Centripetal Force Calculator
Centripetal Force Calculator
When an object moves in a circle at constant speed, its direction continuously changes, so it is accelerating toward the center of the circle. The net force causing that inward acceleration is the centripetal force, given by Fc = mv²/r. Our Centripetal Force Calculator solves for force, velocity, or radius whenever you know the other two, making it handy for analyzing car turns, spinning rides, orbiting satellites, and laboratory centrifuges.
What Is Centripetal Force?
Centripetal ("center-seeking") force is the inward force required to keep an object in circular motion. It is not a new kind of force; it is provided by tension, gravity, friction, or a normal force. Without it, the object would fly off in a straight line (Newton's first law).
Why It Matters
Centripetal force explains why roads are banked, why a string can break if you swing a weight too fast, and how geostationary satellites stay aloft. The dependence on v² means doubling speed quadruples the required force.
It is important to understand that centripetal force is not a separate type of force like gravity or magnetism. The label describes the role a force plays, not its origin. In a given situation the centripetal force might be the tension in a string, the frictional grip of a tire, the gravitational pull of a planet, or the normal force from a banked wall. What makes it "centripetal" is simply that it points toward the center of the circular path and has magnitude mv²/r. When you analyze a real problem, first identify which physical force is supplying it, then set that force equal to mv²/r.
Select what to solve for with the Solve For radio buttons, enter the known quantities, and click Calculate.
Solve for Force
- Enter Mass in kg.
- Enter Velocity in m/s.
- Enter Radius in m.
- Click Calculate → Centripetal force in N.
Worked Example 1: A 1000 kg Car at 20 m/s Around a 50 m Radius Curve
Fc = (1000 × 20²) / 50 = (1000 × 400) / 50 = 400,000 / 50 = 8000 N. The road must supply 8 kN of inward force.
Solve for Velocity
- Enter Centripetal Force in N.
- Enter Mass in kg.
- Enter Radius in m.
- Click Calculate → Velocity in m/s (the maximum safe speed for that force).
Worked Example 2: Same Car, 8000 N, Radius 50 m
v = √(Fc·r / m) = √(8000 × 50 / 1000) = √400 = 20 m/s.
Solve for Radius
- Enter Centripetal Force in N.
- Enter Mass in kg.
- Enter Velocity in m/s.
- Click Calculate → Radius in m. A larger radius needs less force for the same speed.
Worked Example 3: 1000 kg at 20 m/s Needs 8000 N
r = (1000 × 20²) / 8000 = 400,000 / 8000 = 50 m.
Edge Cases
- Zero radius: force becomes infinite (undefined) → no result.
- Zero velocity: force is 0 (no circular motion needed).
- Zero mass: force is 0 and velocity mode is undefined.
- Negative force is rejected; centripetal force magnitude is non-negative.
Worked Example 4: A 60 kg Child on a 3 m Radius Merry-Go-Round at 4 m/s
Fc = (60 × 4²)/3 = (60 × 16)/3 = 960/3 = 320 N. The child feels pushed outward with an apparent "centrifugal" sensation of 320 N, but the real inward force is supplied by the floor and the rail.
Worked Example 5: A 0.5 kg Ball on a 1 m String at 10 m/s
Fc = (0.5 × 10²)/1 = (0.5 × 100)/1 = 50 N. The string must withstand 50 N of tension at the bottom of the swing (plus the ball's weight if swinging vertically). This is why a thin string snaps if you spin a mass too fast.
Worked Example 6: A Geostationary Satellite (m = 1000 kg, r = 42,164 km)
Orbital speed v = √(GM/r). With Earth's GM ≈ 3.986×10¹⁴ m³/s² and r = 4.2164×10⁷ m, v ≈ 3074 m/s. Fc = (1000 × 3074²)/4.2164×10⁷ ≈ 224 N, supplied entirely by gravity. The satellite's period is 24 h, matching Earth's rotation.
Centripetal force for uniform circular motion:
Solving for the other variables:
The associated centripetal acceleration is a_c = v²/r = Fc/m, directed toward the center of the circle.
You can also relate centripetal force to angular speed ω (radians per second) using v = ωr. Substituting gives Fc = mω²r, which is convenient for rotating machinery where rpm is known. Likewise, the period T of one revolution satisfies v = 2πr/T, so Fc = 4π²mr/T² — useful when you measure rotation by how long a full turn takes rather than by linear speed. All three forms (mv²/r, mω²r, 4π²mr/T²) describe the same inward force.
Centripetal force for a 1000 kg mass at 20 m/s across radii:
| Radius (m) | Centripetal Force (N) |
|---|---|
| 25 | 16000 |
| 50 | 8000 |
| 100 | 4000 |
| 200 | 2000 |
| 500 | 800 |
Force is inversely proportional to radius: doubling the radius halves the required force.
Centripetal force for a 1000 kg mass on a 50 m radius curve at various speeds:
| Velocity (m/s) | Centripetal Force (N) |
|---|---|
| 10 | 2000 |
| 20 | 8000 |
| 30 | 18000 |
| 40 | 32000 |
| 50 | 50000 |
Because of the v² term, a small speed increase causes a large force jump (20 → 40 m/s quadruples the force).
Centripetal force for a 50 m radius at 20 m/s across masses:
| Mass (kg) | Centripetal Force (N) |
|---|---|
| 500 | 4000 |
| 1000 | 8000 |
| 1500 | 12000 |
| 2000 | 16000 |
| 3000 | 24000 |
Force scales linearly with mass for fixed speed and radius.
Across all three tables the dominant message is the v² dependence. Compare the velocity table: going from 20 to 50 m/s (2.5× speed) raises the force from 8000 N to 50,000 N — a 6.25× jump, exactly 2.5². Engineers exploit the inverse-radius relationship when designing cloverleaf interchanges: a larger loop radius lets cars take the curve at speed with manageable lateral force, improving comfort and safety.
- Banked curves help: banking provides part of the centripetal force via the normal force, reducing reliance on friction.
- Watch the v² term: speed limits on curves exist because force grows with the square of speed.
- Units: kg, m/s, m, N. Convert mph or km/h before entering.
- Provide the force: centripetal force is supplied by something real (friction, tension, gravity) — identify it in your problem.
- Centrifuge use: higher r or v increases the effective g-force on samples.
- Check the mode: pick the right variable to solve for before entering values.
- Identify the supplying force: ask what physically provides Fc — friction, tension, gravity, or the normal force from a banked surface — and confirm it can actually reach the magnitude you computed.
- Skid risk on curves: a car stays on a flat curve only while static friction can supply mv²/r. If required force exceeds μ_s·m·g, the car slides outward. Lower speed or larger radius keeps you safe.
- Convert rpm to rad/s: for rotating equipment, ω = 2π·(rpm/60); then use Fc = mω²r.
- Vertical loops add weight: at the top of a vertical circle the tension is smallest (gravity helps supply Fc); at the bottom it is largest (tension must also support weight). Add m·g appropriately.
- Effective g in centrifuges: the ratio Fc/(mg) tells you the multiples of gravity a sample experiences; high-speed centrifuges reach thousands of g.
- Uniform circular motion: assumes constant speed; accelerating or decelerating turns add tangential force.
- Ideal force: the calculator returns the magnitude; direction is always toward the center.
- No friction limit: real tires can supply only so much friction before skidding.
- Non-relativistic: valid for everyday speeds, not near light speed.
- Point mass: treats the object as a particle; rotating rigid bodies need moment-of-inertia analysis.
- SI units only: enter values already in kg, m/s, m.
- Tangential acceleration ignored: during speeding-up or slowing-down turns the total force is the vector sum of centripetal and tangential components; this tool gives only the centripetal part.
- Non-circular paths: for elliptical or arbitrary curves, use the instantaneous radius of curvature ρ: Fc = mv²/ρ at each point, not a fixed r.
- Rotating frames: in the rotating frame a fictitious centrifugal force appears; it is not a real force but a bookkeeping term for the inertial effect.
What is centripetal force?
The inward net force needed to keep an object moving in a circle at constant speed, directed toward the center.
Is centripetal force a new kind of force?
No. It is provided by existing forces such as tension, gravity, friction, or the normal force.
Why does doubling speed quadruple the force?
Because Fc = mv²/r contains v²; doubling v makes v² four times larger, so force quadruples.
What happens if there is no centripetal force?
The object moves in a straight line tangent to the circle (Newton's first law) — it "flies off."
How is centripetal acceleration related?
a_c = v²/r = Fc/m, directed inward. Force equals mass times that acceleration.
Can radius be zero?
No — radius approaches zero, required force approaches infinity, which is physically impossible.
Is this the same as centrifugal force?
Centrifugal is the apparent outward force in a rotating reference frame; centripetal is the real inward force in an inertial frame.
What unit is centripetal force in?
Newtons (N), the same as any force.
How do banked roads help?
The road's normal force has an inward component that supplies part of the centripetal force, reducing the needed friction.
Can I use this for satellite orbits?
Yes, for circular orbits: gravity supplies Fc, so you can relate orbital speed, radius, and mass (with the gravitational force formula).
What is the angular form of the formula?
Using angular speed ω (rad/s) with v = ωr, centripetal force is Fc = mω²r. For a known rotation period T, use Fc = 4π²mr/T².
How do I know if a car will skid on a curve?
Compare the needed Fc = mv²/r with the maximum static friction μ_s·m·g. If mv²/r > μ_s·m·g, the car skids. Note mass cancels, so the safe speed depends on μ_s, r, and g, not on vehicle mass.
Why do clothes stick to a spinning drum in a centrifuge?
The drum wall supplies the inward Fc; from the laundry's frame it feels an outward "centrifugal" push. Higher ω (or smaller r) raises the effective g-force, flinging water outward through the holes.
What is the minimum speed at the top of a vertical loop?
At the top, gravity can supply part of Fc. The minimum speed to maintain contact is v = √(gr); below that, the object falls away from the track.
Can centripetal force do work?
No. Because it is always perpendicular to the instantaneous velocity, it changes the direction of motion but not the speed, so it does zero work on the object.
Last updated: July 18, 2026
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