How do I use this calculator?

Enter the required values; results and step-by-step derivation update automatically.

Where is Centripetal Force used?

Mechanics, engineering, research, and related scientific disciplines.

What formulas does this use?

All standard centripetal force formulas are applied and shown step-by-step in the results.

Can I get step-by-step solutions?

Yes. The calculator shows a full step-by-step derivation when results are displayed.

How accurate are the results?

Results use standard floating-point arithmetic (~15 significant digits). Suitable for education and professional use.

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Centripetal Force - Circular Motion

Centripetal force F_c = mv²/r = mω²r keeps objects in circular path. It points toward the center. Centripetal acceleration a_c = v²/r. Essential for orbits, car turns, and rotating systems.

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F_c = mv²/r: force toward center; without it, object flies off tangent Orbital velocity: v = √(GM/r) for circular orbit Car turn: friction provides centripetal force; limit is μmg ω = 2π/T = 2πf links angular velocity to period and frequency

Key quantities

mv²/r = mω²r

F_c

Key relation

v²/r = ω²r

a_c

Key relation

v/r = 2π/T

Key relation

2πr/v

Key relation

Ready to run the numbers?

Why: Centripetal force explains orbits, car cornering, roller coasters, and particle accelerators. Without it, objects move in straight lines. Tension, friction, or gravity often provide centripetal force. Essential for Newtonian mechanics.

How: F_c = mv²/r using linear velocity, or F_c = mω²r using angular velocity. a_c = v²/r = ω²r. ω = v/r = 2π/T = 2πf. Period T = 2πr/v. Force always points toward center. 'Centrifugal force' is fictitious—reaction in rotating frame.

F_c = mv²/r: force toward center; without it, object flies off tangentOrbital velocity: v = √(GM/r) for circular orbit

Sources:HyperPhysics - Circular MotionPhysics Classroom - Circles

Run the calculator when you are ready.

Calculate Centripetal ForceEnter mass, velocity, and radius to find centripetal force and acceleration.

🛰️ Satellite in LEO

500 kg satellite at 7.8 km/s, altitude 400 km

🚗 Car on Highway Curve

1200 kg car at 90 km/h on 100m curve

🏐 Tetherball

0.4 kg ball on 2m rope at 1.5 rad/s

🎡 Ferris Wheel Rider

70 kg person on 25m wheel, 2 RPM

🌙 Moon Orbiting Earth

Moon mass at 384,400 km, 27.3 day period

🚴 Cyclist Turning

85 kg cyclist+bike at 8 m/s, 15m turn

🎠 Amusement Spin Ride

60 kg rider at 3m radius, 15 RPM

🏅 Hammer Throw

7.26 kg hammer at 2.0 Hz rotation, 1.2m wire

⚛️ Electron in Hydrogen

Electron at Bohr radius (simplified)

⛸️ Ice Skater Spin

55 kg skater with 0.5m arm radius at 3 rev/s

Enter Your Values

Calculation Method

Parameters

Mass (m)

Mass Unit

Radius (r)

Radius Unit

Linear Velocity (v)

Velocity Unit

For educational and informational purposes only. Verify with a qualified professional.

🔬 Physics Facts

🔄

F_c = mv²/r: centripetal force points toward center

— HyperPhysics

🌍

Orbital velocity v = √(GM/r) for circular orbit around mass M

— Physics Classroom

🚗

Car cornering: friction supplies F_c; max speed v = √(μgr)

— NIST

⚛️

Electron in atom: Coulomb force provides centripetal force

— APS

📋 Key Takeaways

• Centripetal force is the real force that causes circular motion, always directed toward the center
• Formula: Fc = mv²/r = mω²r, where m is mass, v is velocity, r is radius, ω is angular velocity
• Centripetal force does no work (perpendicular to velocity), so kinetic energy remains constant in uniform circular motion
• Common sources include tension (ball on string), gravity (orbits), friction (car on curve), and normal force (loop-the-loop)

💡 Did You Know?

🌍Earth's centripetal acceleration at the equator is only 0.034 m/s² — just 0.3% of gravitySource: NIST

🚗A car traveling at 60 mph around a 100m curve experiences ~2.4g of centripetal accelerationSource: Physics Classroom

🛰️The ISS orbits Earth at 7.67 km/s, requiring ~4,000 N of centripetal force (gravity) for a 500 kg satelliteSource: NASA

🎢Roller coaster loops require minimum speed v = √(gr) at the top to prevent riders from fallingSource: HyperPhysics

⚛️Electrons in atoms experience centripetal force from electrostatic attraction — ~8×10⁻⁸ N for hydrogenSource: MIT OCW

🌙The Moon's centripetal force from Earth's gravity is approximately 2×10²⁰ N — keeping it in orbitSource: Wolfram Physics

🔬Ultracentrifuges can generate forces up to 400,000g — separating particles by densitySource: NIST

📖 How Centripetal Force Works

Centripetal force is the net force that causes an object to move in a circular path. Unlike centrifugal force (which is fictitious), centripetal force is real and measurable.

The Physics Behind Circular Motion

When an object moves in a circle, its velocity direction constantly changes even if speed is constant. This change in velocity requires acceleration toward the center — centripetal acceleration. According to Newton's second law (F = ma), this acceleration requires a net force pointing toward the center.

Key Relationships

• Linear velocity: v = ωr = 2πr/T = 2πrf
• Centripetal acceleration: ac = v²/r = ω²r
• Centripetal force: Fc = mac = mv²/r = mω²r
• Period: T = 2π/ω = 2πr/v

🎯 Expert Tips

💡 Identify the Force Source

Always identify what provides the centripetal force — tension, gravity, friction, or normal force. Draw free-body diagrams to visualize forces.

💡 Use Consistent Units

Convert all values to SI units (kg, m, s) before calculating. Remember: angular velocity ω is in rad/s, not degrees/s.

💡 Remember v² Relationship

Centripetal force scales with the square of velocity. Doubling speed quadruples the required force — critical for safety calculations.

💡 Check Force Direction

Centripetal force always points toward the center. In vertical circles, gravity affects the net force differently at different positions.

⚖️ Centripetal vs Centrifugal Force

Aspect	Centripetal Force	Centrifugal Force
Type	Real force	Fictitious force
Direction	Toward center	Away from center
Reference frame	Inertial frame	Rotating frame only
Causes motion	Yes — causes circular motion	No — apparent effect
Measurable	Yes	No
Formula	Fc = mv²/r	Fcentrifugal = -mv²/r

❓ Frequently Asked Questions

What is the difference between centripetal and centrifugal force?

Centripetal force is real and points inward (causes circular motion). Centrifugal force is fictitious and only appears in rotating reference frames, pointing outward. They're equal in magnitude but opposite in direction.

What provides centripetal force for a satellite?

Gravity provides the centripetal force for satellites and planets. The gravitational attraction between the satellite and Earth pulls the satellite toward Earth's center, keeping it in orbit.

Why do cars need friction to turn?

When a car turns, friction between the tires and road provides the centripetal force. Without friction (ice), the car would continue in a straight line. Banked curves reduce the friction needed by using a component of the normal force.

Can centripetal force do work?

No! Centripetal force is always perpendicular to velocity, so it does no work (W = F·d·cos90° = 0). It changes direction of motion but not speed. This is why uniform circular motion has constant kinetic energy.

How do you calculate centripetal force from RPM?

Convert RPM to angular velocity: ω = RPM × 2π/60 rad/s. Then use Fc = mω²r. Alternatively, find linear velocity: v = ωr, then use Fc = mv²/r.

What happens if centripetal force disappears?

The object immediately moves in a straight line tangent to the circle at that point (Newton's first law). This is why a car skids straight when friction is lost on a curve.

Why does doubling speed quadruple centripetal force?

Because Fc = mv²/r. The v² term means force scales with the square of velocity. Double v → 4× force. This is why high-speed curves are so dangerous!

What is the minimum speed for a vertical loop?

At the top of a vertical loop, minimum speed is v = √(gr) when normal force equals zero. Gravity alone provides all centripetal force at this critical point.

📊 Centripetal Force by the Numbers

0.034 m/s²

Earth Equator

2.4g

Car at 60 mph

400,000g

Ultracentrifuge

2×10²⁰ N

Moon Orbit

📚 Official Data Sources

NIST Physical Constants ↗

Official physical constants and units

Updated: 2026-02-01

HyperPhysics - Circular Motion ↗

Comprehensive physics reference for circular motion

Updated: 2026-02-06

Khan Academy - Centripetal Force ↗

Educational content on centripetal force

Updated: 2026-02-06

MIT OpenCourseWare - Classical Mechanics ↗

MIT course materials on classical mechanics

Updated: 2026-02-06

Physics Classroom - Circular Motion ↗

Educational resources on circular motion

Updated: 2026-02-06

Wolfram Physics ↗

Physics calculations and examples

Updated: 2026-02-06

⚠️ Disclaimer: This calculator provides estimates based on ideal circular motion assumptions. Real-world applications may involve friction, air resistance, and other factors. Always verify calculations for safety-critical applications. Not a substitute for professional engineering analysis.

What is Centripetal Force?

Centripetal force is the real force that acts on an object moving in a circular path, directed toward the center of the circle. Unlike centrifugal force (which is apparent), centripetal force is what actually causes circular motion - it constantly changes the direction of velocity, keeping the object on its curved path.

🎯

A Real Force

Unlike centrifugal force, centripetal force is a real force that can be provided by tension, gravity, friction, or other physical interactions.

Direction:

Always points toward the center of rotation

🔗

Sources of Centripetal Force

Different physical forces can provide the centripetal force depending on the situation.

Tension (ball on string)
Gravity (planets, satellites)
Friction (car on curve)
Normal force (loop-the-loop)

📚

Newton's Second Law

Centripetal force follows F = ma, where the acceleration is directed toward the center of the circular path.

Key Insight:

No force → straight line motion
Centripetal force → circular motion

How to Calculate Centripetal Force

🧮 Primary Formulas

Using Linear Velocity

Fc = mv²/r

m = mass, v = tangential velocity, r = radius

Using Angular Velocity

Fc = mω²r

ω = angular velocity in rad/s

📊 Related Quantities

Centripetal Acceleration

ac = v²/r = ω²r

Always toward center

Period & Frequency

T = 2π/ω ; f = 1/T

Time for one complete revolution

When to Use Centripetal Force Calculations

🚗

Vehicle Dynamics

Design safe curves, calculate friction requirements, and analyze vehicle stability.

Highway curve banking
Race track design
Safe speed limits

🛰️

Orbital Mechanics

Calculate satellite orbits, planetary motion, and space mission trajectories.

Satellite orbital velocity
Gravitational attraction
Escape velocity

⚙️

Mechanical Systems

Analyze rotating machinery, flywheels, and mechanical components.

Bearing loads
Shaft stress analysis
Pulley systems

Complete Formula Reference

Centripetal Force

Fc = mv²/r = mω²r = 4π²mrf²

Centripetal Acceleration

ac = v²/r = ω²r = 4π²rf²

Orbital Velocity

v = √(GM/r) for gravity-bound orbits

Banking Angle

tan(θ) = v²/(rg) for frictionless banking

Frequently Asked Questions

What's the difference between centripetal and centrifugal force?

Centripetal force is real and points inward (causes circular motion). Centrifugal force is a fictitious force that only appears in rotating reference frames, pointing outward. They're equal in magnitude but opposite in direction.

What provides centripetal force for a satellite?

Gravity provides the centripetal force for satellites and planets. The gravitational attraction between the satellite and Earth pulls the satellite toward Earth's center, keeping it in orbit.

Why do cars need friction to turn?

Can centripetal force do work?

Centripetal Force Examples in Nature and Technology

Scenario	Force Provider	Typical Values
Ball on string	Tension	1-50 N
Car on curve	Friction	1,000-10,000 N
Satellite orbit	Gravity	4,000-5,000 N (for 500 kg)
Electron in atom	Electrostatic	~8×10⁻⁸ N
Roller coaster loop	Normal + Gravity	2,000-4,000 N
Moon around Earth	Gravity	~2×10²⁰ N

Tips and Common Mistakes

✅ Best Practices

• Identify what provides the centripetal force
• Use consistent SI units
• Remember force points toward center
• Consider all forces at play

❌ Common Mistakes

• Confusing centripetal with centrifugal
• Using diameter instead of radius
• Forgetting the v² relationship
• Ignoring direction (it's a vector!)

Practice Problems

Problem 1: Car on a Curve

A 1500 kg car travels around a curve with radius 50 m at a speed of 15 m/s. Calculate the centripetal force and determine if static friction coefficient of 0.6 is sufficient.

Solution:

Fc = mv²/r = 1500 × 15² / 50 = 6,750 N
Max friction = μmg = 0.6 × 1500 × 9.81 = 8,829 N
Since 8,829 N > 6,750 N, the friction is sufficient! ✓

Problem 2: Satellite Orbital Velocity

A 400 kg satellite orbits Earth at altitude 400 km (Earth radius = 6371 km). Find its orbital velocity if centripetal force equals gravitational force.

Solution:

Total radius r = 6371 + 400 = 6771 km = 6.771 × 10⁶ m
v = √(GM/r) = √(3.986×10¹⁴ / 6.771×10⁶)
v ≈ 7,672 m/s ≈ 27,620 km/h

Problem 3: Roller Coaster Loop

A roller coaster car (with rider, total 600 kg) goes through a vertical loop of radius 12 m. What minimum speed at the top prevents the rider from falling?

Solution:

At minimum speed, normal force N = 0, gravity provides all centripetal force
mg = mv²/r → v = √(gr) = √(9.81 × 12)
v ≈ 10.85 m/s ≈ 39 km/h minimum

Mathematical Derivation

Step 1: Position in Circular Motion

For an object moving in a circle of radius r with angular velocity ω:

x = r·cos(ωt), y = r·sin(ωt)

Step 2: Velocity (First Derivative)

Taking the time derivative of position:

vx = -rω·sin(ωt), vy = rω·cos(ωt)
|v| = rω (tangential, perpendicular to r)

Step 3: Acceleration (Second Derivative)

Taking another derivative gives centripetal acceleration:

ax = -rω²·cos(ωt) = -ω²x
ay = -rω²·sin(ωt) = -ω²y
|a| = rω² = v²/r (pointing toward center)

Step 4: Apply Newton's Second Law

Using F = ma with centripetal acceleration:

Fc = mac = m(v²/r) = mω²r

Historical Context

🍎 Isaac Newton (1687)

In Principia Mathematica, Newton formalized the concept of centripetal force, showing that gravity provides the centripetal force keeping planets in orbit. This unified terrestrial and celestial mechanics.

🔭 Christiaan Huygens (1659)

Huygens derived the formula for centripetal acceleration (v²/r) while studying pendulum clocks and published it in Horologium Oscillatorium. He called it "centrifugal force" initially.

Engineering Applications

🛣️ Road Design

Highway curves are banked at specific angles calculated using centripetal force equations to allow safe navigation without relying solely on friction.

✈️ Aerospace

Aircraft turns, satellite orbits, and rocket trajectories all require precise centripetal force calculations for safe and efficient operation.

🏭 Machinery

Centrifuges, turbines, and rotating equipment must be designed to handle centripetal forces without structural failure.

👈 START HERE

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