Centripetal Force Calculator
Last Updated: October 29, 2025
Calculate centripetal force from mass, velocity, radius, angular velocity, frequency, and period using circular motion equations. Essential tool for physics circular motion, rotating systems, and mechanical engineering analysis.
Calculator
Enter your values below to calculate centripetal force instantly.
Choose your calculation method and enter the required values for accurate centripetal force calculations.
Mass of the rotating object
Tangential linear velocity
Distance from axis of rotation
Results
Enter values to see results
Table of Contents
What is Centripetal Force Calculator?
Understanding Circular Motion and Inward Forces
The Centripetal Force Calculator is a specialized physics tool that calculates the centripetal force required to keep an object moving in a circular path. Centripetal force is the real inward force that points toward the center of the circular motion, necessary for maintaining circular motion and preventing objects from moving in straight lines.
This calculator is essential for analyzing circular motion in physics, engineering, and mechanical systems. It provides accurate calculations for centripetal force using multiple methods, including mass-velocity-radius relationships, angular velocity conversions, frequency and period relationships.
Key Concepts
Centripetal Force (F): Inward force toward center of circular path, measured in Newtons (N). This is the REAL force required to keep an object moving in a circle, always pointing toward the center. It's the force that prevents objects from flying off in straight lines.
Mass (m): Mass of the object in circular motion, measured in kilograms (kg). Heavier objects require more centripetal force to maintain the same circular motion at the same speed.
Velocity (v): Linear tangential velocity along the circular path, measured in meters per second (m/s). The speed at which the object moves along its circular trajectory. Faster speeds require more centripetal force.
Angular Velocity (ω): Rate of rotation about an axis, measured in radians per second (rad/s). Related to linear velocity by v = rω. Higher angular velocity means faster rotation and greater centripetal force requirement.
Radius (r): Distance from center of rotation to the object, measured in meters (m). Smaller radii require more centripetal force for the same velocity - this is why tighter curves require more force.
Physical Interpretation
Centripetal force is not a separate type of force - it's the name given to whatever force is providing the inward acceleration needed for circular motion. This could be tension (a ball on a string), friction (car on a curved road), gravity (planets orbiting), or normal force (roller coaster loops). Without centripetal force, objects would move in straight lines due to inertia (Newton's first law).
The formula F = mv²/r shows that centripetal force is proportional to mass and the square of velocity, but inversely proportional to radius. This means doubling the speed quadruples the required force, while doubling the radius halves it. This explains why high-speed turns require strong banking or friction, and why rotating systems must be carefully designed to handle the forces involved.
Relationship to Centrifugal Force
Centripetal force is often confused with centrifugal force. Centripetal force is the real inward force causing circular motion (what you provide). Centrifugal force is the apparent outward force felt in a rotating reference frame (an inertial effect). When you swing a ball on a string, the string provides the centripetal force; you feel the ball pulling outward, which is the centrifugal force in your rotating frame. In inertial frames, only centripetal force is real.
Centripetal Acceleration
Centripetal force produces centripetal acceleration (a = v²/r = ω²r), which is always directed toward the center. This acceleration changes the direction of velocity but not its magnitude (for uniform circular motion). The centripetal acceleration accounts for why velocity is always tangent to the circle but the object moves along the curved path.
Historical Development
The concept of centripetal force was developed by Isaac Newton in the 17th century as part of understanding circular motion and universal gravitation. Newton realized that planets orbit the Sun because of gravitational centripetal force, and that any circular motion requires an inward force. The relationship F = mv²/r is fundamental to understanding circular motion in physics and engineering. This concept was crucial in explaining planetary orbits, the design of rotating machinery, and understanding phenomena from roller coasters to particle accelerators.
Units and Conversions
SI Unit: N (Newton) - force unit, equivalent to kg·m/s²
Alternative Units: dyne (cgs), pound-force (lb-f) - less common in modern physics
Relationship: F = mv²/r = mω²r = m(2πf)²r (multiple equivalent forms)
Practical Note: Centripetal force can be provided by any real force: tension, friction, gravity, normal force, electromagnetic force, etc.
Formulas and Equations
Centripetal Force Calculation Methods
1. From Mass, Velocity, and Radius
F = mv²/r
Where:
- • F = Centripetal force (N)
- • m = Mass (kg)
- • v = Linear velocity (m/s)
- • r = Radius (m)
Use case: Calculate centripetal force when you know the mass, linear velocity, and radius. This is the most direct and commonly used formula. Essential for analyzing cars on curves, objects on strings, and any situation where tangential velocity is measured. The squared velocity means doubling speed quadruples the required force.
2. From Mass, Angular Velocity, and Radius
F = mω²r
Where:
- • F = Centripetal force (N)
- • m = Mass (kg)
- • ω = Angular velocity (rad/s)
- • r = Radius (m)
Use case: Calculate centripetal force using angular velocity, which is convenient for rotating systems where rotation rate is measured (like motors, flywheels, or rotating platforms). Derived from F = mv²/r using v = rω. Particularly useful in mechanical engineering applications involving rotating machinery.
3. From Mass, Frequency, and Radius
F = m(2πf)²r
Where:
- • F = Centripetal force (N)
- • m = Mass (kg)
- • f = Frequency (Hz)
- • r = Radius (m)
- • π ≈ 3.14159
Use case: Calculate centripetal force when frequency of rotation is known. Useful for systems where rotation frequency is measured (like rotating equipment, centrifuges, or oscillatory systems). Converts frequency to angular velocity using ω = 2πf, then applies the angular velocity formula.
4. From Mass, Period, and Radius
F = m(2π/T)²r
Where:
- • F = Centripetal force (N)
- • m = Mass (kg)
- • T = Period (s)
- • r = Radius (m)
- • π ≈ 3.14159
Use case: Calculate centripetal force when period (time for one complete rotation) is known. Useful in experimental physics where period measurements are common, or for analyzing periodic circular motion like planetary orbits or pendulum bobs in circular paths. Converts period to angular velocity using ω = 2π/T.
Applications of Centripetal Force
Real-World Uses Across Industries
Centripetal force calculations are essential across numerous physics, engineering, and mechanical fields. Here's a comprehensive overview of practical applications:
| Industry | Applications | Importance |
|---|---|---|
| Automotive Engineering | Vehicle turning, banked curves, circular tracks, tire design, suspension systems, cornering forces | Critical for safety, road design, and vehicle handling |
| Aerospace | Orbital mechanics, satellite orbits, curved flight paths, centrifuge training, spacecraft maneuvering | Essential for space missions and navigation |
| Amusement Parks | Roller coasters, spinning rides, circular tracks, loop-the-loops, rotating platforms | Vital for ride safety and passenger experience |
| Mechanical Engineering | Rotating machinery, flywheels, centrifuges, grinding wheels, spindles, rotating tools | Key for design, safety, and performance analysis |
| Physics Research | Circular motion analysis, planetary orbits, particle accelerators, atomic models, rotational dynamics | Fundamental for understanding motion and mechanics |
| Sports Engineering | Track design, cycling, racing circuits, discus throwing, hammer throw, curved athletic tracks | Important for performance optimization and safety |
Examples of Centripetal Force Calculation
Real-World Applications and Use Cases
Example 1: Car Turning a Corner
Given:
- • Mass: m = 1200 kg
- • Velocity: v = 25 m/s (90 km/h)
- • Turn radius: r = 50 m
Step-by-step calculation:
Step 1: Apply centripetal force formula
F = mv²/r
F = (1200 kg) × (25 m/s)² / (50 m)
F = (1200 × 625) / 50
F = 15,000 N
Final Answer
15,000 N
This is the force required from friction between tires and road to keep the car turning safely
Example 2: Satellite in Circular Orbit
Given:
- • Mass: m = 500 kg
- • Angular velocity: ω = 0.001 rad/s
- • Orbital radius: r = 7,000,000 m
Step-by-step calculation:
Step 1: Apply centripetal force formula with angular velocity
F = mω²r
F = (500 kg) × (0.001 rad/s)² × (7,000,000 m)
F = 500 × 0.000001 × 7,000,000
F = 3,500 N
Final Answer
3,500 N
This centripetal force is provided by Earth's gravitational attraction, keeping the satellite in orbit
Example 3: Ball Swinging on a String (Frequency Method)
Given:
- • Mass: m = 0.5 kg
- • Frequency: f = 2 Hz (2 rotations per second)
- • String length (radius): r = 1.0 m
Step-by-step calculation:
Step 1: Apply centripetal force formula with frequency
F = m(2πf)²r
F = (0.5 kg) × (2π × 2 Hz)² × (1.0 m)
F = 0.5 × (4π)² × 1.0
F = 0.5 × 157.9 × 1.0
F = 78.95 N
Step 2: Alternative calculation using angular velocity
ω = 2πf = 2π × 2 = 12.57 rad/s
F = mω²r = 0.5 × (12.57)² × 1.0 = 78.95 N
Final Answer
78.95 N
This is the tension in the string required to keep the ball moving in a circular path. The string provides the centripetal force needed for the circular motion.