Circular Motion & Centripetal Force Calculator
Calculate angular velocity ω, period T, rpm, tangential velocity v, centripetal acceleration, and force F for uniform circular motion. Features centrifuge RCF (×g) conversion, vehicle bank angle, and vertical loop dynamics.
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Circular Motion Inputs
F = m・r・ω² = m・v²/rCircular Motion & Vectors Simulation
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Circular Motion & Centripetal Force Calculator
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How to use
- 1
Select calculation mode
Choose Uniform Circular Motion (basic force & kinematics), Centrifuge & RCF (rpm ⇄ ×g), or Bank Angle & Vertical Loop.
- 2
Input radius and velocity parameters
Enter rotation radius r (mm/cm/m/km), mass m, and velocity (period T, angular velocity ω, tangential speed v, or rpm).
- 3
Inspect acceleration, forces & animation
Review centripetal acceleration a, G-ratio, force F, kinetic energy K, live vector orbit simulation, and loop status.
Features
- Interlinked evaluation of angular velocity ω, tangential velocity v, period T, frequency f, and rpm
- Instant calculation of centripetal acceleration a (m/s²) and Earth gravity ratio (G-ratio / ×g)
- Accurate calculation of centripetal/centrifugal force F (N, kN, kgf), kinetic energy K, and angular momentum
- Bi-directional conversion between centrifuge rotational speed (rpm) and Relative Centrifugal Field (RCF / ×g)
- Automatic protocol scaling across differing rotor radii (mm/cm) and centrifugal time integrals (g・min)
- Vehicle curve mechanics solving ideal road bank angle θ, lateral G, and friction slip limits
- Vertical loop dynamics verifying full loop completion (v₀ ≥ √(5gr)) and top/bottom tension forces
- Interactive SVG animation visualizing orbital motion, tangential velocity, and inward centripetal acceleration
- Preconfigured presets for physics labs, microcentrifuges, highway curves, loop coasters, and ISS orbit
- Comprehensive unit conversions across metric prefixes (mm, cm, m, km; g, kg, t; km/h, m/s)
- One-click laboratory report export formatted in clean Markdown for assignments and documentation
Use cases
Physics & Engineering Mechanics Coursework
Avoid unit conversion pitfalls between angular velocity, period, and rpm while inspecting vector relationships.
Life Science & Chemistry Centrifugation Protocols
Accurately convert protocols between RCF (×g) and centrifuge rpm based on actual rotor radii to ensure reproducible pelleting.
Vehicle Dynamics & Roller-Coaster Engineering
Calculate safe bank angles, maximum non-slip cornering speeds, and vertical loop G-forces and safety margins.
Details
Circular motion describes the movement of an object along a circular trajectory, forming the foundation of planetary orbits, laboratory centrifuges, vehicle cornering, roller-coaster loops, and rotational machinery.
In uniform circular motion, while tangential speed v = rω remains constant, the velocity vector continuously changes direction toward the center, producing centripetal acceleration a = rω² = v²/r. According to Newton’s second law, this requires an inward centripetal force F = mrω² = m(v²/r). In the rotating reference frame of the moving body, an equal and opposite fictitious inertial force—centrifugal force—is experienced.
In biology, chemistry, and clinical laboratories, centrifuges separate suspended particles, organelles, and DNA using Relative Centrifugal Field (RCF, measured in ×g). Following RCF = 1.118 × 10⁻⁵ × r[cm] × (rpm)², this tool allows bi-directional conversion between rpm and RCF across different rotor radii to ensure protocol reproducibility.
The calculator also models advanced mechanics: optimal highway bank angles (tan θ = v² / (rg)) for frictionless turns, maximum non-slip cornering speeds (v_max = √(μgr)), and critical vertical loop thresholds (v₀ ≥ √(5gr)) with bottom and top normal/tension forces.
FAQ
What is the physical difference between centripetal force and centrifugal force?
They share identical magnitude and direction line but belong to different reference frames. In an inertial (stationary) frame, the real inward force keeping the object in circular motion is the centripetal force. In the rotating reference frame of the moving object, an apparent outward inertial force—the centrifugal force—is observed balancing the inward force. They describe the same physical reality from two reference perspectives.
Why should centrifugation protocols specify RCF (×g) rather than rpm?
Centrifugal sedimentation force depends directly on rotor radius as well as speed (RCF ∝ r × rpm²). At 10,000 rpm, a small 60 mm rotor generates ~6,700 × g, whereas a 120 mm rotor generates ~13,400 × g—double the force. Specifying RCF (×g) ensures exact protocol reproducibility across different centrifuge models.
What is a banked curve, and why are roads tilted on turns?
A banked curve has its outer edge elevated above its inner edge. Tilting the surface angles the normal force toward the center of curvature, providing centripetal force without relying on tire friction. At the ideal bank angle tan θ = v² / (rg), vehicles negotiate the turn smoothly even on frictionless ice.
Why is a minimum velocity of √(5gr) required at the bottom to complete a vertical loop?
To prevent string slack or rail separation at the highest point (height 2r), inward centripetal acceleration must at least equal gravity, requiring v_top ≥ √(gr). Applying conservation of energy (½mv₀² = ½mv_top² + mg(2r)) with v_top² = gr yields ½mv₀² = 2.5mgr, proving that initial speed must be v₀ ≥ √(5gr).
Does Earth’s rotation reduce body weight via centrifugal force?
Yes, slightly. With an equatorial radius of ~6,378 km and a 24-hour rotation period, equatorial centrifugal acceleration is ~0.0337 m/s², which is approximately 0.34% of standard gravity. Consequently, an object weighs about 0.3% less at the equator than at the poles (about 200 grams less for a 60 kg person).
Related guide
GuideCircular Motion & Centripetal Force: Formulas, Centrifuge RCF, Bank Angles & Loops
Master uniform circular motion calculations: angular velocity ω, period T, rpm, speed v, centripetal acceleration a, and force F. Understand centripetal vs centrifugal force, laboratory centrifuge RCF (×g) conversion, highway bank angles, and vertical loop clearance physics.
GuideSimple Harmonic Motion & Pendulum Period: Equations, Experiments & Energy
Master simple harmonic motion: pendulum period T = 2π√(L/g), Borda’s finite amplitude correction, lab gravity inversion g = 4π²L/T², spring series and parallel stiffness, vertical equilibrium stretch, and phase relationships (90°/180°) with energy conservation.
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Verified: Verified uniform circular motion (0.5m, 0.1kg, 1s -> 6.28rad/s, 3.14m/s, 19.74m/s², 1.97N), centrifuge RCF (60mm, 15000rpm -> 15096×g), vehicle bank angle (200m, 80km/h -> 14.1°), vertical loop (8m, 22m/s -> loop cleared, T_bot=35.15kN), vector animation, and English interface with automated tests
Did you know?
Dutch physicist Christiaan Huygens first derived the centripetal acceleration formula a = v²/r in 1659. Sir Isaac Newton famously combined Huygens’s circular motion dynamics with Kepler’s planetary laws to discover that gravitational attraction between celestial bodies follows an exact inverse-square law.