rotation, and the orbital radius follows from the period formula — combined with the gravitational constraint that sets orbital speed at each altitude. The target radius works out to approximately 42,164 km from Earth's centre, or about 35,786 km above the surface.
Placing the satellite at a smaller radius increases the orbital speed and shortens the circumference proportionally less than the speed increases, yielding a period shorter than 24 hours — the satellite laps the ground below it rather than hovering. A larger radius produces the opposite error.
The simulator isolates the circumference contribution. Setting v = 3 m/s and r = 3 m gives T = 2π·3 / 3 = 6.28 s and f = 0.159 Hz, as confirmed by the Period and Frequency readouts. Increasing r to 6 m at the same speed yields T = 12.57 s — period doubles because circumference doubles. Engineers applying this at orbital scales must also account for the speed change with altitude, but the linear r-to-T link at fixed speed is the conceptual foundation.
Why does a figure skater spin faster when pulling their arms inward?
A spinning skater conserves angular momentum when no external torque acts. Angular momentum L = I·ω, where I is the moment of inertia and ω is the angular velocity. When the skater pulls their arms inward, the mass distribution moves closer to the rotation axis, reducing I. Because L is conserved, ω must increase — the skater spins faster.
In circular-motion terms, each arm moves from a large orbital radius to a small one. The tangential speed of the arm is not fixed; instead angular velocity is fixed by conservation, and the arm's tangential speed actually decreases even as ω increases. The simulator demonstrates the radius-period relationship at fixed tangential speed, which is the complementary scenario.
With v = 4 m/s and r = 1 m, the simulator reports T = 1.57 s; changing r to 0.5 m at the same speed gives T = 0.79 s, cutting the period in half as the orbit tightens. The shorter period corresponds to higher angular velocity — ω = 2π / 0.79 ≈ 7.95 rad/s versus ω = 2π / 1.57 ≈ 4.00 rad/s — consistent with the higher spin rate a skater achieves by drawing inward.
Further Reading
- Circular motion — the forces required to maintain circular paths, including centripetal acceleration and the role of net inward force.
- Circular motion vectors — how velocity and acceleration vectors rotate continuously in uniform circular motion even when their magnitudes stay constant.
- Orbital motion — extending circular-motion kinematics to gravitationally bound orbits, where the speed and radius are linked by the inverse-square law.
- Simple pendulum — another periodic system whose period depends on a geometric parameter (length), with a similar linear scaling relationship.