Centripetal Force Calculator

How to use this calculator

Enter three pieces of information: mass, radius, and a velocity value. The calculator supports three types of velocity input.

Mass (kg): The mass of the object moving in a circle. For a car on a curve, this is the car’s mass. For a satellite, it is the satellite’s mass. The object can have any mass, and centripetal force scales linearly with it.

Radius (m): The radius of the circular path. For a ball on a string, this is the string length. For a car on a curved road, this is the radius of curvature of the road.

Velocity input type: Choose from tangential velocity (m/s), angular velocity in rad/s, or RPM. The calculator converts between them automatically. If you select RPM, enter how many times the object completes a full circle per minute.

Click Calculate to see the centripetal force in Newtons, centripetal acceleration in m/s², the tangential velocity, period, and angular velocity. The Force vs Velocity chart shows how force grows as a parabola with speed, and the red dot marks your input point.

What centripetal force is (and what it is not)

Centripetal force is often misunderstood as a separate type of force, like gravity or friction. It is not. Centripetal force is a role that a force plays, not a unique force in its own right. When a force is directed toward the center of a circular path, we call it the centripetal force for that situation.

Consider three examples:

• A ball on a string: the tension in the string pulls the ball toward the center. Tension is the centripetal force.
• The Moon orbiting Earth: gravity pulls the Moon toward Earth’s center. Gravity is the centripetal force.
• A car on a flat curve: static friction from the road pulls the car toward the center of the curve. Friction is the centripetal force.

In each case, the underlying force is different (tension, gravity, friction), but they all play the same role: keeping the object moving in a circle by constantly redirecting its velocity toward the center.

The direction of centripetal force is always inward, toward the center of the circular path. This is why objects in circular motion do not fly outward: the real force is inward. The sensation of being pushed outward (centrifugal) is a fictitious force experienced only in the rotating reference frame.

Deriving F = mv²/r from first principles

To derive the centripetal force formula, start with the definition of centripetal acceleration.

For an object moving at speed v in a circle of radius r, the velocity vector continuously changes direction even though its magnitude stays constant. The rate of change of the velocity vector’s direction gives the centripetal acceleration.

Consider the object moving a small arc in time Δt. The velocity vector rotates by a small angle Δθ. The change in velocity magnitude is |Δv| ≈ v × Δθ for small angles. Since the arc length traveled is v × Δt, and for a full circle the arc is 2πr for angle 2π, we get Δθ = v × Δt / r. Therefore:

Applying Newton’s second law (F = ma) gives:

Since ω = v/r (angular velocity relates linear speed and radius), substituting v = ωr gives the equivalent form:

Both forms are used depending on which quantity is known. The first form is useful when you know tangential speed. The second form is useful for rotating systems where angular velocity is directly measured.

Centripetal force in real-world examples

Moon orbiting Earth

The Moon orbits Earth at an orbital radius of about 384,400 km and completes one orbit in 27.3 days (2.36 × 10⁶ s). The Moon’s mass is 7.35 × 10²² kg.

Orbital speed: v = 2πr / T = 2π × 3.844 × 10⁸ / 2.36 × 10⁶ ≈ 1,023 m/s

Centripetal force required: F = mv²/r = 7.35 × 10²² × 1023² / 3.844 × 10⁸ ≈ 2.0 × 10²⁰ N

This enormous force is provided entirely by Earth’s gravitational pull, confirming that gravity acts as the centripetal force for orbital motion.

Car on a flat curve

A 1,500 kg car traveling at 20 m/s on a curve with radius 50 m needs:

F_c = 1500 × 20² / 50 = 1500 × 400 / 50 = 12,000 N

This force must be provided by static friction between the tires and road. On a dry road with μ_s ≈ 0.7, the maximum static friction is 0.7 × 1500 × 9.81 ≈ 10,300 N. The car cannot safely take this curve at 20 m/s on a flat road.

Roller coaster loop

At the top of a vertical loop of radius r, both gravity and the normal force from the track point downward (toward the center). The minimum speed condition occurs when the normal force equals zero:

mg = mv²/r → v_min = √(gr)

For a loop with r = 10 m: v_min = √(9.81 × 10) ≈ 9.9 m/s (about 36 km/h). Below this speed, the rider would fall away from the track at the top.

Orbital mechanics as centripetal force

Satellites in circular orbit follow directly from the centripetal force framework. For a satellite of mass m at altitude h above Earth (radius R_E), the orbital radius is r = R_E + h.

Gravitational force provides the centripetal force:

Solving for orbital velocity:

Higher orbits require lower orbital speeds. The International Space Station at 400 km altitude travels at about 7,660 m/s. GPS satellites at 20,200 km altitude travel at about 3,870 m/s.

The period of orbit follows from T = 2πr/v:

This is Kepler’s third law (T² ∝ r³), derived directly from the centripetal force requirement and Newton’s law of gravitation.

Banked curves

A banked curve uses the normal force to contribute to the centripetal force, reducing the reliance on friction. For a curve banked at angle θ:

The normal force N is perpendicular to the surface. Its horizontal component N sinθ points toward the center of the curve. Its vertical component N cosθ balances gravity: N cosθ = mg.

Therefore, the horizontal centripetal component is:

N sinθ = mg × tanθ

Setting this equal to mv²/r:

At this speed, no friction is required. The road provides all the centripetal force through the normal force alone. For speeds above v_ideal, friction is needed to prevent sliding outward. For speeds below v_ideal, friction is needed to prevent sliding inward.

Centripetal force vs centrifugal force

The apparent outward force felt in circular motion (centrifugal force) is a fictitious force that appears only when you analyze the situation from a rotating reference frame. It has no real physical source.

From the ground frame (inertial reference frame): The car on a curve is being pushed inward by friction. The passenger tends to continue in a straight line (Newton’s first law). The car seat pushes the passenger inward, and the passenger feels this as a push.

From the car’s reference frame (rotating, non-inertial): The passenger appears to be pushed outward by a centrifugal force of magnitude mω²r. This is mathematically convenient for analyzing rotating systems, but the centrifugal force is not real in the sense that it has no physical agent. It is a consequence of being in an accelerating reference frame.

Engineers often use the rotating frame analysis for centrifuges, planetary rotation, and vehicle dynamics because it simplifies the equations of equilibrium.

Solving centripetal force problems: a checklist

1. Identify the circular path and find the radius r.
2. Identify the mass m of the object.
3. Determine the speed v or angular velocity ω (from period, RPM, or direct measurement).
4. Calculate F_c = mv²/r or F_c = mω²r.
5. Identify which real force (or combination of forces) provides this centripetal force.
6. Check that the available force (friction limit, tension limit, gravitational force) is sufficient.

For problems involving vertical circles (loops, swings), analyze both the top and bottom of the loop separately. At the bottom, the normal force must exceed gravity to provide both support and centripetal force. At the top, gravity and normal force both contribute to centripetal force.

Centripetal force and artificial gravity

One of the most important applications of centripetal force in aerospace engineering is the concept of artificial gravity. In a space station or spacecraft far from Earth’s gravitational field, there is no natural gravity to keep crew members oriented or to allow normal physiological processes. Prolonged weightlessness causes bone density loss, muscle atrophy, and fluid shifts in the body.

Artificial gravity can be created by rotating the spacecraft or a section of it. If the station rotates with angular velocity ω, an astronaut standing against the outer wall experiences a normal force directed outward (from the wall pushing them inward to maintain circular motion). From the astronaut’s perspective in the rotating frame, this feels exactly like gravity.

For a ring-shaped station with radius r, rotating at ω rad/s, the “gravitational” acceleration felt at the rim is:

To simulate Earth’s gravity (9.81 m/s²) at r = 100 m: ω = √(9.81/100) ≈ 0.313 rad/s, which corresponds to about 3 RPM. This is the same physics as a centrifuge, but on a human scale.

The concept was popularized by science fiction (the rotating stations in Arthur C. Clarke’s and Stanley Kubrick’s 2001: A Space Odyssey and Andy Weir’s The Martian). NASA and private companies have studied rotating habitats for long-duration missions to Mars, where the 6-9 month transit time would otherwise cause significant bone and muscle loss.

A practical complication is the Coriolis effect: in a rotating frame, moving objects appear to deflect sideways. For small radii and high rotation rates, this effect is noticeable and can cause motion sickness. Larger radii (lower rotation rate for the same g) reduce the Coriolis problem, but require a much larger structure.

Centripetal force in everyday spinning systems

Many ordinary devices rely on centripetal force to function.

Washing machine spin cycle: During the spin cycle, the drum rotates at 800-1600 RPM. Clothes press against the drum wall, and water is forced outward through the drum’s perforations. The centripetal acceleration at the drum wall (radius ~0.25 m) at 1200 RPM: ω = 1200 × 2π/60 = 125.7 rad/s, a_c = 125.7² × 0.25 ≈ 3,950 m/s², about 400 g. The drum wall provides the centripetal force keeping the clothes in circular motion. Water droplets, not constrained by the drum holes, exit through them.

Centrifuge: Laboratory centrifuges use exactly this principle at much higher accelerations. Ultracentrifuges reach 500,000 g, allowing separation of proteins, viruses, and even large molecules by density. The sedimentation rate depends on the centripetal acceleration: higher g means faster separation.

Roundabouts and carousels: Children on a roundabout are kept in circular motion by the static friction of the seat or their grip on the handle. The centripetal force needed is F = mω²r. For a child (m = 30 kg) at r = 1 m spinning at 1 rev/s (ω = 2π rad/s): F = 30 × (2π)² × 1 ≈ 1,184 N, more than twice the child’s weight. This is why children can feel it strongly and why they must hold on tightly.