Centripetal Force Calculator
Calculate the centripetal force required to keep an object moving in a circular path using Fc = mv²/r. Enter mass, velocity, and radius to find the centripetal force, acceleration, angular velocity, and period. See also our Force Calculator and Angular Acceleration Calculator.
How to Calculate Centripetal Force
Centripetal force is the net force directed toward the center of a circular path that keeps an object moving in a circle. The word "centripetal" comes from Latin meaning "center-seeking." Without centripetal force, an object would fly off in a straight line tangent to the circle (Newton's first law). Centripetal force is not a new type of force — it is provided by whatever force keeps the object on its circular path: tension in a string, gravity for orbits, friction for cars turning, or normal force in a loop.
To calculate centripetal force, use the formula Fc = mv²/r, where m is the mass of the object, v is its speed (tangential velocity), and r is the radius of the circular path. Alternatively, using angular velocity ω: Fc = mω²r, since v = ωr. The centripetal acceleration is ac = v²/r = ω²r, directed toward the center. The force required increases with the square of speed — doubling the speed quadruples the required centripetal force.
It is important to distinguish centripetal force from the commonly misunderstood "centrifugal force." Centripetal force is real — it is the inward force that causes circular motion. Centrifugal force is a fictitious (pseudo) force that appears only in a rotating reference frame. When you feel "pushed outward" in a turning car, that sensation is your body's inertia trying to continue in a straight line, not an actual outward force. In an inertial (non-rotating) frame, only the inward centripetal force exists.
The centripetal force concept applies to everything from atoms (electrons orbiting nuclei) to galaxies (stars orbiting galactic centers). In engineering, it determines the banking angle of roads, the design of centrifuges, the maximum safe speed for curves, and the structural requirements for rotating machinery. Understanding centripetal force is essential for aerospace, automotive, and mechanical engineering.
Centripetal Force Formula
Centripetal Force:
Fc = mv²/r = mω²r
Centripetal Acceleration:
ac = v²/r = ω²r = 4π²r/T²
Angular velocity:
ω = v/r = 2π/T = 2πf
Period of revolution:
T = 2πr/v
Maximum speed on banked curve:
v_max = √(rg × tan(θ + φ))
θ = bank angle, φ = friction angle
Orbital velocity:
v = √(GM/r) (gravity provides Fc)
Example Calculation
A 5 kg ball moves at 10 m/s in a circle of radius 2 m. Calculate the centripetal force:
Given: m = 5 kg, v = 10 m/s, r = 2 m
Fc = mv²/r = 5 × 10² / 2 = 5 × 100 / 2 = 250 N
Centripetal acceleration: ac = v²/r = 100/2 = 50 m/s²
In g-forces: 50/9.81 = 5.1 g
Angular velocity: ω = v/r = 10/2 = 5 rad/s
Period: T = 2πr/v = 2π×2/10 = 1.257 s
Frequency: f = 1/T = 0.796 Hz = 47.7 RPM
If speed doubles to 20 m/s:
Fc = 5 × 400 / 2 = 1000 N (4× the force!)
Centripetal Force Reference Table
| Scenario | Centripetal Force |
|---|---|
| Car turning (1500kg, 20m/s, r=50m) | 12000 N |
| Ball on string (0.5kg, 5m/s, r=1m) | 12.5 N |
| Satellite orbit (1000kg, 7700m/s, r=6.7×10⁶m) | 8850 N |
| Washing machine (2kg, 10m/s, r=0.3m) | 667 N |
| Roller coaster loop (80kg, 15m/s, r=10m) | 1800 N |
| Centrifuge (0.01kg, 100m/s, r=0.1m) | 10000 N |
| Earth orbit Sun (6×10²⁴kg) | 3.54×10²² N N |
| Bicycle turn (80kg, 8m/s, r=5m) | 1024 N |
| Hammer throw (7kg, 25m/s, r=2m) | 2188 N |
| Merry-go-round (30kg, 3m/s, r=3m) | 90 N |
Frequently Asked Questions
What provides centripetal force?
Centripetal force is not a separate type of force — it is provided by whatever real force acts toward the center of the circular path. For a ball on a string, tension provides the centripetal force. For a car turning on a road, friction between tires and road provides it. For planets orbiting the Sun, gravity provides it. For a roller coaster in a loop, the normal force and gravity combine to provide it. The centripetal force is simply the net inward component of all forces acting on the object.
Is centrifugal force real?
Centrifugal force is a fictitious (pseudo) force that only appears in a rotating reference frame. In an inertial frame, there is no outward force — objects tend to move in straight lines (inertia), and the centripetal force curves their path inward. The sensation of being "pushed outward" in a turning car is your body's inertia resisting the change in direction. However, in a rotating frame, centrifugal force is mathematically useful and gives correct results for calculations within that frame.
Why does centripetal force depend on v²?
The v² dependence comes from the geometry of circular motion. In a small time Δt, the velocity vector rotates by angle Δθ = v×Δt/r. The change in velocity magnitude is Δv ≈ v×Δθ = v²×Δt/r. The acceleration is Δv/Δt = v²/r, and force is F = ma = mv²/r. Physically, faster objects need more force to curve their path because they cover more arc length per unit time and their velocity direction changes more rapidly. This is why high-speed turns are so much more demanding than slow ones.
How do banked curves reduce the need for friction?
On a banked curve, the road surface is tilted inward. The normal force (perpendicular to the road) now has a horizontal component pointing toward the center of the curve. This horizontal component provides some or all of the centripetal force, reducing the friction needed. At the "design speed," the banking angle provides exactly the right centripetal force with zero friction: tan(θ) = v²/(rg). This is why highway curves and racetracks are banked — it allows safe turning even on wet or icy surfaces.
What are g-forces in circular motion?
G-forces in circular motion express centripetal acceleration as a multiple of gravitational acceleration (9.81 m/s²). A centripetal acceleration of 49.05 m/s² equals 5g. Fighter pilots experience up to 9g in tight turns. Astronauts experience about 3g during launch. Centrifuges can produce thousands of g for separating materials. The human body can tolerate about 5g sustained (with a g-suit) or 9g briefly. At high g-forces, blood pools away from the brain, causing tunnel vision and eventually blackout.
How does centripetal force apply to orbits?
For orbital motion, gravity provides the centripetal force: GMm/r² = mv²/r. This gives the orbital velocity: v = √(GM/r). Faster orbits require smaller radii (closer to the central body). The International Space Station orbits at about 7,700 m/s at 400 km altitude. Geostationary satellites orbit at 3,070 m/s at 35,786 km altitude. The Moon orbits at about 1,022 m/s at 384,400 km. In each case, gravity provides exactly the centripetal force needed for the circular orbit.
Applications of Centripetal Force
Centripetal force principles are applied extensively in engineering and technology. Centrifuges use high centripetal acceleration to separate materials by density — from blood components in medical labs to uranium isotopes in nuclear enrichment. Washing machine spin cycles use centripetal force to extract water from clothes. Centrifugal governors regulate engine speed by using the outward tendency of rotating masses.
In transportation, understanding centripetal force is critical for road design (curve radii, banking angles, speed limits), railway engineering (cant/superelevation of tracks), and aerospace (aircraft turn rates, spacecraft orbital mechanics). Amusement park rides are designed with precise centripetal force calculations to create thrilling but safe experiences. Even in sports, centripetal force determines the radius of a discus thrower's spin, the curve of a soccer ball, and the lean angle of a motorcycle in a turn.