Centripetal Force & Circular Motion Calculator

Circular Motion 101: Formulas & Intuition

When an object moves along a circular path, its direction changes constantly—even if its speed stays the same. That continuous change of direction requires an inward (radial) acceleration called centripetal acceleration. For uniform circular motion (constant speed), the magnitude is a = v² / r, where v is the speed and r is the path radius. To produce that acceleration, some agent must supply a centripetal force toward the center: F_c = m v² / r. Dividing a by standard gravity g₀ = 9.80665 m/s² converts acceleration into g-forces—a familiar way to compare ride intensity to everyday weight on Earth.

Two quick, equivalent viewpoints help build intuition. (1) As speed increases, the path tries to “straighten out,” so you need more inward pull to bend it—hence the v² factor. (2) As radius grows, the curve becomes gentler; less inward pull is needed, so the same speed feels milder on a big turn than on a tight corner. Designers of rollercoasters and racetracks exploit this trade-off constantly.

• Units: Use SI for clean results—mass in kilograms (kg), speed in meters per second (m/s), radius in meters (m) → acceleration in m/s², force in newtons (N).
• Direction: Both acceleration and the required force point toward the center (inward). The velocity is always tangent to the circle.
• Rollercoaster tip: Higher speed or smaller radius increases g-force. Comfortable sustained levels are typically a few g; brief peaks can be higher, but real designs follow strict safety margins.

It is also useful to connect linear and angular quantities. The angular speed is ω = v / r (radians per second), the period for one full revolution is T = 2π / ω = 2πr / v, and revolutions per minute follow as rpm = 60 / T. These let you translate a measured lap time or spin rate into forces and g-levels quickly.

Worked example

Consider a 75 kg rider taking a 15 m radius turn at 20 m/s. The inward acceleration is a = 20² / 15 = 26.67 m/s², which is about 2.72 g. The required inward force is F_c = m a = 75 × 26.67 ≈ 2.0 × 10³ N. The angular speed is ω = v/r = 1.333 rad/s, so the period is T = 2π/ω ≈ 4.71 s and the spin rate is about 12.7 rpm.

Common pitfalls & quick checks

• “Outward force” confusion: Riders feel pressed outward due to inertia, but the actual required force is inward. The apparent “centrifugal” sensation is your body resisting the inward turn.
• Top vs. bottom of a loop: In vertical loops, weight mg either adds to or subtracts from the normal force. At the bottom, N − mg = m v²/r; at the top, N + mg = m v²/r. That’s why g-loads vary around the loop.
• Scale sanity check: Doubling speed quadruples a and F. Doubling radius halves them. If your numbers don’t follow these trends, recheck inputs and units.
• Uniform vs. non-uniform: The formulas above give the radial part. If speed changes, there’s also a tangential acceleration from the net tangential force.

Where does the force come from?

The calculator reports the required inward force; in practice this may be supplied by friction (tires on road), track normal force (coasters), string tension (whirling a mass), or gravity itself in orbital motion. In all cases, the numbers let you reason about design limits, comfort, and safety with clear, testable physics.