Uniform Circular Motion Calculator

Uniform circular motion is the motion of an object traveling along a circular path at a constant speed. Although the speed is constant, the velocity is not — it continuously changes direction, meaning the object is always accelerating toward the center of the circle. This inward acceleration is called centripetal acceleration.

In uniform circular motion, the object's speed (magnitude of velocity) and the radius of the circular path remain constant. The direction of velocity, however, changes continuously — always tangent to the circle. The centripetal acceleration and centripetal force also remain constant in magnitude but change in direction.

Centripetal acceleration (aₙ) is calculated using the formula aₙ = v² / r, where v is the linear speed and r is the radius of the circular path. For example, an object moving at 10 m/s on a 5 m radius circle has a centripetal acceleration of 100 / 5 = 20 m/s².

Linear velocity (v) and angular velocity (ω) are related by the equation v = ω × r, where r is the radius. Angular velocity is measured in radians per second (rad/s) and represents how fast the object rotates around the center. Rearranging: ω = v / r.

Centripetal force is the net inward force required to keep an object moving in a circular path. It is calculated as F = m × aₙ = m × v² / r, where m is the object's mass, v is its speed, and r is the radius. This force is always directed toward the center of the circle.

Period (T) is the time it takes for one complete revolution, while frequency (f) is the number of revolutions per second. They are reciprocals: T = 1 / f and f = 1 / T. Period can also be calculated as T = 2πr / v, where r is the radius and v is the speed.

Angular velocity (ω) can be derived from period or frequency using: ω = 2π / T = 2πf. So if the period is 3.14 seconds, ω = 2π / 3.14 ≈ 2 rad/s. Alternatively, if you know the linear speed and radius, ω = v / r.

Common examples include a satellite orbiting Earth at constant altitude, a car navigating a circular roundabout at steady speed, the tip of a clock's second hand, and a ball whirled on a string at constant speed. In all cases, the speed is constant but direction changes continuously.