Centripetal Force Calculator

The Centripetal Force Calculator finds the inward force that keeps any object moving on a circular path. Pick the unknown — force, mass, radius, or speed — type the other three quantities in any common unit, and read the result together with the centripetal acceleration, the equivalent g-force, the angular velocity, the period, and (for vehicle turns) the minimum tire-friction coefficient required to stay on the road. A live SVG animation actually rotates the mass at the computed angular velocity so you can see, not just read, what the numbers mean.

How to Use This Centripetal Force Calculator

1. Select the unknown in the Solve for dropdown — F, m, v, or r. The matching field hides itself, and the others become required.
2. Enter the mass in any familiar unit (kg, g, lb, t) and the radius in m, cm, km, ft, or in.
3. Choose Linear if you know the tangential speed, or switch to Angular if you have RPM, rad/s, period, or frequency. The two modes describe the same motion — the calculator converts between them automatically.
4. Pick a scenario (car turn, orbit, spinning machinery, amusement ride, or generic). The scenario tunes the contextual notes — for example, a car-turn scenario adds a tire-friction sufficiency check.
5. Press Calculate and read the result, the g-force meter, the rotating animation, the step-by-step derivation, and any contextual warnings.

What Makes This Calculator Different

The Centripetal Force Formula

For any object of mass \(m\) moving along a circular path of radius \(r\) at constant tangential speed \(v\), the inward (centripetal) force required to bend its straight-line motion into the circle is

\[ F \;=\; \dfrac{m\,v^{2}}{r} \quad=\quad m\,\omega^{2}\,r \]

where ω = v/r is the angular velocity in radians per second. The corresponding centripetal acceleration is

\[ a \;=\; \dfrac{v^{2}}{r} \;=\; \omega^{2}\,r \]

Both forms describe exactly the same physics — pick whichever is convenient for the problem at hand. Spinning machinery is usually quoted in RPM (revolutions per minute), which converts to angular velocity by \( \omega = \mathrm{RPM} \cdot 2\pi / 60 \). The period of one full revolution is \( T = 2\pi/\omega \), and the rotation frequency is \( f = 1/T \).

Worked Example: Car on a Freeway Curve

A 1500 kg car travels at 100 km/h (≈ 27.78 m/s) along a flat curve of radius 120 m.

• \( F = m v^{2}/r = 1500 \times 27.78^{2} / 120 \approx 9645\) N.
• Centripetal acceleration \(a = v^{2}/r \approx 6.43\) m/s² ≈ 0.66 g.
• Minimum tire-friction coefficient: \( \mu = a/g \approx 0.66 \). Achievable on dry asphalt (μ_dry ≈ 0.7–0.9) but borderline on a wet road, where μ_wet ≈ 0.4–0.6 — exactly why drivers are warned to slow down on wet curves.

Worked Example: International Space Station

The ISS orbits at an altitude of about 408 km, giving an orbital radius of \(r \approx 6783\) km from Earth's center. Its orbital speed is roughly 7660 m/s.

• For a 1 kg payload, \( F = (1)(7660)^{2}/6783000 \approx 8.65\) N — exactly the gravitational pull at that altitude. The ISS is in continuous free-fall around Earth, which is what makes it weightless inside.
• Centripetal acceleration \( a \approx 8.65\) m/s² ≈ 0.88 g, which is gravity at that altitude (gravity at sea level is 9.81 m/s²).
• The orbital period works out to \( T = 2\pi r / v \approx 5564\) s ≈ 92.7 minutes — the ISS circles Earth roughly every hour and a half.

Centripetal vs Centrifugal Force

The two are often confused. Centripetal force is real: it is whatever physical interaction (string tension, gravity, normal force, friction, magnetic force) actually pulls the object toward the center of the circle. Centrifugal force is a "fictitious" force that appears only in a rotating reference frame — it's what you feel pushing you outward when a car turns sharply, but from the perspective of a stationary observer outside the car you're simply continuing in a straight line while the car bends underneath you. The centripetal force from the seat and seat belt is what actually accelerates you into the turn.

Everyday and Engineering Examples

Why Banked Turns Need Less Friction

On a flat curve the only inward force comes from tire-road friction, so the maximum cornering speed is \( v_{max} = \sqrt{\mu g r} \). On a banked turn the road's normal force tilts inward and contributes to the centripetal force, so much less friction is needed. That's why high-speed turns on race tracks are banked — the bank angle does the work that friction would otherwise have to do, allowing higher safe speeds and reducing tire wear.

Frequently Asked Questions

What is the centripetal force formula?
F = m·v²/r where m is the mass, v is the linear speed along the circle, and r is the radius. Equivalently, using the angular velocity ω, F = m·ω²·r. Both expressions give exactly the same value.

Is centripetal force the same as centrifugal force?
No. Centripetal force is a real inward force that keeps an object on its circular path. Centrifugal force is a fictitious outward force that only appears in a rotating reference frame. From an outside non-rotating observer, only the centripetal force exists.

How do I convert RPM to angular velocity?
Multiply RPM by 2π/60. So 600 RPM equals 600 × 2π / 60 ≈ 62.83 rad/s. The calculator does this automatically when you switch to angular input.

What is g-force in this context?
The centripetal acceleration divided by 9.80665 m/s². 1 g equals normal gravity, 4 g feels like a hard roller-coaster turn, and trained pilots can sustain about 9 g for short periods.

How much friction does a car need to take a turn?
μ = v²/(r·g). The calculator shows this automatically when you pick the car-turn scenario, and compares it to typical friction ranges for dry and wet asphalt.

What does the rotating animation show?
It shows the mass tracing a circle of radius r at the angular velocity computed from your inputs. The orange arrow is the centripetal force pointing to the center, and the teal arrow is the tangential velocity. The visual rotation period is clamped to a watchable range so very slow or very fast rotations are still visible.

Can I solve for the radius or the maximum safe speed?
Yes. Set Solve for to "Radius r" or "Linear speed v" and the corresponding field hides itself. The other three values become the inputs and the calculator solves the rearranged formula for you.