Circular Motion Calculator

Calculate centripetal force, acceleration, angular velocity, period, and frequency for uniform circular motion.

Circular Motion Formulas: Centripetal Force: Fc = m × v² / r Centripetal Acceleration: ac = v² / r = ω² × r Angular Velocity: ω = v / r (rad/s) ω = 2π / T (T = period) Period and Frequency: T = 2πr / v f = 1 / T = v / (2πr) Conversions: • v = ω × r (linear velocity) • RPM to rad/s: multiply by 2π/60 • 1 revolution = 2π radians = 360° Where: • Fc = centripetal force (N) • m = mass (kg) • v = tangential velocity (m/s) • r = radius (m) • ac = centripetal acceleration (m/s²) • ω = angular velocity (rad/s) • T = period (s) • f = frequency (Hz)
Example 1 (Car on curve): m = 1500 kg, v = 20 m/s (72 km/h), r = 50 m Fc = 1500 × 400 / 50 = 12,000 N ac = 400 / 50 = 8 m/s² This force must come from tire friction. Example 2 (Satellite orbit): m = 1000 kg, v = 7800 m/s, r = 6,700,000 m (400 km altitude) Fc = 1000 × 60,840,000 / 6,700,000 = 9,081 N Period = 2π × 6,700,000 / 7800 ≈ 5,400 s ≈ 90 minutes Gravity provides the centripetal force. Example 3 (Ball on string): m = 0.5 kg, v = 5 m/s, r = 1 m Fc = 0.5 × 25 / 1 = 12.5 N ω = 5 / 1 = 5 rad/s ≈ 47.7 RPM Period = 2π / 5 ≈ 1.26 s String tension provides centripetal force. Example 4 (Centrifuge): m = 0.01 kg, v = 50 m/s, r = 0.2 m Fc = 0.01 × 2500 / 0.2 = 125 N ac = 12,500 m/s² ≈ 1,275 g ω = 250 rad/s = 2,387 RPM Period = 0.025 s (40 rev/s) Example 5 (Amusement ride): m = 70 kg (person), v = 8 m/s, r = 4 m Fc = 70 × 64 / 4 = 1,120 N ac = 16 m/s² ≈ 1.6 g Person experiences 1.6× normal weight force.

What is circular motion?

Circular motion is movement along a circular path. Uniform circular motion has constant speed but continuously changing velocity direction. The acceleration points toward the center (centripetal acceleration).

What is centripetal force?

Centripetal force is the net force causing circular motion, directed toward the center of the circle. It's calculated as Fc = mv²/r, where m is mass, v is velocity, and r is radius. It's not a separate force but the net result of other forces.

What is the difference between centripetal and centrifugal force?

Centripetal force is real and pulls toward the center. Centrifugal force is a fictitious force experienced in a rotating reference frame, appearing to push outward. Only centripetal force exists in an inertial reference frame.

What provides centripetal force in different scenarios?

Examples: Cars turning (friction), planets orbiting (gravity), ball on string (tension), roller coaster loop (normal force + gravity), electrons in atoms (electromagnetic force). The specific force depends on the situation.

How do I calculate centripetal acceleration?

Centripetal acceleration ac = v²/r, where v is tangential velocity and r is radius. It always points toward the center. Alternatively, ac = ω²r, where ω is angular velocity in rad/s.

What is the relationship between linear and angular velocity?

Linear (tangential) velocity v = ωr, where ω is angular velocity (rad/s) and r is radius. Also, ω = 2π/T, where T is period (time for one revolution). Common: RPM to rad/s: multiply by 2π/60.

What happens if centripetal force is insufficient?

If centripetal force is too weak, the object cannot maintain circular motion and moves in a larger radius or flies off tangentially. Examples: car skidding on icy curve, satellite escaping orbit, ball breaking string.

How does speed affect required centripetal force?

Centripetal force increases with the square of velocity (Fc ∝ v²). Doubling speed requires 4× the force. This is why cars are more likely to skid at higher speeds on curves, even with the same radius.

Can I use this for vertical circular motion?

Yes, but remember gravity affects the force. At the top of a vertical circle, gravity helps provide centripetal force. At the bottom, tension or normal force must overcome gravity plus provide centripetal force.

What is banking in curves?

Banking tilts the road/track inward so the normal force has a component toward the center, reducing reliance on friction. Optimal banking angle θ = arctan(v²/rg), where g is gravity. Race tracks and highways use banking for safer high-speed turns.

How do I calculate period and frequency?

Period T = 2πr/v (time for one revolution). Frequency f = 1/T = v/(2πr) (revolutions per second). Also, ω = 2πf. For Earth's orbit: T ≈ 365.25 days, v ≈ 30 km/s, r ≈ 150 million km.

What are practical applications?

Applications include: centrifuges (separating materials), washing machine spin cycle, planetary motion, satellite orbits, particle accelerators, vehicle dynamics, amusement park rides, and rotating machinery design.