What Is a Circular Motion Calculator?

A circular motion calculator computes the centripetal acceleration, centripetal force, angular velocity, period, frequency, and G-force for an object moving in a circular path. Pre-filled with a 5 kg object at radius 2 m moving at 10 m/s, the centripetal acceleration is 50 m/s² (5.1g) and the centripetal force is 250 N.

Core Formulas

Centripetal acceleration: ac = v²/r = ω²r. Centripetal force: Fc = mac = mv²/r. Angular velocity: ω = v/r = 2πf. Period: T = 2πr/v = 1/f. Frequency: f = v/(2πr) = ω/(2π). All assume uniform circular motion — constant speed along a circular path.

Understanding Centripetal vs Centrifugal

Centripetal ("center-seeking") force is the real, inward force that causes circular motion. It is provided by gravity (orbits), tension (ball on a string), friction (car turning), or normal force (roller coaster loop). Centrifugal ("center-fleeing") force is a fictitious force that appears only in the rotating (non-inertial) reference frame. In the lab frame, objects don't fly outward — they simply continue in a straight line unless an inward force (centripetal) curves their path.

G-Force in Circular Motion

G-force = centripetal acceleration / 9.81 m/s². Humans can tolerate 3–5g sustained (fighter pilots with g-suits). Brief exposure: 9g (F1 car crashes). Centrifuges for astronaut training: 3–8g. Roller coasters: 2–6g. Washing machine spin cycle: ~300g. Uranium enrichment centrifuges: >100,000g. G-force tolerance depends on direction relative to the body — vertical (head-to-foot) is the hardest to sustain.

Circular Motion in Orbital Mechanics

For a circular orbit, the centripetal force equals gravitational force: mv²/r = GMm/r². This gives orbital velocity: v = √(GM/r). The ISS orbits at ~7.7 km/s at altitude ~400 km (r = 6,771 km from Earth's center). Geostationary orbits require r = 42,164 km with v = 3.07 km/s, where the orbital period matches Earth's rotation (23h 56m).

Banked Curves in Road Design

When a road curves, friction provides centripetal force. Banked curves supplement friction with a component of the normal force: tan(θ) = v²/(rg). For a 50 m/s car on a 200m radius curve: θ = arctan(50²/(200×9.81)) = 51.8°. Most highway curves are designed for speeds of 80–120 km/h with 5–10° banking angles. Ice and rain reduce available friction, requiring lower speeds or steeper banking.