AkCalculators

Centripetal force and circular motion — concepts, formulas and practical use

Circular motion is among the most common dynamics problems in physics — from a swing on a playground to satellites orbiting planets. The centripetal force is the name given to the net force required to keep an object moving along a circular path. It is always directed toward the center of the circle and has a magnitude which depends on the object's mass, its speed along the path and the radius of curvature. This page gathers the relevant formulas, explains unit conversions and offers worked examples for both classroom and practical engineering uses. It also shows how centripetal force compares with gravitational force in orbital mechanics and gives straightforward recipes for solving for any missing variable.

Core formulas

For an object of mass m moving at linear speed v on a path of radius r, the centripetal force F is

F = m v² / r

When the motion is described by angular velocity ω (in radians per second), note that v = ω r, so the same force can be written

F = m ω² r

These two expressions are algebraically equivalent; in practice choose whichever variables are known. Angular-mode input (ω) is convenient for rotating machinery where rotational speed is often expressed in rpm. Use the conversion ω (rad/s) = rpm × 2π / 60.

Solving for unknown variables

Both formulas are readily rearranged to isolate any single variable:

• m = F r / v²
• v = sqrt(F r / m)
• r = m v² / F
• ω = sqrt(F / (m r))

When solving, ensure units are consistent: SI units (kg, m, s, N) avoid unit errors. The calculators on this page accept blank inputs for a single unknown and will attempt to solve for it automatically.

Angular units and conversions

Angular speed appears in three common units: radians per second (rad/s), revolutions per minute (rpm), and hertz (cycles per second, Hz). The relationships are

ω (rad/s) = rpm × 2π / 60 = 2π × f, where f is frequency in Hz

Use the rpm option when working with motors or rotating shafts; the tool converts to rad/s internally before computing forces.

Orbital context — centripetal vs gravitational force

For a small body of mass m orbiting a much larger body of mass M (e.g. a satellite around Earth), the gravitational force provides the centripetal acceleration necessary for circular orbit. Newtonian gravity gives

F_g = G M m / r²

Setting the centripetal force equal to the gravitational force yields orbital speed

v_orbit = sqrt(G M / r)

and angular speed ω = v / r = sqrt(G M / r³). The calculator provides these values and compares F_c and F_g so you can see how tightly bound the orbit is. Note: r must be the distance from the primary's center (radius + altitude if you specify altitude above surface).

Practical worked examples

Example 1 — theme park ride: A 75 kg rider on a circular swing has radius 3.0 m and speed 12.5 m/s. Compute centripetal force: F = 75 × 12.5² / 3.0 = 75 × 156.25 / 3 = 75 × 52.0833 = 3906.25 N ≈ 3.91×10³ N. The rider feels this inward force as the tension in the swing's chain.

Example 2 — rotating disk (angular): A puck of mass 0.5 kg sits 0.2 m from center of a rotating disk with ω = 120 rpm. Convert ω: ω = 120 × 2π / 60 = 4π ≈ 12.566 rad/s. Then F = m ω² r = 0.5 × (12.566)² × 0.2 ≈ 7.896 N.

Example 3 — low Earth orbit satellite: For Earth, M ≈ 5.9722×10²⁴ kg and radius R ≈ 6.371×10⁶ m. For a satellite at 400 km altitude (r = R + 4.00×10⁵ m), orbital speed v ≈ sqrt(G M / r) — the calculator computes this and shows F_g and required centripetal force on the satellite mass.

Measurement, rounding and reporting

Report forces and derived quantities with consistent significant figures. For engineering, 3–4 significant digits are common; for classroom problems 2–3 may suffice. Always state the temperature and reference frames when results are sensitive to precision (e.g., atmospheric drag influence on low orbits, non-circular orbits, etc.).

Limitations and notes

The orbit mode uses Newtonian gravity and assumes circular orbits. Real orbits are often elliptical and require orbital mechanics beyond the circular assumption for exact mission design. Tidal forces, atmospheric drag, and non-uniform gravity fields (oblateness) are ignored in this simple model.

Use the built-in presets for Earth and Moon for fast checks — the tool also allows custom primary mass and radius. Export CSVs for lab notebooks and include the step-by-step derivations in reports for reproducibility.