Calculate centripetal force (F = mv²/r), acceleration, and G-force for circular motion — or solve for any missing variable (mass, velocity, radius, force). Plus banked curve angle and angular velocity modes, all Newton’s Law verified.
✓Verified: Newton’s Second Law & NIST Physics Reference 2026
F = mv²/r — enter mass, linear velocity, and radius:
Mass of object in kgEnter valid mass.
Linear speed at radius rEnter valid speed.
Radius of circular pathEnter valid radius.
Solve for any unknown: enter 3 values, find the 4th:
Select which variable to find
Centripetal forceEnter valid force.
Mass of objectEnter valid mass.
Linear speedEnter valid speed.
Radius of circular pathEnter valid radius.
Ideal banked angle: θ = arctan(v²/rg) — no friction needed at this speed:
Design speed of the curveEnter valid speed.
Curve radius in metersEnter valid radius.
Vehicle mass for force calculationEnter valid mass.
F = mω²r — calculate centripetal force from rotation speed and radius:
Mass of rotating objectEnter valid mass.
Rotational speed in RPMEnter valid RPM.
Distance from rotation axisEnter valid radius.
Centripetal Force
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⚠️ Disclaimer: Results use Newton’s Second Law for circular motion. Real-world applications must account for friction, air resistance, and other forces. Banked curve calculations assume ideal frictionless conditions. Always apply appropriate safety factors in engineering design.
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📚 Sources & Methodology
All centripetal force formulas verified against authoritative sources:
NIST Physics Reference Data — SI unit definitions, physical constants (g = 9.80665 m/s²), and fundamental mechanics relationships physics.nist.gov
Halliday, Resnick & Krane, Physics 5th Ed. — Chapter 6: Force and Motion II — centripetal acceleration, Newton’s Second Law applied to circular motion wiley.com
ISO 80000-3:2019 — Quantities and units, Space and time — angular velocity, centripetal acceleration definitions iso.org
Complete Centripetal Force Guide — Formula, Acceleration & Real Applications
What Is Centripetal Force?
Centripetal force is the net force directed toward the centre of a circular path that keeps an object in circular motion. It is not a new type of force — it is the net inward component of forces already acting on the object: gravity (for satellites), string tension (for a ball on a string), friction (for a car on a curve), or normal force (for a roller coaster loop). Without this inward force, the object would travel in a straight line (Newton’s First Law).
The centripetal acceleration is always directed inward (toward the rotation centre) and equals v²/r = ω²r. Multiplied by mass, this gives the centripetal force F = mv²/r = mω²r. As speed doubles, centripetal force quadruples — this is why high-speed circular motion requires very large forces.
Centripetal Force Formulas (Newton’s Second Law)
F = m * v^2 / r [force from mass, speed, radius]F = m * omega^2 * r [force from mass, angular velocity, radius]a = v^2 / r = omega^2 * r [centripetal acceleration, m/s^2]G-force = a / 9.80665 [acceleration in multiples of g]Solve for v: v = sqrt(F*r/m)Solve for r: r = m*v^2/FSolve for m: m = F*r/v^2Banked angle: theta = arctan(v^2 / (r*g)) [ideal, no friction]Min loop speed: v_min = sqrt(g*r) [top of vertical loop]
Real-World Centripetal Force Examples
Scenario
Mass
Speed
Radius
Force
G-force
Car on 100m curve
1,500 kg
50 km/h (13.9 m/s)
100 m
2,890 N
0.20 g
Car on 50m curve
1,500 kg
80 km/h (22.2 m/s)
50 m
14,787 N
1.01 g
Washing machine drum
5 kg
1200 RPM, r=0.25m
0.25 m
19,739 N
403 g
Fighter jet turning
10,000 kg
300 m/s
1,500 m
600,000 N
6.1 g
Ball on 1m string
0.2 kg
5 m/s
1 m
5 N
2.55 g
Satellite orbit (LEO)
1,000 kg
7,800 m/s
6,771,000 m
8,988 N
0.917 g
Centripetal vs Centrifugal Force
Centripetal force is real — it exists in all reference frames and is the inward net force required for circular motion (Newton’s Second Law: F = ma, with a pointing inward). Centrifugal force is a fictitious force that appears only in a rotating (non-inertial) reference frame — it equals −mω²r and acts outward, balancing centripetal force in the rotating frame. From inside a car on a curve, you feel pushed outward (centrifugal) — but from outside (inertial frame), your seat provides inward centripetal force to keep you moving in a circle.
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Banked curve design: The ideal bank angle θ = arctan(v²/rg) eliminates the need for friction at the design speed. At 100 km/h (27.8 m/s) on a 200 m radius highway ramp: θ = arctan(772/1962) = arctan(0.394) = 21.5°. Real roads are banked to 5–10° and rely on friction for the remainder. Formula racing circuits may use 30–40° banking for high-speed corners.
❓ Frequently Asked Questions
Centripetal force is the net inward force keeping an object on a circular path. Formula: F = mv^2/r. It is not a new type of force — it is the inward component of existing forces (friction, tension, gravity, normal force). Without it, the object travels in a straight line (Newton's First Law). Example: a 1000 kg car at 20 m/s on a 50 m radius curve needs F = 1000 x 400/50 = 8,000 N of inward centripetal force from tyre friction.
F = mv^2/r = m*omega^2*r. Where m = mass (kg), v = linear speed (m/s), r = radius (m), omega = angular velocity (rad/s). Centripetal acceleration: a = v^2/r = omega^2*r. G-force = a/9.81. Use the Find Force tab above — enter mass, speed, and radius for instant results.
Centripetal: real, inward, exists in all frames (F=ma toward centre). Centrifugal: fictitious, outward, exists only in rotating reference frames. In a car turning left — inertial frame: seat pushes you left (centripetal). Rotating frame (inside car): you feel pushed right (centrifugal). Same physics, different perspective. For calculations, always use centripetal force (F=mv^2/r) in the inertial frame.
a = v^2/r = omega^2*r (m/s^2), directed toward centre. G-force = a/9.81. Example: car at 90 km/h (25 m/s) on 150 m radius: a = 625/150 = 4.17 m/s^2 = 0.43 g. Washing machine spin at 1400 RPM (146.6 rad/s) with 0.25 m drum radius: a = 146.6^2 x 0.25 = 5,373 m/s^2 = 548 g. The results tab above shows G-force automatically.
Ideal angle theta = arctan(v^2/(r*g)). This gives the angle where no friction is needed. Example: 120 km/h (33.3 m/s) on 300 m radius: theta = arctan(33.3^2/(300*9.81)) = arctan(0.377) = 20.7 degrees. Use the Banked Curve tab above — enter speed and radius for the ideal bank angle plus design notes.
r = mv^2/F. Example: 2 kg mass at 6 m/s, tension 72 N: r = 2 x 36/72 = 1.0 m. Use the Solve for Any Variable tab — select "Radius r" from the dropdown, enter F, m, and v for instant result.
Car on curve: tyre friction (static). Ball on string: string tension. Satellite orbit: gravity (F=GMm/r^2=mv^2/r). Electron in atom: electrostatic Coulomb force. Roller coaster loop: normal force + gravity component. Centrifuge: tube wall on sample. Washing machine: drum wall on clothes. The source changes; the formula F=mv^2/r always applies to the net inward force.
At the top of a vertical loop, centripetal force = gravity + normal force. Minimum speed when N=0: mg = mv^2/r, so v_min = sqrt(g*r). For a 10 m radius loop: v_min = sqrt(9.81 x 10) = 9.9 m/s (35.7 km/h). Below this speed, the object loses contact with the track at the top. Roller coasters calculate this to ensure safe minimum speeds.
G-force = centripetal acceleration / 9.81 m/s^2. A car at 100 km/h (27.8 m/s) on a 80 m radius corner: a = 27.8^2/80 = 9.66 m/s^2 = 0.98 g. A centrifuge at 5000 RPM (omega=523.6 rad/s) with r=0.1 m: a = 523.6^2 x 0.1 = 27,416 m/s^2 = 2,795 g. The results tab above shows G-force automatically for every calculation.
Without sufficient centripetal force, the object follows a straight-line path (Newton's First Law) instead of the circular one. Car on curve: tyres slide outward when friction is exceeded. Ball on string: flies off tangentially if string breaks. Satellite: spirals inward if speed drops. The critical factor is always: is the available inward force (friction, tension, gravity) at least equal to mv^2/r?
F = m*omega^2*r, since v = omega*r, so F = m*(omega*r)^2/r = m*omega^2*r. Example: 0.5 kg at r=0.3 m spinning at 600 RPM (omega=62.83 rad/s): F = 0.5 x 62.83^2 x 0.3 = 591.6 N. Use the From RPM and Radius tab above for rotating machinery calculations.
For a satellite orbit, gravity provides centripetal force: GMm/r^2 = mv^2/r. Orbital speed: v = sqrt(GM/r). ISS at 408 km altitude (r=6,786 km): v = sqrt(3.986e14/6.786e6) = 7,665 m/s (27,594 km/h). Centripetal acceleration = g at that altitude = 8.66 m/s^2 (0.883 g). The satellite is in continuous free fall — its orbital speed matches the rate Earth's surface curves away beneath it.