Physics · Classical Mechanics · Kinematics
Circular Motion Calculator
Calculates centripetal acceleration, centripetal force, period, frequency, and linear velocity for an object in uniform circular motion.
Calculator
Formula
a_c is the centripetal acceleration (m/s²); v is the linear (tangential) speed (m/s); r is the radius of the circular path (m); ω is the angular velocity (rad/s); F_c is the centripetal force (N); m is the mass of the object (kg); T is the period — time for one full revolution (s); f is the frequency (Hz).
Source: Halliday, Resnick & Krane, Physics, 5th Edition, Chapter 4; NIST Reference on Constants, Units, and Uncertainty.
How it works
Uniform circular motion describes an object moving at constant speed along a circular path. Although the speed is constant, the velocity is not — it continuously changes direction toward the center of the circle. This inward change in velocity constitutes a centripetal (center-seeking) acceleration, and by Newton's second law, requires a net inward force called the centripetal force. Understanding these quantities is fundamental to analyzing everything from car cornering dynamics to satellite orbits and centrifuge design.
The core relationships are: centripetal acceleration ac = v² / r = ω² r, centripetal force Fc = m v² / r, linear speed v = 2πr / T = ω r, and period T = 1 / f. Here, v is the tangential speed (m/s), r is the radius (m), ω is the angular velocity in radians per second, T is the period in seconds, f is the frequency in hertz, and m is the mass in kilograms. The calculator accepts any one of these motion descriptors — linear speed, angular velocity, period, or frequency — and derives all remaining quantities internally.
Practical applications span a wide range of disciplines. Mechanical engineers use centripetal force calculations to determine bearing loads in rotating machinery, ensure the structural integrity of flywheels, and design centrifuges for industrial separation processes. Physicists apply these formulas when modeling planetary orbits (using gravitational force as the centripetal force provider) and when analyzing particle motion in cyclotrons and synchrotrons. Civil engineers use circular motion principles to design banked road curves, and aerospace engineers rely on them for calculating the forces experienced during aircraft turns or space station rotation for artificial gravity.
Worked example
Problem: A ball of mass 0.5 kg is swung on a string in a horizontal circle of radius 1.2 m at a frequency of 3 Hz. Find its linear speed, angular velocity, period, centripetal acceleration, and the tension in the string.
Step 1 — Period: T = 1 / f = 1 / 3 ≈ 0.3333 s
Step 2 — Angular Velocity: ω = 2π f = 2π × 3 ≈ 18.8496 rad/s
Step 3 — Linear Speed: v = 2π r f = 2π × 1.2 × 3 ≈ 22.619 m/s
Step 4 — Centripetal Acceleration: ac = v² / r = (22.619)² / 1.2 ≈ 426.0 m/s² (or equivalently ω² r = 18.8496² × 1.2 ≈ 426.0 m/s²)
Step 5 — Centripetal Force (Tension): Fc = m × ac = 0.5 × 426.0 ≈ 213.0 N. This is the tension that the string must supply to keep the ball on its circular path.
Limitations & notes
This calculator assumes uniform circular motion — constant speed along a perfectly circular path with no tangential acceleration. If the speed is changing (non-uniform circular motion), a tangential acceleration component must be added separately, and the net force calculation becomes more complex. The formula also assumes the radius is constant and the motion is in a single plane. For vertical circular motion (e.g., a ball on a string swinging in a vertical loop), gravity must be included in the force analysis and the tension varies with position around the loop. At relativistic speeds (v approaching the speed of light), Newtonian mechanics breaks down and relativistic corrections are required. The centripetal force is not a new type of force — it is the net inward resultant of whatever real forces are acting (tension, gravity, normal force, friction), so care must be taken to identify the correct physical force providing the centripetal acceleration in each specific problem.
Frequently asked questions
What is the difference between centripetal and centrifugal force?
Centripetal force is a real net inward force that acts on the object moving in a circle, directed toward the center of curvature — it is provided by a real physical agent such as tension, gravity, or friction. Centrifugal force is a fictitious (pseudo) force that appears only in a rotating (non-inertial) reference frame, apparently pushing the object outward. In the inertial (ground) frame, there is no outward centrifugal force; the object's tendency to travel in a straight line is simply mistaken for one.
What provides the centripetal force in different real-world scenarios?
The centripetal force is always supplied by a real physical force or a combination of forces. For a ball on a string, it is the string's tension. For a satellite orbiting Earth, it is gravity. For a car turning on a flat road, it is static friction between the tires and road. For a banked curve, it is the horizontal component of the normal force. For an electron orbiting a nucleus (classically), it is electrostatic attraction. The formula F_c = mv²/r tells you how large this net inward force must be.
How do I convert between angular velocity (ω) and RPM?
Angular velocity in rad/s and rotational speed in RPM (revolutions per minute) are related by ω = 2π × RPM / 60. For example, 3000 RPM corresponds to ω = 2π × 3000 / 60 ≈ 314.16 rad/s. To go the other way, RPM = ω × 60 / (2π). Simply enter the converted value into the calculator's angular velocity input.
Does centripetal acceleration depend on mass?
No — centripetal acceleration a_c = v²/r depends only on the speed and the radius of curvature, not on the mass of the object. However, the centripetal force F_c = ma_c does depend on mass, meaning a heavier object moving at the same speed on the same circular path requires a proportionally larger inward force to maintain that motion.
Why does doubling the speed quadruple the centripetal force?
Because centripetal force is proportional to the square of the speed: F_c = mv²/r. If you double v, you get (2v)² = 4v² in the numerator, so the force increases by a factor of four. This quadratic dependence is why high-speed cornering is so demanding — a car taking a turn at twice the speed needs four times the friction force, which may exceed the available tire grip and cause skidding.
Last updated: 2025-01-15 · Formula verified against primary sources.